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16162:, or even of parallel Ricci curvature. Hyperbolic space can be given a Lie group structure relative to which the metric is left-invariant. Any bi-invariant Riemannian metric on a Lie group has nonnegative sectional curvature, giving a variety of such metrics: a Lie group can be given a bi-invariant Riemannian metric if and only if it is the product of a
3450:
17852:(a metric that does not separate points), but it may not be a metric. In the finite-dimensional case, the proof that the Riemannian distance function separates points uses the existence of a pre-compact open set around any point. In the infinite case, open sets are no longer pre-compact, so the proof fails.
3109:
604:, a measuring stick that gives tangent vectors a concept of length and angle. This is an important deficiency because calculus teaches that to calculate the length of a curve, the length of vectors tangent to the curve must be defined. A Riemannian metric puts a measuring stick on every tangent space.
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That is, the entire structure of a smooth
Riemannian manifold can be encoded by a diffeomorphism to a certain embedded submanifold of some Euclidean space. Therefore, one could argue that nothing can be gained from the consideration of abstract smooth manifolds and their Riemannian metrics. However,
167:
proved that every
Riemannian manifold arises as a submanifold of Euclidean space, and although some Riemannian manifolds are naturally exhibited or defined in that way, the idea of a Riemannian manifold emphasizes the intrinsic point of view, which defines geometric notions directly on the abstract
12554:
are curves with no intrinsic acceleration. Equivalently, geodesics are curves that locally take the shortest path between two points. They are the generalization of straight lines in
Euclidean space to arbitrary Riemannian manifolds. An ant living in a Riemannian manifold walking straight ahead
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However, a geodesically complete strong
Riemannian manifold might not be metrically complete and it might have closed and bounded subsets that are not compact. Further, a strong Riemannian manifold for which all closed and bounded subsets are compact might not be geodesically complete.
8853:
must escape. By selecting this open set to be contained in a coordinate chart, one can reduce the claim to the well-known fact that, in
Euclidean geometry, the shortest curve between two points is a line. In particular, as seen by the Euclidean geometry of a coordinate chart around
16437:. Every Riemannian symmetric space is homogeneous, and consequently is geodesically complete and has constant scalar curvature. However, Riemannian symmetric spaces also have a much stronger curvature property not possessed by most homogeneous Riemannian manifolds, namely that the
1164:
16377:. This reduces the study of the curvature of homogeneous Riemannian manifolds largely to algebraic problems. This reduction, together with the flexibility of the above construction, makes the class of homogeneous Riemannian manifolds very useful for constructing examples.
16511:, any Riemannian manifold which is simply-connected and irreducible is either a symmetric space or has Riemannian holonomy belonging to a list of only seven possibilities. Six of the seven exceptions to symmetric spaces in Berger's classification fall into the fields of
15884:, along with the Euclidean plane. Unlike the case of two-dimensional spherical space forms, in some cases two space form structures on the same manifold are not homothetic. The case of two-dimensional hyperbolic space forms is even more complicated, having to do with
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The case of
Euclidean and hyperbolic space forms can likewise be reduced to group theory, based on study of the isometry group of Euclidean space and hyperbolic space. For example, the class of two-dimensional Euclidean space forms includes Riemannian metrics on the
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The
Riemann curvature tensor measures precisely the extent to which parallel transporting vectors around a small rectangle is not the identity map. The Riemann curvature tensor is 0 at every point if and only if the manifold is locally isometric to Euclidean space.
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is the trivial subgroup containing only the identity element. The calculations of the Levi-Civita connection and the curvature referenced there can be generalized to this context, where now the computations are formulated in terms of the inner product on
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293:. Call a property of a surface an intrinsic property if it is preserved by local isometries and call it an extrinsic property if it is not. In this language, the Theorema Egregium says that the Gaussian curvature is an intrinsic property of surfaces.
16531:, each of which study Riemannian manifolds equipped with certain extra structures and symmetries. The seventh exception is the study of 'generic' Riemannian manifolds with no particular symmetry, as reflected by the maximal possible holonomy group.
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This is a fundamental result. Although much of the basic theory of
Riemannian metrics can be developed using only that a smooth manifold is a locally Euclidean topological space, for this result it is necessary to use that smooth manifolds are
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is an additional structure on a
Riemannian manifold that defines differentiation of one vector field with respect to another. Connections contain geometric data, and two Riemannian manifolds with different connections have different geometry.
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says that any simply-connected spherical space form is homothetic to the sphere, any simply-connected
Euclidean space form is homothetic to Euclidean space, and any simply-connected hyperbolic space form is homothetic to hyperbolic space.
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15917:. Using the group structure, any inner product on the tangent space at the identity (or any other particular tangent space) can be transported to all other tangent spaces to define a Riemannian metric. Formally, given an inner product
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5151:, cannot be isometrically embedded in 3-dimensional Euclidean space (right), because it is necessary to bend and stretch the sheet in doing so. Thus the intrinsic geometry of a flat torus is different from that of an embedded torus.
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15850:. There are many more odd-dimensional spherical space forms, although there are known algorithms for their classification. The list of three-dimensional spherical space forms is infinite but explicitly known, and includes the
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phenomena, in which a simply-connected Riemannian manifold can have small volume without having large curvature. They also give an example of a Riemannian metric which has constant scalar curvature but which is not
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is a torsion-free connection that preserves the metric. Once a Riemannian metric is fixed, there exists a unique Levi-Civita connection. Note that the definition of preserving the metric uses the regularity of
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If one works with Riemannian metrics that are merely continuous but possibly not smooth, the length of an admissible curve and the Riemannian distance function are defined exactly the same, and, as before,
151:
are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport.
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This is not the case without the completeness assumption; for counterexamples one could consider any open bounded subset of a Euclidean space with the standard Riemannian metric. It is also not true that
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and hyperbolic space, of which any representation as a submanifold of Euclidean space will fail to represent their remarkable symmetries and properties as clearly as their abstract presentations do.
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is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Given a fixed connection, there is a unique way to do parallel transport.
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3445:{\displaystyle (g_{ij}^{\text{can}})={\begin{pmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{pmatrix}}.}
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An isometry is a function between Riemannian manifolds which preserves all of the structure of Riemannian manifolds. If two Riemannian manifolds have an isometry between them, they are called
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if the curvature is negative. In any dimension, the sphere with its standard Riemannian metric, Euclidean space, and hyperbolic space are Riemannian space forms of constant curvature
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15888:. In three dimensions, the Euclidean space forms are known, while the geometry of hyperbolic space forms in three and higher dimensions remains an area of active research known as
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also carry natural Riemannian metrics making them into symmetric spaces. Among the Lie groups with left-invariant Riemannian metrics, those which are bi-invariant are symmetric.
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Two Riemannian manifolds with different connections have different geometry. Thankfully, there is a natural connection associated to a Riemannian manifold called the
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In Euclidean space, all tangent spaces are canonically identified with each other via translation, so it is easy to move vectors from one tangent space to another.
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7648:. Because the atlas is locally finite, at every point the sum contains only finitely many nonzero terms, so the sum converges. It is straightforward to check that
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3104:{\displaystyle g^{\text{can}}\left(\sum _{i}a_{i}{\frac {\partial }{\partial x^{i}}},\sum _{j}b_{j}{\frac {\partial }{\partial x^{j}}}\right)=\sum _{i}a_{i}b_{i}}
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15846:. For instance, this can be used to show directly that every even-dimensional spherical space form is homothetic to the standard metric on either the sphere or
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is non-differentiable, and it can be remarkably difficult to even determine the location or nature of these points, even in seemingly simple cases such as when
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13310:. Since the Earth is approximately a sphere, this means that the shortest path a plane can fly between two locations on Earth is a segment of a great circle.
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Riemannian metrics are defined in a way similar to the finite-dimensional case. However, there is a distinction between two types of Riemannian metrics:
14819:
1381:{\displaystyle g_{ij}|_{p}:=g_{p}\left(\left.{\frac {\partial }{\partial x^{i}}}\right|_{p},\left.{\frac {\partial }{\partial x^{j}}}\right|_{p}\right)}
7156:
16491:
Based on their algebraic formulation as special kinds of homogeneous spaces, Cartan achieved an explicit classification of symmetric spaces which are
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16449:. Riemannian manifolds with this curvature property, which could loosely be phrased as "constant Riemann curvature tensor" (not to be confused with
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are exactly the straight lines. This agrees the fact from Euclidean geometry that the shortest path between two points is a straight line segment.
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14505:. This second metric has a singularity at the origin, so it does not extend past the puncture, but the first metric extends to the entire plane.
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Any smooth surface in three-dimensional Euclidean space is a Riemannian manifold with a Riemannian metric coming from the way it sits inside the
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in which one wants to consider non-smooth Riemannian metrics. See for instance (Gromov 1999) and (Shi and Tam 2002). However, in this article,
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naturally carries a Euclidean metric, obtained by identifying opposite sides of a square (left). The resulting Riemannian manifold, called a
20455:
15842:. Up to homothety, every spherical space form arises in this way; this largely reduces the study of spherical space forms to problems in
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is a vector space under pointwise vector addition and scalar multiplication. One can also pointwise multiply a smooth vector field along
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The above example of Lie groups with left-invariant Riemannian metrics arises as a very special case of this construction, namely when
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1159:{\displaystyle \left\{{\frac {\partial }{\partial x^{1}}}{\Big |}_{p},\dotsc ,{\frac {\partial }{\partial x^{n}}}{\Big |}_{p}\right\}}
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The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the
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of a simply-connected Riemannian space form, modulo a certain group action of isometries. For example, the isometry group of the
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19182:. Mathematics: Theory & Applications (Translated from the second Portuguese edition of 1979 original ed.). Boston, MA:
1637:
369:
A tangent plane of the sphere with two vectors in it. A Riemannian metric allows one to take the inner product of these vectors.
172:, Riemannian metrics are more naturally defined or constructed using the intrinsic point of view. Additionally, many metrics on
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16771:
16503:; among them only the one-dimensional manifolds have zero scalar curvature. These spaces are important from the perspective of
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The Levi-Civita connection and curvature of a general left-invariant Riemannian metric can be computed explicitly in terms of
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Although the length of a curve is given by an explicit formula, it is generally impossible to write out the distance function
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18951:{\displaystyle (x_{1},\ldots ,x_{n})\cdot (y_{1},\ldots ,y_{n})=(x_{1}+y_{n}x_{1},\ldots ,x_{n-1}+y_{n}x_{n-1},x_{n}y_{n}).}
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complete metric space of finite diameter must be compact; it matters that the metric space came from a Riemannian manifold.
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These can be extended, to a certain degree, to infinite-dimensional manifolds; that is, manifolds that are modeled after a
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4685:{\displaystyle \left\{x\in \mathbb {R} ^{3}:{\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}+{\frac {z^{2}}{c^{2}}}=1\right\}}
1884:
are smooth in any smooth coordinate chart. One can consider many other types of Riemannian metrics in this spirit, such as
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1940:
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Michor, Peter W.; Mumford, David (2005). "Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms".
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The statements and theorems above are for finite-dimensional manifolds—manifolds whose charts map to open subsets of
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11863:
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801:
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15696:{\displaystyle {\frac {dx_{1}^{2}+\cdots +dx_{n}^{2}}{(1+{\frac {\kappa }{4}}(x_{1}^{2}+\cdots +x_{n}^{2}))^{2}}}}
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8491:{\displaystyle d_{g}(p,q)=\inf\{L(\gamma ):\gamma {\text{ an admissible curve with }}\gamma (0)=p,\gamma (1)=q\}.}
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is transitive. Every homogeneous Riemannian manifold is geodesically complete and has constant scalar curvature.
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6969:{\displaystyle \varphi _{\alpha }\colon U_{\alpha }\to \varphi _{\alpha }(U_{\alpha })\subseteq \mathbf {R} ^{n}}
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For example, translations and rotations are both isometries from Euclidean space (to be defined soon) to itself.
184:
to transport an inner product on a single tangent space to the entire manifold, and many special metrics such as
147:
of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of
19465:
Magnani, Valentino; Tiberio, Daniele (2020). "A remark on vanishing geodesic distances in infinite dimensions".
16142:(that is, simultaneously left- and right-invariant). All left-invariant metrics have constant scalar curvature.
2719:. The Riemannian volume form is preserved by orientation-preserving isometries. The volume form gives rise to a
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16484:, quaternionic hyperbolic space, and Cayley hyperbolic space, which are instead analogues of hyperbolic space.
16108:; right-invariant Riemannian metrics could be constructed likewise using the right multiplication map instead.
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the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.
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Many of the fundamental examples of Riemannian manifolds are symmetric. The most basic include the sphere and
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and Euclidean lengths measured in a smooth coordinate chart, also verifies that the metric space topology of
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of a surface embedded in 3-dimensional space only depends on local measurements made within the surface (the
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Up to isometry, all homogeneous Riemannian manifolds arise by the following construction. Given a Lie group
14806:{\displaystyle R:{\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\to {\mathfrak {X}}(M)}
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In analogy to how an inner product on a vector space induces an isomorphism between a vector space and its
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Left- and bi-invariant metrics on Lie groups are an important source of examples of Riemannian manifolds.
5632:{\displaystyle {\widetilde {g}}_{p,q}((u_{1},u_{2}),(v_{1},v_{2}))=g_{p}(u_{1},v_{1})+h_{q}(u_{2},v_{2}).}
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in 1854. However, they would not be formalized until much later. In fact, the more primitive concept of a
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is the dimension of the manifold. In particular, every Riemannian manifold of constant curvature is an
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has a maximum, since it is a continuous function on a compact metric space. This proves the following.
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space itself without referencing an ambient space. In many instances, such as for hyperbolic space and
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In the case of strong Riemannian metrics, one part of the finite-dimensional Hopf–Rinow still holds.
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on which many geometric notions such as distance, angles, length, volume, and curvature are defined.
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proved in the 1920s that a locally symmetric Riemannian manifold which is geodesically complete and
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only allows this "coordinate chart geometry" to distort the "true geometry" by some bounded factor.
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on the tangent space at the identity, the inner product on the tangent space at an arbitrary point
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Burtscher, Annegret (2015). "Length structures on manifolds with continuous Riemannian metrics".
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10842:{\displaystyle \nabla _{f_{1}X_{1}+f_{2}X_{2}}Y=f_{1}\,\nabla _{X_{1}}Y+f_{2}\,\nabla _{X_{2}}Y,}
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must first pass though a certain "inner radius." The assumed continuity of the Riemannian metric
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The integrand is bounded and continuous except at finitely many points, so it is integrable. For
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19226:. Progress in Mathematics. Vol. 152. Translated by Bates, Sean Michael. With appendices by
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12997:. Any two such geodesics agree on their common domain. Taking the union over all open intervals
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contains information about the lengths and angle between the vectors. The dot products on every
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6254:{\displaystyle ({\widetilde {g}}_{ij})={\begin{pmatrix}g_{U}&0\\0&h_{V}\end{pmatrix}}.}
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11023:
10283:
9527:
9507:
8767:
4967:
2720:
1536:
is a positive-definite inner product then says exactly that this matrix-valued function is a
62:
18700:
16924:
16745:
14206:
12811:
9181:
8972:
8908:
7727:
4025:
3955:
3920:
3711:
2688:
2519:
2406:
2099:
1857:
1798:
1202:
1172:
573:
500:
426:
400:
202:, the study of Riemannian manifolds, has deep connections to other areas of math, including
20595:
20329:
20267:
20115:
19819:
19809:
19781:
19756:
19666:
19575:
19428:
19378:
19301:
19264:
19201:
19159:
19105:
18114:
18082:
18008:
17976:
17949:
17899:
17820:
17384:
17237:
16897:
16138:. These formulas simplify considerably in the special case of a Riemannian metric which is
15737:
15556:
in his 1854 lecture introducing Riemannian geometry, the locally-defined Riemannian metric
14976:
13852:
13773:
13282:
12517:
11504:
9938:
9844:
8876:
8289:
6377:
6350:
5955:
5908:
5876:
5829:
5698:
5646:
5192:
5160:
4820:
4526:
4486:
3827:
3682:
3678:
3616:
3550:
2634:
2176:
2144:
2070:
1885:
1512:
1395:
910:
271:
164:
19583:
19436:
19386:
19309:
19272:
19209:
19183:
19167:
19113:
18165:
is a weak Riemannian metric, then no notion of completeness implies the other in general.
17336:
15885:
2138:, and they are considered to be the same manifold for the purpose of Riemannian geometry.
1225:. Relative to this basis, one can define the Riemannian metric's components at each point
8:
20652:
20609:
20580:
20563:
20524:
20467:
20149:
20027:
19941:
19352:
19280:
18746:
In the upper half-space model of hyperbolic space, the Lie group structure is defined by
18210:
18180:
16504:
16154:
15889:
15751:
15371:
15339:
15191:
10584:{\displaystyle \nabla :{\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\to {\mathfrak {X}}(M)}
5063:
5057:
4733:
1928:
1889:
199:
20440:
17360:
17313:
16646:
16472:
with their standard metrics, along with hyperbolic space. The complex projective space,
9662:{\displaystyle L(\gamma )\geq {\sqrt {\lambda }}\int _{0}^{\delta }\|\gamma '(t)\|\,dt.}
7876:
6706:
6551:
4262:
2076:
20750:
20647:
20616:
20409:
20364:
20261:
20132:
19936:
19624:
19531:
19513:
19492:
19474:
19125:
18205:
18148:
17929:
17879:
17859:
17849:
17444:
17083:
16974:
16954:
16877:
16676:
16450:
16446:
15363:
15317:
15225:
15042:
14694:
14296:
14256:
13958:
13949:
13935:
13908:
13886:
13487:
13000:
12905:
12764:
12581:
12352:
11998:
11954:
11774:
11308:
10981:
10961:
10597:
10491:
10060:
9871:
9816:
9547:
9054:
8998:
8698:
8592:
7827:
7803:
7703:
7683:
7651:
7552:
7133:
7113:
7026:
6770:
6749:
6620:
6270:
5935:
5856:
5730:
5678:
5273:
5253:
4977:
4920:
4292:
4242:
4005:
3985:
3761:
3741:
3688:
3529:
2772:
2746:
2726:
2667:
2614:
2545:
1986:
1902:
1896:
1833:
1730:
1546:
1492:
1228:
1038:
886:
866:
653:
633:
613:
550:
530:
480:
460:
376:
365:
341:
323:
275:
262:
Riemannian manifolds were first conceptualized by their namesake, German mathematician
219:
211:
203:
81:
20657:
19946:
19566:
19547:
16512:
14940:
11264:
8761:
8081:
5222:
20662:
20344:
20324:
20319:
20226:
20137:
19951:
19931:
19786:
19725:
19496:
19453:
19414:
19364:
19329:
19250:
19187:
19175:
19145:
19091:
18220:
16500:
16191:
16167:
16163:
15914:
15826:
15780:
15718:
can be covered by coordinate charts relative to which the metric has the above form.
15549:
15215:
15183:
10444:
10438:
9672:
The integral which appears here represents the Euclidean length of a curve from 0 to
7898:
4435:{\displaystyle S^{n}=\{x\in \mathbb {R} ^{n+1}:(x^{1})^{2}+\cdots +(x^{n+1})^{2}=1\}}
1934:
1032:
671:
315:
283:
227:
177:
17973:
may fail to separate points. In fact, it may even be identically 0. For example, if
258:
20729:
20724:
20600:
20553:
20482:
20276:
20231:
20154:
20125:
19983:
19916:
19911:
19906:
19896:
19688:
19671:
19579:
19561:
19523:
19484:
19432:
19406:
19382:
19356:
19305:
19268:
19242:
19205:
19163:
19137:
19109:
19083:
18185:
17817:
are defined in a way similar to the finite-dimensional case. The distance function
16654:
16528:
16462:
15788:
15553:
15527:
15351:
14927:{\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{}Z}
6856:
5112:. Hence isometric immersions and isometric embeddings are Riemannian submanifolds.
1980:
1537:
297:
263:
243:
231:
207:
169:
121:
101:
15378:. This is equivalent to the condition that, relative to any coordinate chart, the
13382:
is not geodesically complete because the maximal geodesic with initial conditions
20558:
20493:
20425:
20334:
20164:
20120:
19886:
19571:
19535:
19424:
19374:
19297:
19260:
19197:
19155:
19101:
19079:
18215:
18175:
16650:
16442:
16404:
16386:
16240:
16159:
15806:
15728:
15499:
15014:
14317:
13877:
13575:
10015:
8547:
7217:{\displaystyle g=\sum _{\alpha \in A}\tau _{\alpha }\cdot {\tilde {g}}_{\alpha }}
6741:
4914:
3296:
394:
333:
301:
296:
Riemannian manifolds and their curvature were first introduced non-rigorously by
136:
74:
70:
31:
15867:
15310:
holds, and a (pseudo-)Riemannian manifold whose metric is Einstein is called an
13218:
that has the shortest length of any admissible curve with the same endpoints as
6153:{\displaystyle {\widetilde {g}}=\sum _{ij}{\widetilde {g}}_{ij}\,dx^{i}\,dx^{j}}
20291:
20216:
20186:
20084:
20077:
20017:
19988:
19858:
19853:
19814:
19394:
19284:
19235:
16225:
15776:
15711:
6976:
are diffeomorphisms. Such an atlas exists because the manifold is paracompact.
2206:
1976:
1567:
290:
185:
105:
19360:
19246:
19087:
19078:. Ergebnisse der Mathematik und ihrer Grenzgebiete (3). Vol. 10. Berlin:
16458:
311:
20780:
20585:
20477:
20301:
20296:
20281:
20271:
20221:
20198:
20072:
20032:
19973:
19921:
19720:
19457:
18200:
17048:
16508:
16496:
16146:
15383:
13940:
13659:
is not geodesically complete as the maximal geodesic with initial conditions
13307:
12542:
11154:{\displaystyle X{\bigl (}g(Y,Z){\bigr )}=g(\nabla _{X}Y,Z)+g(Y,\nabla _{X}Z)}
10485:
2764:
940:
601:
454:
181:
144:
140:
54:
19239:
18716:
16480:
are analogues of the real projective space which are also symmetric, as are
12555:
without making any effort to accelerate or turn would trace out a geodesic.
7700:
into Euclidean space and then pulls back the metric from Euclidean space to
7290:{\displaystyle {\tilde {g}}_{\alpha }=\varphi _{\alpha }^{*}g^{\text{can}}.}
4186:{\displaystyle i^{*}g_{p}(v,w)=g_{i(p)}{\big (}di_{p}(v),di_{p}(w){\big )},}
37:
20404:
20399:
20241:
20208:
20181:
20089:
19730:
19231:
19227:
19121:
19071:
17307:
16477:
16434:
16221:
15863:
15843:
15838:
inheriting a geodesically complete Riemannian metric of constant curvature
10229:{\displaystyle \operatorname {diam} (M,d_{g})=\sup\{d_{g}(p,q):p,q\in M\}.}
10011:
8543:
7614:{\displaystyle \operatorname {supp} (\tau _{\alpha })\subseteq U_{\alpha }}
7074:{\displaystyle \operatorname {supp} (\tau _{\alpha })\subseteq U_{\alpha }}
5134:
947:
741:
in a smooth way (see the section on regularity below). This induces a norm
305:
223:
12117:{\displaystyle D_{t}:{\mathfrak {X}}(\gamma )\to {\mathfrak {X}}(\gamma )}
946:
A Riemannian metric is not to be confused with the distance function of a
57:, packaged together into one mathematical object, are a Riemannian metric.
20247:
20236:
20193:
20094:
19695:
19543:
19488:
19317:
18988:
18111:
be a strong Riemannian manifold. Then metric completeness (in the metric
17354:
16520:
16300:
16255:
16131:
8275:{\displaystyle L(\gamma )=\int _{0}^{1}\|\gamma '(t)\|_{\gamma (t)}\,dt.}
6745:
2664:
235:
160:
42:
15174:
is the trace. The Ricci curvature tensor is a covariant 2-tensor field.
14511:
Parallel transports on the punctured plane under Levi-Civita connections
14366:. The curve the parallel transport is done along is the unit circle. In
20703:
20472:
20430:
20256:
20169:
19801:
19705:
19616:
19410:
19141:
18688:
17923:
separates points (hence is a metric) and induces the original topology.
15851:
15814:
15727:
is a Riemannian manifold with constant curvature which is additionally
15723:
15714:, and so it follows that any Riemannian manifold of constant curvature
6403:
5148:
4966:
automatically inherits a Riemannian metric. By the same principle, any
1571:
173:
113:
14518:
12514:(left), the maximal geodesics are straight lines. In the round sphere
10415:
is complete, then it is compact if and only if it has finite diameter.
20286:
20251:
19956:
19843:
19527:
19518:
17303:
15910:
12551:
4556:
3521:
349:
289:
A map that preserves the local measurements of a surface is called a
109:
1975:, a Riemannian metric induces an isomorphism of bundles between the
20575:
20450:
20445:
20435:
19826:
19647:
19479:
15875:
15799:
14586:
13944:
Parallel transport of a tangent vector along a curve in the sphere.
12457:
6845:{\displaystyle \{(U_{\alpha },\varphi _{\alpha })\}_{\alpha \in A}}
6728:
4289:
3526:
117:
78:
15710:. Any two Riemannian manifolds of the same constant curvature are
13802:
be a connected Riemannian manifold. The following are equivalent:
12476:
6736:
Every smooth manifold admits a (non-canonical) Riemannian metric.
4937:. This is an immersion (since it is locally a diffeomorphism), so
19132:(Revised reprint of the 1975 original ed.). Providence, RI:
19022:
19020:
19018:
17742:{\displaystyle G_{f}(u,v)=\int _{M}g_{f(x)}(u(x),v(x))\,d\mu (x)}
15219:
12467:
9269:
denotes the Euclidean norm induced by the local coordinates. Let
6267:
5125:
215:
18676:
18664:
16059:{\displaystyle g_{p}(u,v)=g_{e}(dL_{p^{-1}}(u),dL_{p^{-1}}(v)),}
7897:
there are many natural smooth Riemannian manifolds, such as the
3201:{\displaystyle g^{\text{can}}=(dx^{1})^{2}+\cdots +(dx^{n})^{2}}
20042:
3574:
with the round metric is an embedded Riemannian submanifold of
46:
19015:
19005:
19003:
16312:
are in one-to-one correspondence with those inner products on
15314:. Examples of Einstein manifolds include Euclidean space, the
15144:{\displaystyle Ric(X,Y)=\operatorname {tr} (Z\mapsto R(Z,X)Y)}
8016:
is nonzero everywhere it is defined. The nonnegative function
5434:{\displaystyle T_{(p,q)}(M\times N)\cong T_{p}M\oplus T_{q}N,}
360:
19037:
19035:
15871:
9770:
The observation about comparison between lengths measured by
6436:{\displaystyle \mathbb {R} \times \cdots \times \mathbb {R} }
5144:
19238:. (Based on the 1981 French original ed.). Boston, MA:
18960:
18654:
18652:
15779:
construction, any Riemannian space form is isometric to the
15488:{\displaystyle R_{ijkl}=\kappa (g_{il}g_{jk}-g_{ik}g_{jl}).}
8838:. Verification of the other metric space axioms is omitted.
120:. Riemannian manifolds are named after German mathematician
19401:(Sixth edition of 1967 original ed.). Providence, RI:
19351:. Vol. 171 (Third edition of 1998 original ed.).
19000:
18972:
18565:
17298:
its diffeomorphism group. The latter is a smooth manifold (
16495:, referring to those which cannot be locally decomposed as
16349:
naturally viewed as a subgroup of the full isometry group.
14645:
Warning: This is parallel transport on the punctured plane
12440:{\displaystyle D_{t}X(t)=\nabla _{\gamma '(t)}{\tilde {X}}}
11840:
is a smooth vector field on a neighborhood of the image of
9813:
coincides with the original topological space structure of
7542:{\displaystyle \tau _{\alpha }\cdot {\tilde {g}}_{\alpha }}
6760:
Proof that every smooth manifold admits a Riemannian metric
6374:
is given the round metric, the product Riemannian manifold
1339:
1300:
1024:{\displaystyle (x^{1},\ldots ,x^{n}):U\to \mathbb {R} ^{n}}
27:
Smooth manifold with an inner product on each tangent space
19223:
Metric structures for Riemannian and non-Riemannian spaces
19032:
17306:. Its tangent bundle at the identity is the set of smooth
15817:, the natural group action of the orthogonal group on the
14370:, the metric on the left is the standard Euclidean metric
1712:{\displaystyle g=\sum _{i,j}g_{ij}\,dx^{i}\otimes dx^{j}.}
19047:
18649:
16828:{\displaystyle g_{x}:T_{x}M\times T_{x}M\to \mathbb {R} }
16365:, and the direct sum decomposition of the Lie algebra of
13247:
The nonconstant maximal geodesics of the Euclidean plane
734:{\displaystyle g_{p}:T_{p}M\times T_{p}M\to \mathbb {R} }
18428:
18271:
18269:
18256:
18254:
18239:
15552:, thereby having constant scalar curvature. As found by
14452:{\displaystyle dx^{2}+dy^{2}=dr^{2}+r^{2}\,d\theta ^{2}}
13623:{\displaystyle \mathbb {R} ^{2}\smallsetminus \{(0,0)\}}
10427:
7399:{\displaystyle \varphi _{\alpha }^{*}g^{\mathrm {can} }}
2393:{\displaystyle g_{p}(u,v)=h_{f(p)}(df_{p}(u),df_{p}(v))}
18728:
16457:. This property nearly characterizes symmetric spaces;
15900:
10286:
and has finite diameter, it is compact. Conversely, if
2743:
which allows measurable functions to be integrated. If
1570:, the Riemannian metric can be written in terms of the
953:
352:, which is a 4-dimensional pseudo-Riemannian manifold.
304:
was first explicitly defined only in 1913 in a book by
18613:
18577:
18500:
18476:
18452:
17760:
Length of curves and the Riemannian distance function
14359:{\displaystyle \mathbb {R} ^{2}\smallsetminus \{0,0\}}
13348:
6203:
4970:
of a Riemannian manifold inherits a Riemannian metric.
3344:
1722:
340:(a generalization of Riemannian manifolds) to develop
18752:
18519:
18517:
18515:
18266:
18251:
18151:
18117:
18085:
18038:
18011:
17979:
17952:
17932:
17902:
17882:
17862:
17823:
17766:
17632:
17564:
17551:{\displaystyle u,v\in T_{f}\operatorname {Diff} (M).}
17507:
17467:
17447:
17414:
17387:
17363:
17339:
17316:
17272:
17240:
17199:
17139:
17106:
17086:
17057:
17040:{\displaystyle (H,\langle \,\cdot ,\cdot \,\rangle )}
17007:
16977:
16957:
16927:
16900:
16880:
16841:
16774:
16748:
16699:
16679:
16615:
16266:. If this subspace is invariant under the linear map
15941:
15565:
15395:
15356:
15320:
15293:
15258:
15228:
15194:
15160:
15074:
15045:
15025:
14979:
14943:
14822:
14723:
14697:
14677:
14598:
14530:
14465:
14376:
14325:
14299:
14279:
14259:
14239:
14209:
14173:
14138:
14103:
14059:
14023:
14002:
13981:
13961:
13911:
13889:
13855:
13812:
13776:
13741:
13703:
13665:
13636:
13583:
13551:
13516:
13490:
13464:
13426:
13388:
13335:
13285:
13253:
13224:
13180:
13137:
13102:
13058:
13023:
13003:
12963:
12928:
12908:
12876:
12840:
12814:
12787:
12767:
12726:
12706:
12671:
12650:
12604:
12584:
12564:
12520:
12491:
12375:
12355:
12326:
12248:
12156:
12132:
12065:
12021:
12001:
11981:
11957:
11926:
11866:
11846:
11817:
11797:
11777:
11732:
11667:
11618:
11598:
11578:
11545:
11507:
11453:
11406:
11385:
11339:
11328:
11311:
11267:
11197:
11170:
11053:
11026:
10984:
10964:
10932:
10857:
10714:
10672:
10620:
10600:
10520:
10494:
10457:
10382:
10331:
10292:
10249:
10139:
10097:
10063:
10024:
9977:
9941:
9894:
9874:
9847:
9819:
9780:
9729:
9712:{\displaystyle x(\partial V)\subset \mathbb {R} ^{n}}
9678:
9588:
9550:
9530:
9510:
9467:
9429:
9403:
9351:
9279:
9249:
9210:
9184:
9127:
9107:
9077:
9057:
9021:
9001:
8975:
8946:
8911:
8879:
8796:
8770:
8727:
8701:
8662:
8623:
8595:
8556:
8509:
8385:
8325:
8292:
8192:
8145:
8116:
8084:
8071:{\displaystyle t\mapsto \|\gamma '(t)\|_{\gamma (t)}}
8022:
7963:
7919:
7879:
7850:
7830:
7806:
7765:
7730:
7706:
7686:
7654:
7627:
7575:
7555:
7506:
7479:
7443:
7416:
7362:
7333:
7306:
7236:
7159:
7136:
7116:
7087:
7035:
6985:
6898:
6865:
6793:
6773:
6709:
6643:
6623:
6577:
6554:
6508:
6471:
6449:
6413:
6380:
6353:
6340:{\displaystyle T^{n}=S^{1}\times \cdots \times S^{1}}
6294:
6273:
6166:
6071:
6019:
5990:
5958:
5938:
5911:
5879:
5859:
5832:
5803:
5753:
5733:
5701:
5681:
5649:
5449:
5358:
5325:
5296:
5276:
5256:
5230:
5195:
5163:
5073:
5029:
5000:
4980:
4943:
4923:
4890:
4855:
4823:
4785:
4742:
4704:
4566:
4529:
4489:
4454:
4318:
4295:
4265:
4245:
4202:
4061:
4028:
4008:
3988:
3958:
3923:
3891:
3865:
3830:
3784:
3764:
3744:
3714:
3691:
3651:
3619:
3580:
3553:
3532:
3461:
3311:
3271:
3220:
3123:
2960:
2930:
2895:
2849:
2796:
2775:
2749:
2729:
2691:
2670:
2637:
2617:
2568:
2548:
2522:
2483:
2435:
2409:
2289:
2250:
2214:
2179:
2147:
2102:
2079:
2012:
1989:
1943:
1905:
1860:
1836:
1801:
1757:
1733:
1640:
1579:
1549:
1515:
1495:
1469:
1425:
1398:
1254:
1231:
1205:
1175:
1064:
1041:
964:
913:
889:
869:
804:
791:{\displaystyle \|\cdot \|_{p}:T_{p}M\to \mathbb {R} }
747:
680:
656:
636:
616:
576:
553:
533:
503:
483:
463:
429:
403:
379:
84:
19548:"Curvatures of left invariant metrics on Lie groups"
18440:
18394:
18392:
18367:
18365:
13375:{\displaystyle \mathbb {R} ^{2}\backslash \{(0,0)\}}
13238:
is a geodesic (in a unit-speed reparameterization).
4698:
is a smooth embedded submanifold of Euclidean space
4448:
is a smooth embedded submanifold of Euclidean space
18350:
18328:
18326:
18313:
18311:
18286:
18284:
16345:. Each such Riemannian metric is homogeneous, with
15895:
13630:with the restriction of the Riemannian metric from
10915:{\displaystyle \nabla _{X}fY=X(f)Y+f\,\nabla _{X}Y}
9761:{\displaystyle L(\gamma )\geq {\sqrt {\lambda }}R.}
9338:{\displaystyle \sup\{r>0:B_{r}(0)\subset x(V)\}}
6497:
20523:
18950:
18637:
18625:
18601:
18589:
18553:
18541:
18529:
18512:
18488:
18464:
18157:
18130:
18103:
18056:
18024:
17997:
17965:
17938:
17915:
17888:
17868:
17836:
17809:
17741:
17616:
17550:
17494:
17453:
17432:
17400:
17372:
17345:
17325:
17290:
17258:
17223:
17185:
17125:
17092:
17072:
17039:
16983:
16963:
16943:
16913:
16886:
16860:
16827:
16760:
16734:
16685:
16633:
16224:as the requirement that the natural action of the
16058:
15695:
15487:
15326:
15299:
15279:
15234:
15206:
15166:
15143:
15051:
15031:
14997:
14961:
14926:
14805:
14703:
14683:
14630:
14572:
14497:
14451:
14358:
14305:
14293:, and then take the value of this vector field at
14285:
14265:
14245:
14225:
14195:
14159:
14124:
14089:
14045:
14008:
13987:
13967:
13917:
13895:
13868:
13837:
13794:
13749:
13727:
13689:
13651:
13622:
13566:
13537:
13496:
13472:
13450:
13412:
13374:
13298:
13268:
13230:
13210:
13163:
13123:
13084:
13044:
13009:
12989:
12949:
12914:
12894:
12862:
12826:
12793:
12773:
12753:
12712:
12692:
12656:
12634:
12590:
12570:
12533:
12506:
12439:
12361:
12341:
12310:
12233:
12138:
12116:
12051:
12007:
11987:
11963:
11941:
11912:
11852:
11832:
11803:
11783:
11760:
11718:
11650:
11604:
11584:
11564:
11531:
11493:
11439:
11391:
11369:
11317:
11285:
11250:
11176:
11153:
11032:
10990:
10970:
10948:
10914:
10841:
10700:
10654:
10606:
10583:
10500:
10476:
10407:
10364:
10317:
10274:
10228:
10122:
10069:
10049:
10002:
9959:
9927:
9880:
9860:
9825:
9805:
9760:
9711:
9661:
9571:
9536:
9516:
9494:
9453:
9415:
9381:
9337:
9261:
9235:
9196:
9170:
9113:
9093:
9063:
9043:
9007:
8987:
8961:
8932:
8897:
8830:
8782:
8752:
8707:
8687:
8648:
8601:
8581:
8534:
8490:
8368:
8310:
8274:
8175:
8131:
8102:
8070:
8008:
7949:
7888:
7865:
7836:
7812:
7792:
7751:
7724:states that, given any smooth Riemannian manifold
7712:
7692:
7660:
7640:
7613:
7561:
7541:
7492:
7465:
7429:
7398:
7348:
7319:
7289:
7216:
7142:
7122:
7099:
7073:
7018:{\displaystyle \{\tau _{\alpha }\}_{\alpha \in A}}
7017:
6968:
6884:
6844:
6779:
6718:
6695:
6629:
6609:
6563:
6540:
6486:
6457:
6435:
6393:
6366:
6339:
6279:
6253:
6152:
6055:
6005:
5976:
5944:
5924:
5897:
5865:
5845:
5818:
5789:
5739:
5719:
5687:
5667:
5631:
5433:
5340:
5311:
5282:
5262:
5242:
5213:
5181:
5104:
5047:
5015:
4986:
4958:
4929:
4905:
4876:
4841:
4806:
4772:{\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }
4771:
4719:
4684:
4547:
4502:
4475:
4434:
4301:
4274:
4251:
4227:
4185:
4044:
4014:
3994:
3974:
3944:
3909:
3877:
3848:
3812:
3770:
3750:
3730:
3697:
3669:
3637:
3601:
3566:
3538:
3495:
3444:
3287:
3257:
3200:
3103:
2943:
2913:
2881:
2822:
2781:
2755:
2735:
2707:
2676:
2655:
2623:
2595:
2554:
2534:
2504:
2466:
2421:
2392:
2272:
2232:
2197:
2165:
2118:
2088:
2058:
1995:
1967:
1911:
1876:
1842:
1822:
1787:
1739:
1711:
1623:
1555:
1528:
1501:
1481:
1455:
1411:
1380:
1237:
1217:
1191:
1158:
1047:
1023:
931:
895:
875:
855:
790:
733:
662:
642:
622:
592:
559:
539:
519:
489:
469:
445:
415:
385:
330:, a special connection on a Riemannian manifold.
238:. Generalizations of Riemannian manifolds include
90:
18416:
18404:
18389:
18377:
18362:
16534:
13574:is geodesically complete. On the other hand, the
12234:{\displaystyle D_{t}(aX+bY)=a\,D_{t}X+b\,D_{t}Y,}
4849:is not simply connected, there is a covering map
3496:{\displaystyle (\mathbb {R} ^{n},g^{\text{can}})}
1968:{\displaystyle v\mapsto \langle v,\cdot \rangle }
1919:is assumed to be smooth unless stated otherwise.
1140:
1095:
20778:
18338:
18323:
18308:
18296:
18281:
16566:but its sources remain unclear because it lacks
16417:there exists some isometry of the manifold with
10171:
9280:
8414:
6729:Every smooth manifold admits a Riemannian metric
3258:{\displaystyle g_{ij}^{\text{can}}=\delta _{ij}}
19279:
18994:
17266:be a compact Riemannian manifold and denote by
16149:, constructed as left-invariant metrics on the
13764:characterizes geodesically complete manifolds.
10365:{\displaystyle d_{g}:M\times M\to \mathbb {R} }
9928:{\displaystyle d_{g}:M\times M\to \mathbb {R} }
9495:{\displaystyle \gamma (\delta )\in \partial V.}
8110:except for at finitely many points. The length
7029:subordinate to the given atlas, i.e. such that
192:are constructed intrinsically using tools from
19464:
19290:Foundations of differential geometry. Volume I
19041:
17810:{\displaystyle d_{g}:M\times M\to [0,\infty )}
17495:{\displaystyle f\in \operatorname {Diff} (M),}
17186:{\displaystyle g_{x}(u,v)=\langle u,v\rangle }
16735:{\displaystyle g:TM\times TM\to \mathbb {R} ,}
16153:SU(2), are among the simplest examples of the
16104:. Riemannian metrics constructed this way are
11913:{\displaystyle X(t)={\tilde {X}}_{\gamma (t)}}
11761:{\displaystyle X\in {\mathfrak {X}}(\gamma ).}
8841:There must be some precompact open set around
8656:is a metric space, and the metric topology on
8369:{\displaystyle d_{g}:M\times M\to [0,\infty )}
8009:{\displaystyle \gamma '(t)\in T_{\gamma (t)}M}
6748:. The reason is that the proof makes use of a
5221:be two Riemannian manifolds, and consider the
1795:are continuous in any smooth coordinate chart
856:{\displaystyle \|v\|_{p}={\sqrt {g_{p}(v,v)}}}
163:of Euclidean space of any dimension. Although
20509:
19632:
19503:
19053:
15915:group of rotations in three-dimensional space
11084:
11059:
6696:{\displaystyle f_{1}g_{1}+\ldots +f_{k}g_{k}}
6407:. As another example, the Riemannian product
4175:
4121:
939:. A Riemannian metric is a special case of a
19598:
17180:
17168:
17031:
17017:
14353:
14341:
13617:
13599:
13369:
13351:
13017:containing 0 on which a geodesic satisfying
10220:
10174:
9888:is compact, there always exist points where
9646:
9626:
9332:
9283:
9256:
9250:
9171:{\displaystyle g(X,X)\geq \lambda \|X\|^{2}}
9159:
9152:
8482:
8417:
8244:
8223:
8050:
8029:
7000:
6986:
6827:
6794:
4429:
4332:
3211:or equivalently by its coordinate functions
2603:is an isometry (and thus a diffeomorphism).
2059:{\displaystyle (p,v)\mapsto g_{p}(v,\cdot )}
1962:
1950:
1618:
1580:
812:
805:
755:
748:
19120:
18706:
18694:
18005:is a compact Riemannian manifold, then the
17755:
16220:. This can be rephrased in the language of
14656:
13510:if the domain of every maximal geodesic is
11251:{\displaystyle \nabla _{X}Y-\nabla _{Y}X=,}
9524:is at least as large as the restriction of
8764:, the most difficult part is checking that
7899:set of rotations of three-dimensional space
361:Riemannian metrics and Riemannian manifolds
20679:Fundamental theorem of Riemannian geometry
20516:
20502:
19639:
19625:
19130:Comparison theorems in Riemannian geometry
15735:. A Riemannian space form is said to be a
6465:has the Euclidean metric, is isometric to
19565:
19517:
19478:
19443:
18434:
17723:
17030:
17020:
16821:
16725:
16618:
16597:Learn how and when to remove this message
15218:, which has applications to the study of
15008:
14435:
14328:
13955:Specifically, call a smooth vector field
13743:
13639:
13586:
13554:
13466:
13338:
13256:
12494:
12291:
12214:
12194:
11644:
11002:
10898:
10815:
10781:
10655:{\displaystyle (X,Y)\mapsto \nabla _{X}Y}
10358:
9921:
9699:
9649:
9454:{\displaystyle \gamma (\delta )\notin V;}
9044:{\displaystyle {\overline {V}}\subset U.}
8262:
7904:
7853:
7780:
7336:
6474:
6451:
6429:
6415:
6136:
6122:
4788:
4765:
4751:
4707:
4580:
4457:
4343:
3583:
3467:
2898:
2512:not assumed to be a diffeomorphism, is a
1781:
1676:
1624:{\displaystyle \{dx^{1},\ldots ,dx^{n}\}}
1449:
1011:
784:
727:
527:are thought of as the vectors tangent to
19646:
19342:
19174:
19026:
18734:
18722:
18710:
18571:
18275:
18260:
18245:
18057:{\displaystyle \operatorname {Diff} (M)}
17617:{\displaystyle x\in M,u(x)\in T_{f(x)}M}
17433:{\displaystyle \operatorname {Diff} (M)}
17291:{\displaystyle \operatorname {Diff} (M)}
14649:the unit circle, not parallel transport
14573:{\displaystyle dr^{2}+r^{2}d\theta ^{2}}
13939:
13324:
13092:exists, one obtains a geodesic called a
12311:{\displaystyle D_{t}(fX)=f'X+f\,D_{t}X,}
11565:{\displaystyle {\mathfrak {X}}(\gamma )}
9719:, and so it is greater than or equal to
8319:a connected Riemannian manifold, define
4483:. The Riemannian metric this induces on
3520:
1788:{\displaystyle g_{ij}:U\to \mathbb {R} }
1456:{\displaystyle g_{ij}:U\to \mathbb {R} }
364:
257:
36:
18068:
16499:. Every such space is an example of an
15821:-sphere restricts to a group action of
15214:plays a defining role in the theory of
11494:{\displaystyle X(t)\in T_{\gamma (t)}M}
7793:{\displaystyle F:M\to \mathbb {R} ^{N}}
14:
20779:
19542:
19216:
18982:
16894:is a weak Riemannian metric such that
16239:which does not contain any nontrivial
15361:A Riemannian manifold is said to have
14592:This transport is given by the metric
14524:This transport is given by the metric
13306:with the round metric are exactly the
7466:{\displaystyle {\tilde {g}}_{\alpha }}
6885:{\displaystyle U_{\alpha }\subseteq M}
5348:which can be described in a few ways.
1922:
423:, there is an associated vector space
20497:
19620:
19070:
19009:
16299:-invariant Riemannian metrics on the
16173:
15177:
13929:
13314:
13279:The nonconstant maximal geodesics of
10428:Connections, geodesics, and curvature
7844:of the standard Riemannian metric on
6617:are any positive smooth functions on
4877:{\displaystyle {\widetilde {M}}\to M}
19393:
19322:Introduction to Riemannian Manifolds
18682:
18670:
18658:
18064:induces vanishing geodesic distance.
16971:is a strong Riemannian metric, then
16538:
15901:Left-invariant metrics on Lie groups
14973:. The Riemann curvature tensor is a
8440: an admissible curve with
5023:, then the immersion (or embedding)
4779:is a smooth embedded submanifold of
954:The Riemannian metric in coordinates
348:are constraints on the curvature of
108:in three-dimensional space, such as
19316:
18978:
18966:
18643:
18631:
18619:
18607:
18595:
18583:
18559:
18547:
18535:
18523:
18506:
18494:
18482:
18470:
18458:
18446:
18422:
18410:
18398:
18383:
18371:
18356:
18344:
18332:
18317:
18302:
18290:
16380:
16212:of the Riemannian manifold sending
15345:
15167:{\displaystyle \operatorname {tr} }
14789:
14770:
14751:
14732:
14631:{\displaystyle dr^{2}+d\theta ^{2}}
14498:{\displaystyle dr^{2}+d\theta ^{2}}
14459:, while the metric on the right is
13504:with its Levi-Civita connection is
13096:of which every geodesic satisfying
12541:(right), the maximal geodesics are
12100:
12081:
11741:
11548:
10701:{\displaystyle f\in C^{\infty }(M)}
10567:
10548:
10529:
10460:
9868:by any explicit means. In fact, if
6610:{\displaystyle f_{1},\ldots ,f_{k}}
6541:{\displaystyle g_{1},\ldots ,g_{k}}
5105:{\displaystyle {\tilde {g}}=i^{*}g}
2889:denote the standard coordinates on
2882:{\displaystyle x^{1},\ldots ,x^{n}}
1723:Regularity of the Riemannian metric
180:are defined intrinsically by using
24:
17801:
16425:and for which the negation of the
15357:Constant curvature and space forms
15026:
14900:
14884:
14874:
14858:
14848:
14678:
13883:All closed and bounded subsets of
13538:{\displaystyle (-\infty ,\infty )}
13529:
13523:
12565:
12402:
11982:
11651:{\displaystyle f:\to \mathbb {R} }
11329:Covariant derivative along a curve
11215:
11199:
11171:
11136:
11099:
11027:
10934:
10900:
10859:
10817:
10783:
10716:
10684:
10640:
10521:
10477:{\displaystyle {\mathfrak {X}}(M)}
9685:
9483:
9345:. Now, given any admissible curve
8905:be a smooth coordinate chart with
8360:
7621:. It takes the value 0 outside of
7430:{\displaystyle \varphi _{\alpha }}
7390:
7387:
7384:
5290:naturally put a Riemannian metric
4807:{\displaystyle \mathbb {R} ^{n+1}}
4476:{\displaystyle \mathbb {R} ^{n+1}}
3602:{\displaystyle \mathbb {R} ^{n+1}}
3047:
3043:
3002:
2998:
2838:
1347:
1343:
1308:
1304:
1121:
1117:
1076:
1072:
149:differential and integral calculus
25:
20808:
19592:
18138:) implies geodesic completeness.
17876:is a strong Riemannian metric on
16634:{\displaystyle \mathbb {R} ^{n}.}
15382:can be expressed in terms of the
14273:to a vector field parallel along
14167:. to parallel transport a vector
6056:{\displaystyle (U\times V,(x,y))}
5790:{\displaystyle (U\times V,(x,y))}
3645:be a Riemannian manifold and let
2914:{\displaystyle \mathbb {R} ^{n}.}
2631:-dimensional Riemannian manifold
1169:form a basis of the vector space
950:, which is also called a metric.
883:endowed with a Riemannian metric
318:, one of the first concepts of a
286:("remarkable theorem" in Latin).
186:constant scalar curvature metrics
124:, who first conceptualized them.
116:, are all examples of Riemannian
16543:
16391:A connected Riemannian manifold
15896:Riemannian metrics on Lie groups
15745:if the curvature is zero, and a
15741:if the curvature is positive, a
14585:
14517:
13652:{\displaystyle \mathbb {R} ^{2}}
13567:{\displaystyle \mathbb {R} ^{2}}
13269:{\displaystyle \mathbb {R} ^{2}}
12507:{\displaystyle \mathbb {R} ^{n}}
12475:
12466:
11719:{\displaystyle (fX)(t)=f(t)X(t)}
9094:{\displaystyle {\overline {V}},}
7866:{\displaystyle \mathbb {R} ^{N}}
7349:{\displaystyle \mathbb {R} ^{n}}
6956:
6703:is another Riemannian metric on
6498:Positive combinations of metrics
6487:{\displaystyle \mathbb {R} ^{n}}
6006:{\displaystyle {\widetilde {g}}}
5797:is a smooth coordinate chart on
5727:is a smooth coordinate chart on
5675:is a smooth coordinate chart on
5312:{\displaystyle {\widetilde {g}}}
5133:
5124:
4994:already has a Riemannian metric
4959:{\displaystyle {\widetilde {M}}}
4906:{\displaystyle {\widetilde {M}}}
4720:{\displaystyle \mathbb {R} ^{3}}
2205:are two Riemannian manifolds, a
19446:New York Journal of Mathematics
18740:
16084:is the left multiplication map
15767:respectively. Furthermore, the
15334:-sphere, hyperbolic space, and
15222:. A (pseudo-)Riemannian metric
12754:{\displaystyle D_{t}\gamma '=0}
11791:be a smooth vector field along
10057:coincides with the topology on
8831:{\displaystyle d_{g}(p,q)>0}
8589:coincides with the topology on
3510:
3302:which together form the matrix
2823:{\displaystyle \int _{M}dV_{g}}
600:does not come equipped with an
282:). This result is known as the
19679:Differentiable/Smooth manifold
18942:
18829:
18823:
18791:
18785:
18753:
18098:
18086:
18051:
18045:
17992:
17980:
17804:
17792:
17789:
17736:
17730:
17720:
17717:
17711:
17702:
17696:
17690:
17685:
17679:
17655:
17643:
17606:
17600:
17586:
17580:
17542:
17536:
17486:
17480:
17427:
17421:
17285:
17279:
17253:
17241:
17231:is a strong Riemannian metric.
17162:
17150:
17034:
17008:
16817:
16721:
16660:
16535:Infinite-dimensional manifolds
16050:
16047:
16041:
16012:
16006:
15980:
15964:
15952:
15681:
15677:
15635:
15616:
15479:
15421:
15138:
15132:
15120:
15114:
15108:
15096:
15084:
14992:
14980:
14956:
14944:
14916:
14904:
14838:
14826:
14800:
14794:
14784:
14781:
14775:
14762:
14756:
14743:
14737:
14148:
14142:
14113:
14107:
14081:
14078:
14066:
13832:
13813:
13789:
13777:
13722:
13710:
13684:
13672:
13614:
13602:
13532:
13517:
13445:
13433:
13407:
13395:
13366:
13354:
13202:
13199:
13187:
13152:
13146:
13112:
13106:
13073:
13067:
13033:
13027:
12978:
12972:
12938:
12932:
12902:defined on some open interval
12895:{\displaystyle \gamma :I\to M}
12886:
12626:
12623:
12611:
12431:
12420:
12414:
12395:
12389:
12333:
12268:
12259:
12185:
12167:
12111:
12105:
12095:
12092:
12086:
12043:
12040:
12028:
11933:
11905:
11899:
11889:
11876:
11870:
11824:
11752:
11746:
11713:
11707:
11701:
11695:
11686:
11680:
11677:
11668:
11640:
11637:
11625:
11572:of smooth vector fields along
11559:
11553:
11526:
11514:
11483:
11477:
11463:
11457:
11428:
11425:
11413:
11361:
11358:
11346:
11280:
11268:
11242:
11230:
11148:
11126:
11117:
11095:
11079:
11067:
10886:
10880:
10695:
10689:
10636:
10633:
10621:
10578:
10572:
10562:
10559:
10553:
10540:
10534:
10471:
10465:
10432:
10402:
10383:
10354:
10325:is compact, then the function
10312:
10293:
10269:
10250:
10199:
10187:
10165:
10146:
10117:
10098:
10044:
10025:
9997:
9978:
9954:
9942:
9917:
9800:
9781:
9739:
9733:
9691:
9682:
9643:
9637:
9598:
9592:
9563:
9551:
9477:
9471:
9439:
9433:
9373:
9370:
9358:
9329:
9323:
9314:
9308:
9143:
9131:
8921:
8915:
8892:
8880:
8819:
8807:
8747:
8728:
8682:
8663:
8643:
8624:
8576:
8557:
8529:
8510:
8473:
8467:
8452:
8446:
8429:
8423:
8408:
8396:
8363:
8351:
8348:
8305:
8293:
8257:
8251:
8240:
8234:
8202:
8196:
8167:
8164:
8152:
8126:
8120:
8097:
8085:
8063:
8057:
8046:
8040:
8026:
7998:
7992:
7978:
7972:
7941:
7938:
7926:
7775:
7743:
7731:
7676:An alternative proof uses the
7595:
7582:
7527:
7451:
7320:{\displaystyle g^{\text{can}}}
7244:
7202:
7055:
7042:
6948:
6935:
6922:
6823:
6797:
6192:
6167:
6050:
6047:
6035:
6020:
5971:
5959:
5892:
5880:
5784:
5781:
5769:
5754:
5714:
5702:
5662:
5650:
5623:
5597:
5581:
5555:
5539:
5536:
5510:
5504:
5478:
5475:
5393:
5381:
5376:
5364:
5352:Considering the decomposition
5208:
5196:
5176:
5164:
5080:
5039:
5007:
4868:
4836:
4824:
4761:
4414:
4394:
4376:
4362:
4222:
4216:
4170:
4164:
4145:
4139:
4114:
4108:
4094:
4082:
4022:. In general, the formula for
3933:
3927:
3901:
3843:
3831:
3807:
3785:
3661:
3632:
3620:
3490:
3462:
3333:
3312:
3189:
3172:
3154:
3137:
2944:{\displaystyle g^{\text{can}}}
2650:
2638:
2590:
2584:
2578:
2493:
2467:{\displaystyle u,v\in T_{p}M.}
2387:
2384:
2378:
2359:
2353:
2337:
2332:
2326:
2312:
2300:
2224:
2192:
2180:
2160:
2148:
2053:
2041:
2028:
2025:
2013:
1947:
1814:
1802:
1777:
1445:
1270:
1006:
997:
965:
926:
914:
848:
836:
780:
723:
194:partial differential equations
45:of two vectors tangent to the
13:
1:
19567:10.1016/S0001-8708(76)80002-3
19349:Graduate Texts in Mathematics
18227:
17946:is a weak Riemannian metric,
17461:, is defined as follows. Let
16995:
16474:quaternionic projective space
15280:{\displaystyle Ric=\lambda g}
14090:{\displaystyle \gamma :\to M}
13241:
13211:{\displaystyle \gamma :\to M}
13164:{\displaystyle \gamma '(0)=v}
13085:{\displaystyle \gamma '(0)=v}
12990:{\displaystyle \gamma '(0)=v}
12693:{\displaystyle D_{t}\gamma '}
12635:{\displaystyle \gamma :\to M}
12059:, there is a unique operator
12052:{\displaystyle \gamma :\to M}
11370:{\displaystyle \gamma :\to M}
9397:, there must be some minimal
9382:{\displaystyle \gamma :\to M}
8176:{\displaystyle \gamma :\to M}
7950:{\displaystyle \gamma :\to M}
2129:
2003:is a Riemannian metric, then
355:
51:3-dimensional Euclidean space
20606:Raising and lowering indices
19399:Spaces of constant curvature
16196:if for every pair of points
14971:Lie bracket of vector fields
14160:{\displaystyle \gamma (1)=q}
14125:{\displaystyle \gamma (0)=p}
13750:{\displaystyle \mathbb {R} }
13473:{\displaystyle \mathbb {R} }
13124:{\displaystyle \gamma (0)=p}
13045:{\displaystyle \gamma (0)=p}
12950:{\displaystyle \gamma (0)=p}
12451:
12342:{\displaystyle {\tilde {X}}}
11942:{\displaystyle {\tilde {X}}}
11833:{\displaystyle {\tilde {X}}}
10949:{\displaystyle \nabla _{X}Y}
9416:{\displaystyle \delta >0}
9236:{\displaystyle X\in T_{r}M,}
9083:
9027:
8695:agrees with the topology on
7913:is a piecewise smooth curve
7100:{\displaystyle \alpha \in A}
6458:{\displaystyle \mathbb {R} }
5016:{\displaystyle {\tilde {g}}}
3878:{\displaystyle N\subseteq M}
3288:{\displaystyle \delta _{ij}}
2273:{\displaystyle g=f^{\ast }h}
1463:can be put together into an
7:
20385:Classification of manifolds
19606:Encyclopedia of Mathematics
19295:John Wiley & Sons, Inc.
19176:do Carmo, Manfredo Perdigão
18995:Kobayashi & Nomizu 1963
18725:, Section 4.4.3 and p. 399.
18168:
16991:must be a Hilbert manifold.
16507:. As found in the 1950s by
16465:must in fact be symmetric.
15856:Poincaré dodecahedral space
15188:The Ricci curvature tensor
14196:{\displaystyle v\in T_{p}M}
12863:{\displaystyle v\in T_{p}M}
12126:covariant derivative along
10080:
9101:there is a positive number
8078:is defined on the interval
7641:{\displaystyle U_{\alpha }}
7493:{\displaystyle U_{\alpha }}
7327:is the Euclidean metric on
7110:Define a Riemannian metric
6494:with the Euclidean metric.
5115:
3982:is just the restriction of
2833:
2596:{\displaystyle f:U\to f(U)}
2477:One says that a smooth map
1631:of the cotangent bundle as
338:pseudo-Riemannian manifolds
240:pseudo-Riemannian manifolds
159:. The same is true for any
10:
20813:
20627:Pseudo-Riemannian manifold
19063:
19042:Magnani & Tiberio 2020
18196:Pseudo-Riemannian manifold
18032:weak Riemannian metric on
17408:weak Riemannian metric on
17224:{\displaystyle x,u,v\in H}
16517:quaternion-Kähler geometry
16384:
16316:which are invariant under
16262:within the Lie algebra of
15805:thereof in which only the
15349:
15181:
15012:
14660:
13933:
13318:
12870:, there exists a geodesic
12455:
11379:smooth vector field along
11006:
10436:
9262:{\displaystyle \|\cdot \|}
8962:{\displaystyle q\notin U.}
8132:{\displaystyle L(\gamma )}
6266:For example, consider the
5819:{\displaystyle M\times N.}
5341:{\displaystyle M\times N,}
3813:{\displaystyle (N,i^{*}g)}
3758:is a Riemannian metric on
3514:
1926:
1489:matrix-valued function on
253:
29:
20756:Geometrization conjecture
20743:
20717:
20671:
20640:
20536:
20461:over commutative algebras
20418:
20377:
20310:
20207:
20103:
20050:
20041:
19877:
19800:
19739:
19659:
19361:10.1007/978-3-319-26654-1
19247:10.1007/978-0-8176-4583-0
19088:10.1007/978-3-540-74311-8
19054:Michor & Mumford 2005
16208:, there is some isometry
14053:identically. Fix a curve
13925:is geodesically complete.
13838:{\displaystyle (M,d_{g})}
11975:Given a fixed connection
10408:{\displaystyle (M,d_{g})}
10318:{\displaystyle (M,d_{g})}
10275:{\displaystyle (M,d_{g})}
10123:{\displaystyle (M,d_{g})}
10050:{\displaystyle (M,d_{g})}
10003:{\displaystyle (M,d_{g})}
9806:{\displaystyle (M,d_{g})}
8753:{\displaystyle (M,d_{g})}
8688:{\displaystyle (M,d_{g})}
8649:{\displaystyle (M,d_{g})}
8582:{\displaystyle (M,d_{g})}
8535:{\displaystyle (M,d_{g})}
7720:. On the other hand, the
7678:Whitney embedding theorem
7549:is defined and smooth on
6787:be a smooth manifold and
6548:be Riemannian metrics on
5932:be the representation of
5853:be the representation of
5250:. The Riemannian metrics
5243:{\displaystyle M\times N}
4814:with its standard metric.
4228:{\displaystyle di_{p}(v)}
4002:to vectors tangent along
2606:
2542:has an open neighborhood
2505:{\displaystyle f:M\to N,}
1482:{\displaystyle n\times n}
20177:Riemann curvature tensor
19343:Petersen, Peter (2016).
18232:
16921:induces the topology on
16872:strong Riemannian metric
16643:topological vector space
16552:This section includes a
16482:complex hyperbolic space
16439:Riemann curvature tensor
16369:into the Lie algebra of
15380:Riemann curvature tensor
15336:complex projective space
15300:{\displaystyle \lambda }
14714:Riemann curvature tensor
14663:Riemann curvature tensor
14657:Riemann curvature tensor
14046:{\displaystyle D_{t}V=0}
13484:The Riemannian manifold
11440:{\displaystyle X:\to TM}
10958:covariant derivative of
9114:{\displaystyle \lambda }
8762:axioms of a metric space
7668:is a Riemannian metric.
5984:. The representation of
5048:{\displaystyle i:N\to M}
3910:{\displaystyle i:N\to M}
3670:{\displaystyle i:N\to M}
3455:The Riemannian manifold
2233:{\displaystyle f:M\to N}
2096:to the cotangent bundle
2073:from the tangent bundle
1895:There are situations in
346:Einstein field equations
248:sub-Riemannian manifolds
30:Not to be confused with
19553:Advances in Mathematics
19240:Birkhäuser Boston, Inc.
18707:Cheeger & Ebin 2008
18695:Cheeger & Ebin 2008
18191:Sub-Riemannian manifold
17126:{\displaystyle T_{x}H.}
17073:{\displaystyle x\in H,}
16861:{\displaystyle T_{x}M.}
16835:is an inner product on
16581:more precise citations.
15706:has constant curvature
15032:{\displaystyle \nabla }
14684:{\displaystyle \nabla }
14286:{\displaystyle \gamma }
14246:{\displaystyle \gamma }
14009:{\displaystyle \gamma }
13988:{\displaystyle \gamma }
13728:{\displaystyle v=(1,1)}
13690:{\displaystyle p=(1,1)}
13451:{\displaystyle v=(1,1)}
13413:{\displaystyle p=(1,1)}
13231:{\displaystyle \gamma }
12922:containing 0 such that
12794:{\displaystyle \gamma }
12713:{\displaystyle \gamma }
12657:{\displaystyle \gamma }
12642:be a smooth curve. The
12571:{\displaystyle \nabla }
12139:{\displaystyle \gamma }
11988:{\displaystyle \nabla }
11853:{\displaystyle \gamma }
11804:{\displaystyle \gamma }
11605:{\displaystyle \gamma }
11585:{\displaystyle \gamma }
11392:{\displaystyle \gamma }
11177:{\displaystyle \nabla }
11033:{\displaystyle \nabla }
9537:{\displaystyle \gamma }
9517:{\displaystyle \gamma }
8845:which every curve from
8783:{\displaystyle p\neq q}
8139:of an admissible curve
1509:. The requirement that
214:. Applications include
190:Kähler–Einstein metrics
20766:Uniformization theorem
20699:Nash embedding theorem
20632:Riemannian volume form
20591:Levi-Civita connection
19969:Manifold with boundary
19684:Differential structure
19599:L.A. Sidorov (2001) ,
19403:AMS Chelsea Publishing
19134:AMS Chelsea Publishing
18952:
18159:
18132:
18105:
18058:
18026:
17999:
17967:
17940:
17917:
17890:
17870:
17838:
17811:
17756:Metric space structure
17743:
17618:
17552:
17496:
17455:
17434:
17402:
17374:
17347:
17327:
17292:
17260:
17225:
17187:
17127:
17094:
17074:
17041:
16985:
16965:
16945:
16944:{\displaystyle T_{x}M}
16915:
16888:
16862:
16829:
16762:
16761:{\displaystyle x\in M}
16736:
16687:
16671:weak Riemannian metric
16635:
16486:Grassmannian manifolds
16470:real projective spaces
16235:with compact subgroup
16178:A Riemannian manifold
16124:adjoint representation
16060:
15697:
15498:This implies that the
15489:
15328:
15301:
15281:
15236:
15208:
15168:
15145:
15062:Ricci curvature tensor
15053:
15033:
15009:Ricci curvature tensor
14999:
14963:
14928:
14807:
14705:
14685:
14632:
14574:
14499:
14453:
14360:
14307:
14287:
14267:
14247:
14227:
14226:{\displaystyle T_{q}M}
14197:
14161:
14126:
14091:
14047:
14010:
13989:
13969:
13945:
13919:
13897:
13870:
13839:
13796:
13751:
13729:
13691:
13653:
13624:
13568:
13539:
13498:
13481:
13474:
13452:
13414:
13376:
13300:
13270:
13232:
13212:
13165:
13125:
13086:
13046:
13011:
12991:
12951:
12916:
12896:
12864:
12828:
12827:{\displaystyle p\in M}
12795:
12775:
12755:
12714:
12694:
12658:
12636:
12592:
12572:
12535:
12508:
12441:
12363:
12343:
12312:
12235:
12140:
12118:
12053:
12009:
11989:
11965:
11943:
11914:
11854:
11834:
11805:
11785:
11762:
11720:
11652:
11606:
11586:
11566:
11533:
11495:
11441:
11393:
11371:
11319:
11302:Levi-Civita connection
11287:
11252:
11178:
11155:
11034:
11015:Levi-Civita connection
11009:Levi-Civita connection
11003:Levi-Civita connection
10992:
10972:
10950:
10916:
10843:
10702:
10656:
10608:
10585:
10502:
10478:
10409:
10366:
10319:
10276:
10230:
10124:
10071:
10051:
10004:
9961:
9929:
9882:
9862:
9827:
9807:
9762:
9713:
9663:
9573:
9538:
9518:
9496:
9455:
9417:
9383:
9339:
9263:
9237:
9198:
9197:{\displaystyle r\in V}
9172:
9115:
9095:
9065:
9045:
9009:
8989:
8988:{\displaystyle V\ni x}
8963:
8934:
8933:{\displaystyle x(p)=0}
8899:
8832:
8784:
8754:
8709:
8689:
8650:
8603:
8583:
8536:
8492:
8370:
8312:
8276:
8177:
8133:
8104:
8072:
8010:
7951:
7905:Metric space structure
7890:
7867:
7838:
7814:
7794:
7759:there is an embedding
7753:
7752:{\displaystyle (M,g),}
7722:Nash embedding theorem
7714:
7694:
7662:
7642:
7615:
7563:
7543:
7494:
7467:
7431:
7400:
7350:
7321:
7291:
7218:
7144:
7124:
7101:
7075:
7019:
6970:
6886:
6846:
6781:
6720:
6697:
6631:
6611:
6565:
6542:
6488:
6459:
6437:
6395:
6368:
6341:
6281:
6255:
6154:
6057:
6007:
5978:
5946:
5926:
5899:
5867:
5847:
5820:
5791:
5741:
5721:
5689:
5669:
5633:
5435:
5342:
5313:
5284:
5264:
5244:
5215:
5183:
5106:
5049:
5017:
4988:
4974:On the other hand, if
4960:
4931:
4907:
4878:
4843:
4808:
4773:
4721:
4686:
4549:
4504:
4477:
4436:
4303:
4276:
4253:
4229:
4187:
4046:
4045:{\displaystyle i^{*}g}
4016:
3996:
3976:
3975:{\displaystyle i^{*}g}
3946:
3945:{\displaystyle i(x)=x}
3911:
3879:
3850:
3822:Riemannian submanifold
3814:
3772:
3752:
3732:
3731:{\displaystyle i^{*}g}
3699:
3671:
3639:
3610:
3603:
3568:
3540:
3517:Riemannian submanifold
3497:
3446:
3289:
3259:
3202:
3105:
2945:
2915:
2883:
2824:
2783:
2757:
2737:
2717:Riemannian volume form
2709:
2708:{\displaystyle dV_{g}}
2678:
2657:
2625:
2597:
2556:
2536:
2535:{\displaystyle p\in M}
2506:
2468:
2423:
2422:{\displaystyle p\in M}
2394:
2274:
2234:
2199:
2167:
2120:
2119:{\displaystyle T^{*}M}
2090:
2060:
1997:
1969:
1913:
1888:Riemannian metrics or
1878:
1877:{\displaystyle g_{ij}}
1844:
1830:The Riemannian metric
1824:
1823:{\displaystyle (U,x).}
1789:
1741:
1727:The Riemannian metric
1713:
1625:
1557:
1530:
1503:
1483:
1457:
1413:
1382:
1239:
1219:
1218:{\displaystyle p\in U}
1193:
1192:{\displaystyle T_{p}M}
1160:
1049:
1025:
933:
897:
877:
857:
792:
735:
664:
644:
624:
594:
593:{\displaystyle T_{p}M}
561:
541:
521:
520:{\displaystyle T_{p}M}
491:
471:
447:
446:{\displaystyle T_{p}M}
417:
416:{\displaystyle p\in M}
387:
370:
328:Levi-Civita connection
280:first fundamental form
267:
92:
58:
20797:Differential geometry
19467:Proc. Amer. Math. Soc
18953:
18160:
18133:
18131:{\displaystyle d_{g}}
18106:
18104:{\displaystyle (M,g)}
18059:
18027:
18025:{\displaystyle L^{2}}
18000:
17998:{\displaystyle (M,g)}
17968:
17966:{\displaystyle d_{g}}
17941:
17918:
17916:{\displaystyle d_{g}}
17891:
17871:
17839:
17837:{\displaystyle d_{g}}
17812:
17744:
17619:
17553:
17497:
17456:
17435:
17403:
17401:{\displaystyle L^{2}}
17375:
17348:
17328:
17293:
17261:
17259:{\displaystyle (M,g)}
17226:
17188:
17128:
17095:
17075:
17042:
16986:
16966:
16946:
16916:
16914:{\displaystyle g_{x}}
16889:
16863:
16830:
16763:
16737:
16693:is a smooth function
16688:
16636:
16361:, the Lie algebra of
16249:complemented subspace
16151:special unitary group
16061:
15848:real projective space
15747:hyperbolic space form
15733:geodesically complete
15724:Riemannian space form
15698:
15490:
15329:
15302:
15282:
15237:
15209:
15169:
15146:
15054:
15034:
15000:
14998:{\displaystyle (1,3)}
14964:
14929:
14808:
14706:
14686:
14633:
14575:
14500:
14454:
14361:
14308:
14288:
14268:
14248:
14228:
14198:
14162:
14127:
14092:
14048:
14011:
13990:
13975:along a smooth curve
13970:
13943:
13920:
13898:
13871:
13869:{\displaystyle d_{g}}
13840:
13797:
13795:{\displaystyle (M,g)}
13752:
13735:does not have domain
13730:
13692:
13654:
13625:
13569:
13540:
13507:geodesically complete
13499:
13475:
13458:does not have domain
13453:
13415:
13377:
13328:
13301:
13299:{\displaystyle S^{2}}
13271:
13233:
13213:
13166:
13126:
13087:
13047:
13012:
12992:
12952:
12917:
12897:
12865:
12829:
12796:
12776:
12756:
12715:
12695:
12659:
12637:
12593:
12573:
12536:
12534:{\displaystyle S^{n}}
12509:
12442:
12364:
12344:
12313:
12236:
12141:
12119:
12054:
12010:
11990:
11966:
11944:
11915:
11855:
11835:
11806:
11786:
11763:
11721:
11653:
11612:by a smooth function
11607:
11587:
11567:
11534:
11532:{\displaystyle t\in }
11496:
11442:
11394:
11377:is a smooth curve, a
11372:
11320:
11288:
11253:
11179:
11156:
11035:
10993:
10973:
10951:
10917:
10844:
10703:
10657:
10609:
10586:
10503:
10479:
10410:
10367:
10320:
10277:
10231:
10125:
10072:
10052:
10005:
9962:
9960:{\displaystyle (M,g)}
9930:
9883:
9863:
9861:{\displaystyle d_{g}}
9828:
9808:
9763:
9714:
9664:
9574:
9539:
9519:
9497:
9456:
9418:
9384:
9340:
9264:
9238:
9199:
9173:
9116:
9096:
9066:
9046:
9010:
8995:be an open subset of
8990:
8964:
8935:
8900:
8898:{\displaystyle (U,x)}
8833:
8785:
8760:satisfies all of the
8755:
8710:
8690:
8651:
8604:
8584:
8537:
8493:
8371:
8313:
8311:{\displaystyle (M,g)}
8277:
8178:
8134:
8105:
8073:
8011:
7952:
7891:
7868:
7839:
7815:
7795:
7754:
7715:
7695:
7663:
7643:
7616:
7564:
7544:
7495:
7468:
7432:
7401:
7351:
7322:
7292:
7219:
7145:
7125:
7102:
7076:
7020:
6971:
6892:are open subsets and
6887:
6847:
6782:
6721:
6698:
6632:
6612:
6566:
6543:
6489:
6460:
6443:, where each copy of
6438:
6396:
6394:{\displaystyle T^{n}}
6369:
6367:{\displaystyle S^{1}}
6342:
6282:
6256:
6155:
6058:
6008:
5979:
5977:{\displaystyle (V,y)}
5947:
5927:
5925:{\displaystyle h_{V}}
5900:
5898:{\displaystyle (U,x)}
5868:
5848:
5846:{\displaystyle g_{U}}
5821:
5792:
5742:
5722:
5720:{\displaystyle (V,y)}
5690:
5670:
5668:{\displaystyle (U,x)}
5634:
5436:
5343:
5314:
5285:
5265:
5245:
5216:
5214:{\displaystyle (N,h)}
5184:
5182:{\displaystyle (M,g)}
5107:
5050:
5018:
4989:
4968:smooth covering space
4961:
4932:
4908:
4879:
4844:
4842:{\displaystyle (M,g)}
4809:
4774:
4736:of a smooth function
4722:
4687:
4550:
4548:{\displaystyle a,b,c}
4505:
4503:{\displaystyle S^{n}}
4478:
4437:
4304:
4277:
4254:
4230:
4188:
4047:
4017:
3997:
3977:
3947:
3912:
3880:
3851:
3849:{\displaystyle (M,g)}
3815:
3773:
3753:
3733:
3700:
3672:
3640:
3638:{\displaystyle (M,g)}
3604:
3569:
3567:{\displaystyle S^{n}}
3541:
3524:
3498:
3447:
3290:
3260:
3203:
3106:
2946:
2916:
2884:
2825:
2784:
2758:
2738:
2710:
2679:
2658:
2656:{\displaystyle (M,g)}
2626:
2598:
2557:
2537:
2507:
2469:
2424:
2395:
2275:
2235:
2200:
2198:{\displaystyle (N,h)}
2168:
2166:{\displaystyle (M,g)}
2121:
2091:
2071:smooth vector bundles
2061:
1998:
1970:
1914:
1879:
1845:
1825:
1790:
1742:
1714:
1626:
1558:
1531:
1529:{\displaystyle g_{p}}
1504:
1484:
1458:
1414:
1412:{\displaystyle n^{2}}
1383:
1240:
1220:
1194:
1161:
1050:
1026:
934:
932:{\displaystyle (M,g)}
898:
878:
858:
793:
736:
665:
645:
625:
595:
562:
542:
522:
492:
472:
448:
418:
388:
368:
261:
93:
63:differential geometry
40:
20792:Riemannian manifolds
20689:Gauss–Bonnet theorem
20596:Covariant derivative
20116:Covariant derivative
19667:Topological manifold
19281:Kobayashi, Shoshichi
18750:
18149:
18115:
18083:
18036:
18009:
17977:
17950:
17930:
17900:
17880:
17860:
17821:
17764:
17630:
17562:
17505:
17465:
17445:
17412:
17385:
17361:
17346:{\displaystyle \mu }
17337:
17314:
17270:
17238:
17197:
17137:
17104:
17084:
17055:
17005:
16975:
16955:
16925:
16898:
16878:
16839:
16772:
16746:
16697:
16677:
16613:
16069:where for arbitrary
15939:
15769:Killing–Hopf theorem
15743:Euclidean space form
15738:spherical space form
15563:
15393:
15318:
15291:
15256:
15226:
15192:
15158:
15072:
15043:
15023:
14977:
14941:
14820:
14721:
14695:
14675:
14596:
14528:
14463:
14374:
14323:
14297:
14277:
14257:
14237:
14207:
14171:
14136:
14101:
14057:
14021:
14000:
13979:
13959:
13909:
13887:
13853:
13810:
13774:
13739:
13701:
13663:
13634:
13581:
13549:
13514:
13488:
13462:
13424:
13386:
13333:
13329:The punctured plane
13283:
13251:
13222:
13178:
13135:
13100:
13056:
13021:
13001:
12961:
12926:
12906:
12874:
12838:
12812:
12785:
12765:
12724:
12704:
12669:
12665:is the vector field
12648:
12602:
12582:
12562:
12518:
12489:
12373:
12353:
12324:
12246:
12154:
12130:
12063:
12019:
11999:
11979:
11955:
11924:
11864:
11844:
11815:
11795:
11775:
11730:
11665:
11616:
11596:
11576:
11543:
11505:
11451:
11404:
11383:
11337:
11309:
11265:
11195:
11168:
11051:
11024:
10982:
10962:
10930:
10855:
10712:
10670:
10618:
10598:
10518:
10492:
10484:denote the space of
10455:
10380:
10329:
10290:
10247:
10137:
10095:
10091:of the metric space
10061:
10022:
9975:
9939:
9892:
9872:
9845:
9817:
9778:
9727:
9676:
9586:
9548:
9528:
9508:
9465:
9427:
9401:
9349:
9277:
9247:
9208:
9182:
9125:
9105:
9075:
9055:
9019:
8999:
8973:
8944:
8909:
8877:
8794:
8768:
8725:
8699:
8660:
8621:
8593:
8554:
8507:
8383:
8323:
8290:
8190:
8143:
8114:
8082:
8020:
7961:
7917:
7877:
7848:
7828:
7804:
7763:
7728:
7704:
7684:
7652:
7625:
7573:
7553:
7504:
7477:
7441:
7414:
7360:
7331:
7304:
7234:
7157:
7134:
7114:
7085:
7033:
7025:be a differentiable
6983:
6896:
6863:
6791:
6771:
6707:
6641:
6621:
6575:
6552:
6506:
6469:
6447:
6411:
6378:
6351:
6292:
6271:
6164:
6069:
6017:
5988:
5956:
5936:
5909:
5877:
5857:
5830:
5801:
5751:
5731:
5699:
5679:
5647:
5447:
5356:
5323:
5294:
5274:
5254:
5228:
5193:
5161:
5071:
5027:
4998:
4978:
4941:
4921:
4888:
4853:
4821:
4783:
4740:
4702:
4564:
4527:
4487:
4452:
4316:
4293:
4263:
4243:
4200:
4059:
4026:
4006:
3986:
3956:
3921:
3889:
3863:
3828:
3782:
3762:
3742:
3712:
3689:
3683:embedded submanifold
3679:immersed submanifold
3649:
3617:
3578:
3551:
3530:
3459:
3309:
3269:
3218:
3121:
2958:
2928:
2893:
2847:
2794:
2773:
2747:
2727:
2689:
2668:
2635:
2615:
2566:
2546:
2520:
2481:
2433:
2407:
2287:
2248:
2212:
2177:
2145:
2100:
2077:
2069:is a isomorphism of
2010:
1987:
1941:
1903:
1892:Riemannian metrics.
1858:
1834:
1799:
1755:
1731:
1638:
1577:
1547:
1513:
1493:
1467:
1423:
1396:
1252:
1229:
1203:
1173:
1062:
1039:
962:
911:
887:
867:
863:. A smooth manifold
802:
745:
678:
654:
634:
614:
574:
551:
531:
501:
481:
461:
427:
401:
377:
344:. Specifically, the
274:discovered that the
272:Carl Friedrich Gauss
82:
20787:Riemannian geometry
20761:Poincaré conjecture
20622:Riemannian manifold
20610:Musical isomorphism
20525:Riemannian geometry
20150:Exterior derivative
19752:Atiyah–Singer index
19701:Riemannian manifold
19601:"Riemannian metric"
19345:Riemannian geometry
19293:. New York–London:
19180:Riemannian geometry
18697:, Proposition 3.18.
18685:, Chapters 2 and 3.
18673:, Chapters 2 and 7.
18622:, pp. 196–197.
18586:, pp. 105–110.
18574:, pp. 146–147.
18509:, pp. 103–104.
18485:, pp. 101–102.
18461:, pp. 122–123.
18211:Symplectic manifold
18181:Riemannian geometry
16505:Riemannian holonomy
16409:if for every point
15890:hyperbolic geometry
15798:. Given any finite
15752:hyperbolic manifold
15676:
15652:
15613:
15586:
15372:sectional curvature
15340:Fubini-Study metric
15248:Einstein's equation
15207:{\displaystyle Ric}
12485:In Euclidean space
12349:is an extension of
12015:and a smooth curve
11042:preserve the metric
10666:For every function
10510:(affine) connection
10445:(affine) connection
9625:
9071:and compactness of
8873:To be precise, let
8222:
7473:is only defined on
7377:
7273:
6013:in the coordinates
5064:isometric embedding
5058:isometric immersion
3332:
3238:
1929:Musical isomorphism
1923:Musical isomorphism
905:Riemannian manifold
336:used the theory of
200:Riemannian geometry
67:Riemannian manifold
20751:General relativity
20694:Hopf–Rinow theorem
20641:Types of manifolds
20617:Parallel transport
20456:Secondary calculus
20410:Singularity theory
20365:Parallel transport
20133:De Rham cohomology
19772:Generalized Stokes
19489:10.1090/proc/14986
19076:Einstein manifolds
18948:
18709:, Corollary 3.19;
18206:Hermitian manifold
18155:
18128:
18101:
18069:Hopf–Rinow theorem
18054:
18022:
17995:
17963:
17936:
17913:
17886:
17866:
17834:
17807:
17739:
17614:
17548:
17492:
17451:
17430:
17398:
17373:{\displaystyle M.}
17370:
17343:
17326:{\displaystyle M.}
17323:
17288:
17256:
17221:
17183:
17123:
17090:
17070:
17037:
16981:
16961:
16941:
16911:
16884:
16858:
16825:
16758:
16742:such that for any
16732:
16683:
16631:
16554:list of references
16453:), are said to be
16451:constant curvature
16337:for every element
16174:Homogeneous spaces
16056:
15693:
15662:
15638:
15599:
15572:
15485:
15374:equals the number
15364:constant curvature
15324:
15297:
15287:for some constant
15277:
15232:
15216:Einstein manifolds
15204:
15178:Einstein manifolds
15164:
15141:
15049:
15029:
14995:
14959:
14924:
14803:
14701:
14681:
14628:
14570:
14495:
14449:
14356:
14303:
14283:
14263:
14243:
14223:
14193:
14157:
14122:
14087:
14043:
14006:
13985:
13965:
13950:Parallel transport
13946:
13936:Parallel transport
13930:Parallel transport
13915:
13893:
13866:
13835:
13792:
13762:Hopf–Rinow theorem
13747:
13725:
13687:
13649:
13620:
13564:
13535:
13494:
13482:
13470:
13448:
13410:
13372:
13321:Hopf–Rinow theorem
13315:Hopf–Rinow theorem
13296:
13266:
13228:
13208:
13171:is a restriction.
13161:
13121:
13082:
13042:
13007:
12987:
12947:
12912:
12892:
12860:
12824:
12791:
12771:
12751:
12710:
12690:
12654:
12632:
12588:
12568:
12531:
12504:
12437:
12359:
12339:
12308:
12231:
12136:
12114:
12049:
12005:
11985:
11961:
11939:
11910:
11850:
11830:
11801:
11781:
11758:
11716:
11648:
11602:
11582:
11562:
11529:
11491:
11437:
11389:
11367:
11315:
11283:
11248:
11174:
11151:
11030:
10988:
10968:
10946:
10912:
10839:
10698:
10652:
10614:is a bilinear map
10604:
10581:
10498:
10474:
10405:
10362:
10315:
10272:
10241:Hopf–Rinow theorem
10226:
10120:
10067:
10047:
10000:
9957:
9925:
9878:
9858:
9823:
9803:
9758:
9709:
9659:
9611:
9569:
9534:
9514:
9492:
9451:
9413:
9379:
9335:
9259:
9233:
9194:
9168:
9111:
9091:
9061:
9041:
9005:
8985:
8959:
8930:
8895:
8828:
8780:
8750:
8721:In verifying that
8705:
8685:
8646:
8617:Proof sketch that
8599:
8579:
8532:
8488:
8366:
8308:
8272:
8208:
8173:
8129:
8100:
8068:
8006:
7947:
7889:{\displaystyle g.}
7886:
7863:
7834:
7810:
7790:
7749:
7710:
7690:
7658:
7638:
7611:
7559:
7539:
7490:
7463:
7427:
7396:
7363:
7346:
7317:
7287:
7259:
7214:
7181:
7140:
7120:
7097:
7071:
7027:partition of unity
7015:
6966:
6882:
6842:
6777:
6750:partition of unity
6719:{\displaystyle M.}
6716:
6693:
6627:
6607:
6564:{\displaystyle M.}
6561:
6538:
6484:
6455:
6433:
6391:
6364:
6347:. If each copy of
6337:
6277:
6251:
6242:
6150:
6099:
6053:
6003:
5974:
5942:
5922:
5895:
5863:
5843:
5816:
5787:
5737:
5717:
5685:
5665:
5629:
5431:
5338:
5309:
5280:
5260:
5240:
5211:
5179:
5102:
5045:
5013:
4984:
4956:
4927:
4903:
4874:
4839:
4804:
4769:
4717:
4682:
4545:
4500:
4473:
4432:
4299:
4275:{\displaystyle i.}
4272:
4249:
4225:
4183:
4042:
4012:
3992:
3972:
3942:
3907:
3875:
3859:In the case where
3846:
3810:
3768:
3748:
3728:
3695:
3667:
3635:
3611:
3599:
3564:
3536:
3493:
3442:
3433:
3315:
3285:
3255:
3221:
3198:
3101:
3080:
3030:
2985:
2941:
2911:
2879:
2820:
2779:
2753:
2733:
2705:
2674:
2653:
2621:
2593:
2552:
2532:
2502:
2464:
2419:
2390:
2270:
2230:
2195:
2163:
2116:
2089:{\displaystyle TM}
2086:
2056:
1993:
1965:
1909:
1897:geometric analysis
1874:
1854:if its components
1840:
1820:
1785:
1751:if its components
1737:
1709:
1662:
1621:
1553:
1526:
1499:
1479:
1453:
1409:
1378:
1235:
1215:
1189:
1156:
1045:
1021:
929:
893:
873:
853:
788:
731:
660:
640:
620:
590:
557:
537:
517:
487:
467:
443:
413:
383:
371:
342:general relativity
276:Gaussian curvature
268:
220:general relativity
212:algebraic geometry
204:geometric topology
178:homogeneous spaces
88:
59:
20774:
20773:
20491:
20490:
20373:
20372:
20138:Differential form
19792:Whitney embedding
19726:Differential form
19420:978-0-8218-5282-8
19370:978-3-319-26652-7
19335:978-3-319-91754-2
19193:978-0-8176-3490-2
19184:Birkhäuser Boston
19151:978-0-8218-4417-5
18997:, Theorem IV.4.5.
18449:, pp. 89–91.
18359:, pp. 12–13.
18248:, pp. 35–36.
18221:Einstein manifold
18158:{\displaystyle g}
17939:{\displaystyle g}
17889:{\displaystyle M}
17869:{\displaystyle g}
17846:geodesic distance
17454:{\displaystyle G}
17302:) and in fact, a
17093:{\displaystyle H}
17080:one can identify
16984:{\displaystyle M}
16964:{\displaystyle g}
16887:{\displaystyle M}
16686:{\displaystyle M}
16655:Hilbert manifolds
16607:
16606:
16599:
16501:Einstein manifold
16455:locally symmetric
16168:abelian Lie group
16164:compact Lie group
15886:Teichmüller space
15827:quotient manifold
15781:quotient manifold
15777:covering manifold
15712:locally isometric
15691:
15633:
15550:Einstein manifold
15327:{\displaystyle n}
15312:Einstein manifold
15235:{\displaystyle g}
15184:Einstein manifold
15052:{\displaystyle M}
15019:Fix a connection
14704:{\displaystyle M}
14671:Fix a connection
14368:polar coordinates
14306:{\displaystyle q}
14266:{\displaystyle v}
13968:{\displaystyle V}
13918:{\displaystyle M}
13896:{\displaystyle M}
13806:The metric space
13497:{\displaystyle M}
13010:{\displaystyle I}
12915:{\displaystyle I}
12774:{\displaystyle t}
12591:{\displaystyle M}
12558:Fix a connection
12434:
12362:{\displaystyle X}
12336:
12008:{\displaystyle M}
11964:{\displaystyle X}
11936:
11892:
11827:
11784:{\displaystyle X}
11318:{\displaystyle g}
10991:{\displaystyle X}
10971:{\displaystyle Y}
10851:The product rule
10607:{\displaystyle M}
10501:{\displaystyle M}
10439:Affine connection
10070:{\displaystyle M}
9967:is an ellipsoid.
9881:{\displaystyle M}
9838:
9837:
9826:{\displaystyle M}
9750:
9723:. So we conclude
9609:
9572:{\displaystyle .}
9086:
9064:{\displaystyle g}
9051:By continuity of
9030:
9008:{\displaystyle U}
8858:, any curve from
8708:{\displaystyle M}
8602:{\displaystyle M}
8441:
7837:{\displaystyle F}
7813:{\displaystyle N}
7713:{\displaystyle M}
7693:{\displaystyle M}
7673:
7672:
7661:{\displaystyle g}
7562:{\displaystyle M}
7530:
7454:
7314:
7281:
7247:
7205:
7166:
7143:{\displaystyle M}
7123:{\displaystyle g}
6780:{\displaystyle M}
6630:{\displaystyle M}
6280:{\displaystyle n}
6180:
6110:
6087:
6081:
6000:
5945:{\displaystyle h}
5866:{\displaystyle g}
5740:{\displaystyle N}
5688:{\displaystyle M}
5460:
5306:
5283:{\displaystyle h}
5263:{\displaystyle g}
5083:
5010:
4987:{\displaystyle N}
4953:
4930:{\displaystyle M}
4900:
4865:
4669:
4642:
4615:
4523:Fix real numbers
4302:{\displaystyle n}
4252:{\displaystyle v}
4015:{\displaystyle N}
3995:{\displaystyle g}
3771:{\displaystyle N}
3751:{\displaystyle g}
3698:{\displaystyle M}
3539:{\displaystyle n}
3487:
3330:
3236:
3131:
3071:
3061:
3021:
3016:
2976:
2968:
2938:
2782:{\displaystyle M}
2756:{\displaystyle M}
2736:{\displaystyle M}
2677:{\displaystyle n}
2624:{\displaystyle n}
2555:{\displaystyle U}
2141:Specifically, if
1996:{\displaystyle g}
1912:{\displaystyle g}
1843:{\displaystyle g}
1740:{\displaystyle g}
1647:
1556:{\displaystyle p}
1541:positive-definite
1502:{\displaystyle U}
1361:
1322:
1238:{\displaystyle p}
1135:
1090:
1048:{\displaystyle M}
1033:local coordinates
896:{\displaystyle g}
876:{\displaystyle M}
851:
672:positive-definite
663:{\displaystyle p}
643:{\displaystyle M}
623:{\displaystyle g}
609:Riemannian metric
560:{\displaystyle p}
540:{\displaystyle M}
490:{\displaystyle p}
470:{\displaystyle M}
397:. For each point
386:{\displaystyle M}
316:Cartan connection
284:Theorema Egregium
244:Finsler manifolds
228:computer graphics
129:Riemannian metric
91:{\displaystyle n}
18:Riemannian metric
16:(Redirected from
20804:
20518:
20511:
20504:
20495:
20494:
20483:Stratified space
20441:Fréchet manifold
20155:Interior product
20048:
20047:
19745:
19641:
19634:
19627:
19618:
19617:
19613:
19587:
19569:
19539:
19521:
19500:
19482:
19473:(1): 3653–3656.
19461:
19440:
19411:10.1090/chel/372
19390:
19339:
19313:
19276:
19213:
19171:
19142:10.1090/chel/365
19117:
19072:Besse, Arthur L.
19057:
19051:
19045:
19039:
19030:
19024:
19013:
19007:
18998:
18992:
18986:
18976:
18970:
18969:, Example 3.16f.
18964:
18958:
18957:
18955:
18954:
18949:
18941:
18940:
18931:
18930:
18918:
18917:
18902:
18901:
18889:
18888:
18864:
18863:
18854:
18853:
18841:
18840:
18822:
18821:
18803:
18802:
18784:
18783:
18765:
18764:
18744:
18738:
18732:
18726:
18720:
18714:
18704:
18698:
18692:
18686:
18680:
18674:
18668:
18662:
18656:
18647:
18641:
18635:
18629:
18623:
18617:
18611:
18605:
18599:
18593:
18587:
18581:
18575:
18569:
18563:
18557:
18551:
18545:
18539:
18533:
18527:
18521:
18510:
18504:
18498:
18492:
18486:
18480:
18474:
18468:
18462:
18456:
18450:
18444:
18438:
18432:
18426:
18420:
18414:
18408:
18402:
18396:
18387:
18381:
18375:
18369:
18360:
18354:
18348:
18342:
18336:
18330:
18321:
18315:
18306:
18300:
18294:
18288:
18279:
18273:
18264:
18258:
18249:
18243:
18186:Finsler manifold
18164:
18162:
18161:
18156:
18137:
18135:
18134:
18129:
18127:
18126:
18110:
18108:
18107:
18102:
18063:
18061:
18060:
18055:
18031:
18029:
18028:
18023:
18021:
18020:
18004:
18002:
18001:
17996:
17972:
17970:
17969:
17964:
17962:
17961:
17945:
17943:
17942:
17937:
17922:
17920:
17919:
17914:
17912:
17911:
17895:
17893:
17892:
17887:
17875:
17873:
17872:
17867:
17843:
17841:
17840:
17835:
17833:
17832:
17816:
17814:
17813:
17808:
17776:
17775:
17748:
17746:
17745:
17740:
17689:
17688:
17670:
17669:
17642:
17641:
17623:
17621:
17620:
17615:
17610:
17609:
17557:
17555:
17554:
17549:
17529:
17528:
17501:
17499:
17498:
17493:
17460:
17458:
17457:
17452:
17439:
17437:
17436:
17431:
17407:
17405:
17404:
17399:
17397:
17396:
17379:
17377:
17376:
17371:
17352:
17350:
17349:
17344:
17332:
17330:
17329:
17324:
17297:
17295:
17294:
17289:
17265:
17263:
17262:
17257:
17230:
17228:
17227:
17222:
17192:
17190:
17189:
17184:
17149:
17148:
17132:
17130:
17129:
17124:
17116:
17115:
17099:
17097:
17096:
17091:
17079:
17077:
17076:
17071:
17046:
17044:
17043:
17038:
16990:
16988:
16987:
16982:
16970:
16968:
16967:
16962:
16950:
16948:
16947:
16942:
16937:
16936:
16920:
16918:
16917:
16912:
16910:
16909:
16893:
16891:
16890:
16885:
16867:
16865:
16864:
16859:
16851:
16850:
16834:
16832:
16831:
16826:
16824:
16813:
16812:
16797:
16796:
16784:
16783:
16768:the restriction
16767:
16765:
16764:
16759:
16741:
16739:
16738:
16733:
16728:
16692:
16690:
16689:
16684:
16640:
16638:
16637:
16632:
16627:
16626:
16621:
16602:
16595:
16591:
16588:
16582:
16577:this section by
16568:inline citations
16547:
16546:
16539:
16529:Spin(7) geometry
16463:simply-connected
16432:
16420:
16416:
16412:
16402:
16381:Symmetric spaces
16376:
16372:
16368:
16364:
16360:
16355:
16348:
16344:
16340:
16336:
16315:
16311:
16298:
16294:
16290:
16287:for any element
16286:
16265:
16261:
16253:
16246:
16238:
16234:
16219:
16215:
16211:
16207:
16203:
16199:
16189:
16137:
16129:
16121:
16103:
16097:
16094:sending a point
16093:
16083:
16072:
16065:
16063:
16062:
16057:
16040:
16039:
16038:
16037:
16005:
16004:
16003:
16002:
15979:
15978:
15951:
15950:
15931:
15927:
15908:
15883:
15841:
15837:
15824:
15820:
15812:
15804:
15797:
15789:orthogonal group
15786:
15766:
15762:
15758:
15717:
15709:
15702:
15700:
15699:
15694:
15692:
15690:
15689:
15688:
15675:
15670:
15651:
15646:
15634:
15626:
15614:
15612:
15607:
15585:
15580:
15567:
15554:Bernhard Riemann
15547:
15543:
15528:scalar curvature
15525:
15494:
15492:
15491:
15486:
15478:
15477:
15465:
15464:
15449:
15448:
15436:
15435:
15414:
15413:
15377:
15369:
15352:Scalar curvature
15346:Scalar curvature
15333:
15331:
15330:
15325:
15306:
15304:
15303:
15298:
15286:
15284:
15283:
15278:
15241:
15239:
15238:
15233:
15213:
15211:
15210:
15205:
15173:
15171:
15170:
15165:
15150:
15148:
15147:
15142:
15058:
15056:
15055:
15050:
15038:
15036:
15035:
15030:
15004:
15002:
15001:
14996:
14968:
14966:
14965:
14962:{\displaystyle }
14960:
14933:
14931:
14930:
14925:
14920:
14919:
14892:
14891:
14882:
14881:
14866:
14865:
14856:
14855:
14812:
14810:
14809:
14804:
14793:
14792:
14774:
14773:
14755:
14754:
14736:
14735:
14710:
14708:
14707:
14702:
14690:
14688:
14687:
14682:
14637:
14635:
14634:
14629:
14627:
14626:
14611:
14610:
14589:
14579:
14577:
14576:
14571:
14569:
14568:
14556:
14555:
14543:
14542:
14521:
14504:
14502:
14501:
14496:
14494:
14493:
14478:
14477:
14458:
14456:
14455:
14450:
14448:
14447:
14434:
14433:
14421:
14420:
14405:
14404:
14389:
14388:
14365:
14363:
14362:
14357:
14337:
14336:
14331:
14312:
14310:
14309:
14304:
14292:
14290:
14289:
14284:
14272:
14270:
14269:
14264:
14252:
14250:
14249:
14244:
14232:
14230:
14229:
14224:
14219:
14218:
14202:
14200:
14199:
14194:
14189:
14188:
14166:
14164:
14163:
14158:
14131:
14129:
14128:
14123:
14096:
14094:
14093:
14088:
14052:
14050:
14049:
14044:
14033:
14032:
14015:
14013:
14012:
14007:
13994:
13992:
13991:
13986:
13974:
13972:
13971:
13966:
13924:
13922:
13921:
13916:
13902:
13900:
13899:
13894:
13875:
13873:
13872:
13867:
13865:
13864:
13844:
13842:
13841:
13836:
13831:
13830:
13801:
13799:
13798:
13793:
13756:
13754:
13753:
13748:
13746:
13734:
13732:
13731:
13726:
13696:
13694:
13693:
13688:
13658:
13656:
13655:
13650:
13648:
13647:
13642:
13629:
13627:
13626:
13621:
13595:
13594:
13589:
13573:
13571:
13570:
13565:
13563:
13562:
13557:
13544:
13542:
13541:
13536:
13503:
13501:
13500:
13495:
13479:
13477:
13476:
13471:
13469:
13457:
13455:
13454:
13449:
13419:
13417:
13416:
13411:
13381:
13379:
13378:
13373:
13347:
13346:
13341:
13305:
13303:
13302:
13297:
13295:
13294:
13275:
13273:
13272:
13267:
13265:
13264:
13259:
13237:
13235:
13234:
13229:
13217:
13215:
13214:
13209:
13170:
13168:
13167:
13162:
13145:
13130:
13128:
13127:
13122:
13094:maximal geodesic
13091:
13089:
13088:
13083:
13066:
13051:
13049:
13048:
13043:
13016:
13014:
13013:
13008:
12996:
12994:
12993:
12988:
12971:
12956:
12954:
12953:
12948:
12921:
12919:
12918:
12913:
12901:
12899:
12898:
12893:
12869:
12867:
12866:
12861:
12856:
12855:
12833:
12831:
12830:
12825:
12800:
12798:
12797:
12792:
12780:
12778:
12777:
12772:
12760:
12758:
12757:
12752:
12744:
12736:
12735:
12719:
12717:
12716:
12711:
12699:
12697:
12696:
12691:
12689:
12681:
12680:
12663:
12661:
12660:
12655:
12644:acceleration of
12641:
12639:
12638:
12633:
12597:
12595:
12594:
12589:
12577:
12575:
12574:
12569:
12540:
12538:
12537:
12532:
12530:
12529:
12513:
12511:
12510:
12505:
12503:
12502:
12497:
12479:
12470:
12446:
12444:
12443:
12438:
12436:
12435:
12427:
12424:
12423:
12413:
12385:
12384:
12368:
12366:
12365:
12360:
12348:
12346:
12345:
12340:
12338:
12337:
12329:
12317:
12315:
12314:
12309:
12301:
12300:
12281:
12258:
12257:
12240:
12238:
12237:
12232:
12224:
12223:
12204:
12203:
12166:
12165:
12145:
12143:
12142:
12137:
12123:
12121:
12120:
12115:
12104:
12103:
12085:
12084:
12075:
12074:
12058:
12056:
12055:
12050:
12014:
12012:
12011:
12006:
11994:
11992:
11991:
11986:
11970:
11968:
11967:
11962:
11948:
11946:
11945:
11940:
11938:
11937:
11929:
11919:
11917:
11916:
11911:
11909:
11908:
11894:
11893:
11885:
11859:
11857:
11856:
11851:
11839:
11837:
11836:
11831:
11829:
11828:
11820:
11810:
11808:
11807:
11802:
11790:
11788:
11787:
11782:
11767:
11765:
11764:
11759:
11745:
11744:
11725:
11723:
11722:
11717:
11657:
11655:
11654:
11649:
11647:
11611:
11609:
11608:
11603:
11591:
11589:
11588:
11583:
11571:
11569:
11568:
11563:
11552:
11551:
11538:
11536:
11535:
11530:
11500:
11498:
11497:
11492:
11487:
11486:
11446:
11444:
11443:
11438:
11400:is a smooth map
11398:
11396:
11395:
11390:
11376:
11374:
11373:
11368:
11324:
11322:
11321:
11316:
11292:
11290:
11289:
11286:{\displaystyle }
11284:
11257:
11255:
11254:
11249:
11223:
11222:
11207:
11206:
11183:
11181:
11180:
11175:
11160:
11158:
11157:
11152:
11144:
11143:
11107:
11106:
11088:
11087:
11063:
11062:
11039:
11037:
11036:
11031:
10997:
10995:
10994:
10989:
10978:with respect to
10977:
10975:
10974:
10969:
10955:
10953:
10952:
10947:
10942:
10941:
10921:
10919:
10918:
10913:
10908:
10907:
10867:
10866:
10848:
10846:
10845:
10840:
10832:
10831:
10830:
10829:
10814:
10813:
10798:
10797:
10796:
10795:
10780:
10779:
10764:
10763:
10762:
10761:
10752:
10751:
10739:
10738:
10729:
10728:
10707:
10705:
10704:
10699:
10688:
10687:
10661:
10659:
10658:
10653:
10648:
10647:
10613:
10611:
10610:
10605:
10590:
10588:
10587:
10582:
10571:
10570:
10552:
10551:
10533:
10532:
10507:
10505:
10504:
10499:
10483:
10481:
10480:
10475:
10464:
10463:
10414:
10412:
10411:
10406:
10401:
10400:
10371:
10369:
10368:
10363:
10361:
10341:
10340:
10324:
10322:
10321:
10316:
10311:
10310:
10281:
10279:
10278:
10273:
10268:
10267:
10235:
10233:
10232:
10227:
10186:
10185:
10164:
10163:
10129:
10127:
10126:
10121:
10116:
10115:
10076:
10074:
10073:
10068:
10056:
10054:
10053:
10048:
10043:
10042:
10009:
10007:
10006:
10001:
9996:
9995:
9966:
9964:
9963:
9958:
9934:
9932:
9931:
9926:
9924:
9904:
9903:
9887:
9885:
9884:
9879:
9867:
9865:
9864:
9859:
9857:
9856:
9832:
9830:
9829:
9824:
9812:
9810:
9809:
9804:
9799:
9798:
9767:
9765:
9764:
9759:
9751:
9746:
9718:
9716:
9715:
9710:
9708:
9707:
9702:
9668:
9666:
9665:
9660:
9636:
9624:
9619:
9610:
9605:
9578:
9576:
9575:
9570:
9543:
9541:
9540:
9535:
9523:
9521:
9520:
9515:
9501:
9499:
9498:
9493:
9460:
9458:
9457:
9452:
9422:
9420:
9419:
9414:
9388:
9386:
9385:
9380:
9344:
9342:
9341:
9336:
9307:
9306:
9268:
9266:
9265:
9260:
9242:
9240:
9239:
9234:
9226:
9225:
9203:
9201:
9200:
9195:
9177:
9175:
9174:
9169:
9167:
9166:
9120:
9118:
9117:
9112:
9100:
9098:
9097:
9092:
9087:
9079:
9070:
9068:
9067:
9062:
9050:
9048:
9047:
9042:
9031:
9023:
9014:
9012:
9011:
9006:
8994:
8992:
8991:
8986:
8968:
8966:
8965:
8960:
8939:
8937:
8936:
8931:
8904:
8902:
8901:
8896:
8837:
8835:
8834:
8829:
8806:
8805:
8789:
8787:
8786:
8781:
8759:
8757:
8756:
8751:
8746:
8745:
8714:
8712:
8711:
8706:
8694:
8692:
8691:
8686:
8681:
8680:
8655:
8653:
8652:
8647:
8642:
8641:
8613:
8612:
8608:
8606:
8605:
8600:
8588:
8586:
8585:
8580:
8575:
8574:
8541:
8539:
8538:
8533:
8528:
8527:
8497:
8495:
8494:
8489:
8442:
8439:
8395:
8394:
8375:
8373:
8372:
8367:
8335:
8334:
8317:
8315:
8314:
8309:
8281:
8279:
8278:
8273:
8261:
8260:
8233:
8221:
8216:
8182:
8180:
8179:
8174:
8138:
8136:
8135:
8130:
8109:
8107:
8106:
8103:{\displaystyle }
8101:
8077:
8075:
8074:
8069:
8067:
8066:
8039:
8015:
8013:
8012:
8007:
8002:
8001:
7971:
7956:
7954:
7953:
7948:
7911:admissible curve
7895:
7893:
7892:
7887:
7872:
7870:
7869:
7864:
7862:
7861:
7856:
7843:
7841:
7840:
7835:
7819:
7817:
7816:
7811:
7799:
7797:
7796:
7791:
7789:
7788:
7783:
7758:
7756:
7755:
7750:
7719:
7717:
7716:
7711:
7699:
7697:
7696:
7691:
7667:
7665:
7664:
7659:
7647:
7645:
7644:
7639:
7637:
7636:
7620:
7618:
7617:
7612:
7610:
7609:
7594:
7593:
7568:
7566:
7565:
7560:
7548:
7546:
7545:
7540:
7538:
7537:
7532:
7531:
7523:
7516:
7515:
7499:
7497:
7496:
7491:
7489:
7488:
7472:
7470:
7469:
7464:
7462:
7461:
7456:
7455:
7447:
7436:
7434:
7433:
7428:
7426:
7425:
7405:
7403:
7402:
7397:
7395:
7394:
7393:
7376:
7371:
7355:
7353:
7352:
7347:
7345:
7344:
7339:
7326:
7324:
7323:
7318:
7316:
7315:
7312:
7296:
7294:
7293:
7288:
7283:
7282:
7279:
7272:
7267:
7255:
7254:
7249:
7248:
7240:
7223:
7221:
7220:
7215:
7213:
7212:
7207:
7206:
7198:
7191:
7190:
7180:
7149:
7147:
7146:
7141:
7129:
7127:
7126:
7121:
7106:
7104:
7103:
7098:
7080:
7078:
7077:
7072:
7070:
7069:
7054:
7053:
7024:
7022:
7021:
7016:
7014:
7013:
6998:
6997:
6975:
6973:
6972:
6967:
6965:
6964:
6959:
6947:
6946:
6934:
6933:
6921:
6920:
6908:
6907:
6891:
6889:
6888:
6883:
6875:
6874:
6851:
6849:
6848:
6843:
6841:
6840:
6822:
6821:
6809:
6808:
6786:
6784:
6783:
6778:
6756:
6755:
6725:
6723:
6722:
6717:
6702:
6700:
6699:
6694:
6692:
6691:
6682:
6681:
6663:
6662:
6653:
6652:
6636:
6634:
6633:
6628:
6616:
6614:
6613:
6608:
6606:
6605:
6587:
6586:
6570:
6568:
6567:
6562:
6547:
6545:
6544:
6539:
6537:
6536:
6518:
6517:
6493:
6491:
6490:
6485:
6483:
6482:
6477:
6464:
6462:
6461:
6456:
6454:
6442:
6440:
6439:
6434:
6432:
6418:
6400:
6398:
6397:
6392:
6390:
6389:
6373:
6371:
6370:
6365:
6363:
6362:
6346:
6344:
6343:
6338:
6336:
6335:
6317:
6316:
6304:
6303:
6286:
6284:
6283:
6278:
6260:
6258:
6257:
6252:
6247:
6246:
6239:
6238:
6215:
6214:
6191:
6190:
6182:
6181:
6173:
6159:
6157:
6156:
6151:
6149:
6148:
6135:
6134:
6121:
6120:
6112:
6111:
6103:
6098:
6083:
6082:
6074:
6062:
6060:
6059:
6054:
6012:
6010:
6009:
6004:
6002:
6001:
5993:
5983:
5981:
5980:
5975:
5951:
5949:
5948:
5943:
5931:
5929:
5928:
5923:
5921:
5920:
5904:
5902:
5901:
5896:
5872:
5870:
5869:
5864:
5852:
5850:
5849:
5844:
5842:
5841:
5825:
5823:
5822:
5817:
5796:
5794:
5793:
5788:
5746:
5744:
5743:
5738:
5726:
5724:
5723:
5718:
5694:
5692:
5691:
5686:
5674:
5672:
5671:
5666:
5638:
5636:
5635:
5630:
5622:
5621:
5609:
5608:
5596:
5595:
5580:
5579:
5567:
5566:
5554:
5553:
5535:
5534:
5522:
5521:
5503:
5502:
5490:
5489:
5474:
5473:
5462:
5461:
5453:
5440:
5438:
5437:
5432:
5424:
5423:
5408:
5407:
5380:
5379:
5347:
5345:
5344:
5339:
5318:
5316:
5315:
5310:
5308:
5307:
5299:
5289:
5287:
5286:
5281:
5269:
5267:
5266:
5261:
5249:
5247:
5246:
5241:
5223:product manifold
5220:
5218:
5217:
5212:
5188:
5186:
5185:
5180:
5137:
5128:
5111:
5109:
5108:
5103:
5098:
5097:
5085:
5084:
5076:
5054:
5052:
5051:
5046:
5022:
5020:
5019:
5014:
5012:
5011:
5003:
4993:
4991:
4990:
4985:
4965:
4963:
4962:
4957:
4955:
4954:
4946:
4936:
4934:
4933:
4928:
4912:
4910:
4909:
4904:
4902:
4901:
4893:
4883:
4881:
4880:
4875:
4867:
4866:
4858:
4848:
4846:
4845:
4840:
4813:
4811:
4810:
4805:
4803:
4802:
4791:
4778:
4776:
4775:
4770:
4768:
4760:
4759:
4754:
4726:
4724:
4723:
4718:
4716:
4715:
4710:
4691:
4689:
4688:
4683:
4681:
4677:
4670:
4668:
4667:
4658:
4657:
4648:
4643:
4641:
4640:
4631:
4630:
4621:
4616:
4614:
4613:
4604:
4603:
4594:
4589:
4588:
4583:
4554:
4552:
4551:
4546:
4509:
4507:
4506:
4501:
4499:
4498:
4482:
4480:
4479:
4474:
4472:
4471:
4460:
4441:
4439:
4438:
4433:
4422:
4421:
4412:
4411:
4384:
4383:
4374:
4373:
4358:
4357:
4346:
4328:
4327:
4308:
4306:
4305:
4300:
4281:
4279:
4278:
4273:
4258:
4256:
4255:
4250:
4234:
4232:
4231:
4226:
4215:
4214:
4192:
4190:
4189:
4184:
4179:
4178:
4163:
4162:
4138:
4137:
4125:
4124:
4118:
4117:
4081:
4080:
4071:
4070:
4051:
4049:
4048:
4043:
4038:
4037:
4021:
4019:
4018:
4013:
4001:
3999:
3998:
3993:
3981:
3979:
3978:
3973:
3968:
3967:
3951:
3949:
3948:
3943:
3916:
3914:
3913:
3908:
3884:
3882:
3881:
3876:
3855:
3853:
3852:
3847:
3820:is said to be a
3819:
3817:
3816:
3811:
3803:
3802:
3777:
3775:
3774:
3769:
3757:
3755:
3754:
3749:
3737:
3735:
3734:
3729:
3724:
3723:
3704:
3702:
3701:
3696:
3676:
3674:
3673:
3668:
3644:
3642:
3641:
3636:
3608:
3606:
3605:
3600:
3598:
3597:
3586:
3573:
3571:
3570:
3565:
3563:
3562:
3545:
3543:
3542:
3537:
3502:
3500:
3499:
3494:
3489:
3488:
3485:
3476:
3475:
3470:
3451:
3449:
3448:
3443:
3438:
3437:
3331:
3328:
3326:
3294:
3292:
3291:
3286:
3284:
3283:
3264:
3262:
3261:
3256:
3254:
3253:
3237:
3234:
3232:
3207:
3205:
3204:
3199:
3197:
3196:
3187:
3186:
3162:
3161:
3152:
3151:
3133:
3132:
3129:
3114:or equivalently
3110:
3108:
3107:
3102:
3100:
3099:
3090:
3089:
3079:
3067:
3063:
3062:
3060:
3059:
3058:
3042:
3040:
3039:
3029:
3017:
3015:
3014:
3013:
2997:
2995:
2994:
2984:
2970:
2969:
2966:
2950:
2948:
2947:
2942:
2940:
2939:
2936:
2923:Euclidean metric
2921:The (canonical)
2920:
2918:
2917:
2912:
2907:
2906:
2901:
2888:
2886:
2885:
2880:
2878:
2877:
2859:
2858:
2829:
2827:
2826:
2821:
2819:
2818:
2806:
2805:
2788:
2786:
2785:
2780:
2762:
2760:
2759:
2754:
2742:
2740:
2739:
2734:
2714:
2712:
2711:
2706:
2704:
2703:
2683:
2681:
2680:
2675:
2662:
2660:
2659:
2654:
2630:
2628:
2627:
2622:
2602:
2600:
2599:
2594:
2561:
2559:
2558:
2553:
2541:
2539:
2538:
2533:
2511:
2509:
2508:
2503:
2473:
2471:
2470:
2465:
2457:
2456:
2428:
2426:
2425:
2420:
2399:
2397:
2396:
2391:
2377:
2376:
2352:
2351:
2336:
2335:
2299:
2298:
2279:
2277:
2276:
2271:
2266:
2265:
2239:
2237:
2236:
2231:
2204:
2202:
2201:
2196:
2172:
2170:
2169:
2164:
2125:
2123:
2122:
2117:
2112:
2111:
2095:
2093:
2092:
2087:
2065:
2063:
2062:
2057:
2040:
2039:
2002:
2000:
1999:
1994:
1981:cotangent bundle
1974:
1972:
1971:
1966:
1918:
1916:
1915:
1910:
1883:
1881:
1880:
1875:
1873:
1872:
1849:
1847:
1846:
1841:
1829:
1827:
1826:
1821:
1794:
1792:
1791:
1786:
1784:
1770:
1769:
1746:
1744:
1743:
1738:
1718:
1716:
1715:
1710:
1705:
1704:
1689:
1688:
1675:
1674:
1661:
1630:
1628:
1627:
1622:
1617:
1616:
1595:
1594:
1566:In terms of the
1562:
1560:
1559:
1554:
1535:
1533:
1532:
1527:
1525:
1524:
1508:
1506:
1505:
1500:
1488:
1486:
1485:
1480:
1462:
1460:
1459:
1454:
1452:
1438:
1437:
1418:
1416:
1415:
1410:
1408:
1407:
1387:
1385:
1384:
1379:
1377:
1373:
1372:
1371:
1366:
1362:
1360:
1359:
1358:
1342:
1333:
1332:
1327:
1323:
1321:
1320:
1319:
1303:
1292:
1291:
1279:
1278:
1273:
1267:
1266:
1244:
1242:
1241:
1236:
1224:
1222:
1221:
1216:
1198:
1196:
1195:
1190:
1185:
1184:
1165:
1163:
1162:
1157:
1155:
1151:
1150:
1149:
1144:
1143:
1136:
1134:
1133:
1132:
1116:
1105:
1104:
1099:
1098:
1091:
1089:
1088:
1087:
1071:
1054:
1052:
1051:
1046:
1030:
1028:
1027:
1022:
1020:
1019:
1014:
996:
995:
977:
976:
938:
936:
935:
930:
902:
900:
899:
894:
882:
880:
879:
874:
862:
860:
859:
854:
852:
835:
834:
825:
820:
819:
797:
795:
794:
789:
787:
776:
775:
763:
762:
740:
738:
737:
732:
730:
719:
718:
703:
702:
690:
689:
669:
667:
666:
661:
650:assigns to each
649:
647:
646:
641:
629:
627:
626:
621:
599:
597:
596:
591:
586:
585:
566:
564:
563:
558:
546:
544:
543:
538:
526:
524:
523:
518:
513:
512:
496:
494:
493:
488:
476:
474:
473:
468:
452:
450:
449:
444:
439:
438:
422:
420:
419:
414:
392:
390:
389:
384:
298:Bernhard Riemann
264:Bernhard Riemann
232:machine learning
208:complex geometry
170:projective space
122:Bernhard Riemann
102:hyperbolic space
97:
95:
94:
89:
21:
20812:
20811:
20807:
20806:
20805:
20803:
20802:
20801:
20777:
20776:
20775:
20770:
20739:
20718:Generalizations
20713:
20667:
20636:
20571:Exponential map
20532:
20522:
20492:
20487:
20426:Banach manifold
20419:Generalizations
20414:
20369:
20306:
20203:
20165:Ricci curvature
20121:Cotangent space
20099:
20037:
19879:
19873:
19832:Exponential map
19796:
19741:
19735:
19655:
19645:
19595:
19590:
19421:
19395:Wolf, Joseph A.
19371:
19336:
19326:Springer-Verlag
19285:Nomizu, Katsumi
19257:
19194:
19152:
19098:
19080:Springer-Verlag
19066:
19061:
19060:
19052:
19048:
19040:
19033:
19025:
19016:
19008:
19001:
18993:
18989:
18977:
18973:
18965:
18961:
18936:
18932:
18926:
18922:
18907:
18903:
18897:
18893:
18878:
18874:
18859:
18855:
18849:
18845:
18836:
18832:
18817:
18813:
18798:
18794:
18779:
18775:
18760:
18756:
18751:
18748:
18747:
18745:
18741:
18733:
18729:
18721:
18717:
18705:
18701:
18693:
18689:
18681:
18677:
18669:
18665:
18657:
18650:
18642:
18638:
18630:
18626:
18618:
18614:
18606:
18602:
18594:
18590:
18582:
18578:
18570:
18566:
18558:
18554:
18546:
18542:
18534:
18530:
18522:
18513:
18505:
18501:
18493:
18489:
18481:
18477:
18469:
18465:
18457:
18453:
18445:
18441:
18433:
18429:
18421:
18417:
18409:
18405:
18397:
18390:
18382:
18378:
18370:
18363:
18355:
18351:
18343:
18339:
18331:
18324:
18316:
18309:
18301:
18297:
18289:
18282:
18274:
18267:
18259:
18252:
18244:
18240:
18235:
18230:
18225:
18216:Kahler manifold
18176:Smooth manifold
18171:
18150:
18147:
18146:
18122:
18118:
18116:
18113:
18112:
18084:
18081:
18080:
18071:
18037:
18034:
18033:
18016:
18012:
18010:
18007:
18006:
17978:
17975:
17974:
17957:
17953:
17951:
17948:
17947:
17931:
17928:
17927:
17907:
17903:
17901:
17898:
17897:
17881:
17878:
17877:
17861:
17858:
17857:
17828:
17824:
17822:
17819:
17818:
17771:
17767:
17765:
17762:
17761:
17758:
17675:
17671:
17665:
17661:
17637:
17633:
17631:
17628:
17627:
17596:
17592:
17563:
17560:
17559:
17524:
17520:
17506:
17503:
17502:
17466:
17463:
17462:
17446:
17443:
17442:
17413:
17410:
17409:
17392:
17388:
17386:
17383:
17382:
17362:
17359:
17358:
17338:
17335:
17334:
17315:
17312:
17311:
17271:
17268:
17267:
17239:
17236:
17235:
17198:
17195:
17194:
17144:
17140:
17138:
17135:
17134:
17111:
17107:
17105:
17102:
17101:
17085:
17082:
17081:
17056:
17053:
17052:
17051:, then for any
17006:
17003:
17002:
16998:
16976:
16973:
16972:
16956:
16953:
16952:
16932:
16928:
16926:
16923:
16922:
16905:
16901:
16899:
16896:
16895:
16879:
16876:
16875:
16846:
16842:
16840:
16837:
16836:
16820:
16808:
16804:
16792:
16788:
16779:
16775:
16773:
16770:
16769:
16747:
16744:
16743:
16724:
16698:
16695:
16694:
16678:
16675:
16674:
16663:
16645:; for example,
16622:
16617:
16616:
16614:
16611:
16610:
16603:
16592:
16586:
16583:
16572:
16558:related reading
16548:
16544:
16537:
16524:
16513:Kähler geometry
16443:Ricci curvature
16430:
16418:
16414:
16410:
16392:
16389:
16387:Symmetric space
16383:
16374:
16370:
16366:
16362:
16358:
16353:
16346:
16342:
16338:
16323:
16317:
16313:
16303:
16296:
16292:
16288:
16273:
16267:
16263:
16259:
16251:
16244:
16241:normal subgroup
16236:
16232:
16217:
16213:
16209:
16205:
16201:
16197:
16179:
16176:
16135:
16127:
16120:
16112:
16099:
16095:
16085:
16082:
16074:
16070:
16030:
16026:
16025:
16021:
15995:
15991:
15990:
15986:
15974:
15970:
15946:
15942:
15940:
15937:
15936:
15929:
15926:
15918:
15906:
15903:
15898:
15878:
15839:
15829:
15822:
15818:
15810:
15807:identity matrix
15802:
15791:
15787:-sphere is the
15784:
15764:
15760:
15756:
15715:
15707:
15684:
15680:
15671:
15666:
15647:
15642:
15625:
15615:
15608:
15603:
15581:
15576:
15568:
15566:
15564:
15561:
15560:
15545:
15531:
15524:
15511:
15503:
15500:Ricci curvature
15470:
15466:
15457:
15453:
15441:
15437:
15428:
15424:
15400:
15396:
15394:
15391:
15390:
15375:
15367:
15359:
15354:
15348:
15319:
15316:
15315:
15292:
15289:
15288:
15257:
15254:
15253:
15244:Einstein metric
15227:
15224:
15223:
15193:
15190:
15189:
15186:
15180:
15159:
15156:
15155:
15073:
15070:
15069:
15044:
15041:
15040:
15024:
15021:
15020:
15017:
15015:Ricci curvature
15011:
15005:-tensor field.
14978:
14975:
14974:
14942:
14939:
14938:
14903:
14899:
14887:
14883:
14877:
14873:
14861:
14857:
14851:
14847:
14821:
14818:
14817:
14788:
14787:
14769:
14768:
14750:
14749:
14731:
14730:
14722:
14719:
14718:
14696:
14693:
14692:
14676:
14673:
14672:
14665:
14659:
14643:
14642:
14641:
14640:
14639:
14622:
14618:
14606:
14602:
14597:
14594:
14593:
14590:
14582:
14581:
14564:
14560:
14551:
14547:
14538:
14534:
14529:
14526:
14525:
14522:
14513:
14512:
14489:
14485:
14473:
14469:
14464:
14461:
14460:
14443:
14439:
14429:
14425:
14416:
14412:
14400:
14396:
14384:
14380:
14375:
14372:
14371:
14332:
14327:
14326:
14324:
14321:
14320:
14318:punctured plane
14298:
14295:
14294:
14278:
14275:
14274:
14258:
14255:
14254:
14253:, first extend
14238:
14235:
14234:
14214:
14210:
14208:
14205:
14204:
14203:to a vector in
14184:
14180:
14172:
14169:
14168:
14137:
14134:
14133:
14102:
14099:
14098:
14058:
14055:
14054:
14028:
14024:
14022:
14019:
14018:
14001:
13998:
13997:
13996:parallel along
13980:
13977:
13976:
13960:
13957:
13956:
13938:
13932:
13910:
13907:
13906:
13888:
13885:
13884:
13878:Cauchy sequence
13860:
13856:
13854:
13851:
13850:
13826:
13822:
13811:
13808:
13807:
13775:
13772:
13771:
13742:
13740:
13737:
13736:
13702:
13699:
13698:
13664:
13661:
13660:
13643:
13638:
13637:
13635:
13632:
13631:
13590:
13585:
13584:
13582:
13579:
13578:
13576:punctured plane
13558:
13553:
13552:
13550:
13547:
13546:
13515:
13512:
13511:
13489:
13486:
13485:
13465:
13463:
13460:
13459:
13425:
13422:
13421:
13387:
13384:
13383:
13342:
13337:
13336:
13334:
13331:
13330:
13323:
13317:
13290:
13286:
13284:
13281:
13280:
13260:
13255:
13254:
13252:
13249:
13248:
13244:
13223:
13220:
13219:
13179:
13176:
13175:
13138:
13136:
13133:
13132:
13101:
13098:
13097:
13059:
13057:
13054:
13053:
13022:
13019:
13018:
13002:
12999:
12998:
12964:
12962:
12959:
12958:
12927:
12924:
12923:
12907:
12904:
12903:
12875:
12872:
12871:
12851:
12847:
12839:
12836:
12835:
12813:
12810:
12809:
12786:
12783:
12782:
12766:
12763:
12762:
12737:
12731:
12727:
12725:
12722:
12721:
12705:
12702:
12701:
12682:
12676:
12672:
12670:
12667:
12666:
12649:
12646:
12645:
12603:
12600:
12599:
12583:
12580:
12579:
12563:
12560:
12559:
12549:
12548:
12547:
12546:
12525:
12521:
12519:
12516:
12515:
12498:
12493:
12492:
12490:
12487:
12486:
12482:
12481:
12480:
12472:
12471:
12460:
12454:
12426:
12425:
12406:
12405:
12401:
12380:
12376:
12374:
12371:
12370:
12354:
12351:
12350:
12328:
12327:
12325:
12322:
12321:
12296:
12292:
12274:
12253:
12249:
12247:
12244:
12243:
12219:
12215:
12199:
12195:
12161:
12157:
12155:
12152:
12151:
12131:
12128:
12127:
12099:
12098:
12080:
12079:
12070:
12066:
12064:
12061:
12060:
12020:
12017:
12016:
12000:
11997:
11996:
11980:
11977:
11976:
11956:
11953:
11952:
11928:
11927:
11925:
11922:
11921:
11895:
11884:
11883:
11882:
11865:
11862:
11861:
11845:
11842:
11841:
11819:
11818:
11816:
11813:
11812:
11796:
11793:
11792:
11776:
11773:
11772:
11740:
11739:
11731:
11728:
11727:
11666:
11663:
11662:
11643:
11617:
11614:
11613:
11597:
11594:
11593:
11577:
11574:
11573:
11547:
11546:
11544:
11541:
11540:
11506:
11503:
11502:
11473:
11469:
11452:
11449:
11448:
11405:
11402:
11401:
11384:
11381:
11380:
11338:
11335:
11334:
11331:
11310:
11307:
11306:
11266:
11263:
11262:
11218:
11214:
11202:
11198:
11196:
11193:
11192:
11169:
11166:
11165:
11139:
11135:
11102:
11098:
11083:
11082:
11058:
11057:
11052:
11049:
11048:
11025:
11022:
11021:
11011:
11005:
10983:
10980:
10979:
10963:
10960:
10959:
10937:
10933:
10931:
10928:
10927:
10926:The expression
10903:
10899:
10862:
10858:
10856:
10853:
10852:
10825:
10821:
10820:
10816:
10809:
10805:
10791:
10787:
10786:
10782:
10775:
10771:
10757:
10753:
10747:
10743:
10734:
10730:
10724:
10720:
10719:
10715:
10713:
10710:
10709:
10683:
10679:
10671:
10668:
10667:
10643:
10639:
10619:
10616:
10615:
10599:
10596:
10595:
10566:
10565:
10547:
10546:
10528:
10527:
10519:
10516:
10515:
10493:
10490:
10489:
10459:
10458:
10456:
10453:
10452:
10441:
10435:
10430:
10396:
10392:
10381:
10378:
10377:
10357:
10336:
10332:
10330:
10327:
10326:
10306:
10302:
10291:
10288:
10287:
10263:
10259:
10248:
10245:
10244:
10181:
10177:
10159:
10155:
10138:
10135:
10134:
10111:
10107:
10096:
10093:
10092:
10083:
10062:
10059:
10058:
10038:
10034:
10023:
10020:
10019:
10016:metric topology
9991:
9987:
9976:
9973:
9972:
9940:
9937:
9936:
9920:
9899:
9895:
9893:
9890:
9889:
9873:
9870:
9869:
9852:
9848:
9846:
9843:
9842:
9839:
9818:
9815:
9814:
9794:
9790:
9779:
9776:
9775:
9745:
9728:
9725:
9724:
9703:
9698:
9697:
9677:
9674:
9673:
9629:
9620:
9615:
9604:
9587:
9584:
9583:
9549:
9546:
9545:
9529:
9526:
9525:
9509:
9506:
9505:
9466:
9463:
9462:
9428:
9425:
9424:
9402:
9399:
9398:
9350:
9347:
9346:
9302:
9298:
9278:
9275:
9274:
9248:
9245:
9244:
9221:
9217:
9209:
9206:
9205:
9183:
9180:
9179:
9162:
9158:
9126:
9123:
9122:
9106:
9103:
9102:
9078:
9076:
9073:
9072:
9056:
9053:
9052:
9022:
9020:
9017:
9016:
9000:
8997:
8996:
8974:
8971:
8970:
8945:
8942:
8941:
8910:
8907:
8906:
8878:
8875:
8874:
8801:
8797:
8795:
8792:
8791:
8769:
8766:
8765:
8741:
8737:
8726:
8723:
8722:
8715:
8700:
8697:
8696:
8676:
8672:
8661:
8658:
8657:
8637:
8633:
8622:
8619:
8618:
8594:
8591:
8590:
8570:
8566:
8555:
8552:
8551:
8548:metric topology
8523:
8519:
8508:
8505:
8504:
8438:
8390:
8386:
8384:
8381:
8380:
8330:
8326:
8324:
8321:
8320:
8291:
8288:
8287:
8247:
8243:
8226:
8217:
8212:
8191:
8188:
8187:
8144:
8141:
8140:
8115:
8112:
8111:
8083:
8080:
8079:
8053:
8049:
8032:
8021:
8018:
8017:
7988:
7984:
7964:
7962:
7959:
7958:
7957:whose velocity
7918:
7915:
7914:
7907:
7878:
7875:
7874:
7857:
7852:
7851:
7849:
7846:
7845:
7829:
7826:
7825:
7805:
7802:
7801:
7784:
7779:
7778:
7764:
7761:
7760:
7729:
7726:
7725:
7705:
7702:
7701:
7685:
7682:
7681:
7674:
7653:
7650:
7649:
7632:
7628:
7626:
7623:
7622:
7605:
7601:
7589:
7585:
7574:
7571:
7570:
7554:
7551:
7550:
7533:
7522:
7521:
7520:
7511:
7507:
7505:
7502:
7501:
7484:
7480:
7478:
7475:
7474:
7457:
7446:
7445:
7444:
7442:
7439:
7438:
7421:
7417:
7415:
7412:
7411:
7383:
7382:
7378:
7372:
7367:
7361:
7358:
7357:
7340:
7335:
7334:
7332:
7329:
7328:
7311:
7307:
7305:
7302:
7301:
7278:
7274:
7268:
7263:
7250:
7239:
7238:
7237:
7235:
7232:
7231:
7208:
7197:
7196:
7195:
7186:
7182:
7170:
7158:
7155:
7154:
7135:
7132:
7131:
7115:
7112:
7111:
7086:
7083:
7082:
7065:
7061:
7049:
7045:
7034:
7031:
7030:
7003:
6999:
6993:
6989:
6984:
6981:
6980:
6960:
6955:
6954:
6942:
6938:
6929:
6925:
6916:
6912:
6903:
6899:
6897:
6894:
6893:
6870:
6866:
6864:
6861:
6860:
6830:
6826:
6817:
6813:
6804:
6800:
6792:
6789:
6788:
6772:
6769:
6768:
6761:
6731:
6708:
6705:
6704:
6687:
6683:
6677:
6673:
6658:
6654:
6648:
6644:
6642:
6639:
6638:
6622:
6619:
6618:
6601:
6597:
6582:
6578:
6576:
6573:
6572:
6553:
6550:
6549:
6532:
6528:
6513:
6509:
6507:
6504:
6503:
6500:
6478:
6473:
6472:
6470:
6467:
6466:
6450:
6448:
6445:
6444:
6428:
6414:
6412:
6409:
6408:
6385:
6381:
6379:
6376:
6375:
6358:
6354:
6352:
6349:
6348:
6331:
6327:
6312:
6308:
6299:
6295:
6293:
6290:
6289:
6272:
6269:
6268:
6241:
6240:
6234:
6230:
6228:
6222:
6221:
6216:
6210:
6206:
6199:
6198:
6183:
6172:
6171:
6170:
6165:
6162:
6161:
6144:
6140:
6130:
6126:
6113:
6102:
6101:
6100:
6091:
6073:
6072:
6070:
6067:
6066:
6018:
6015:
6014:
5992:
5991:
5989:
5986:
5985:
5957:
5954:
5953:
5937:
5934:
5933:
5916:
5912:
5910:
5907:
5906:
5878:
5875:
5874:
5858:
5855:
5854:
5837:
5833:
5831:
5828:
5827:
5802:
5799:
5798:
5752:
5749:
5748:
5732:
5729:
5728:
5700:
5697:
5696:
5680:
5677:
5676:
5648:
5645:
5644:
5617:
5613:
5604:
5600:
5591:
5587:
5575:
5571:
5562:
5558:
5549:
5545:
5530:
5526:
5517:
5513:
5498:
5494:
5485:
5481:
5463:
5452:
5451:
5450:
5448:
5445:
5444:
5441:one may define
5419:
5415:
5403:
5399:
5363:
5359:
5357:
5354:
5353:
5324:
5321:
5320:
5298:
5297:
5295:
5292:
5291:
5275:
5272:
5271:
5255:
5252:
5251:
5229:
5226:
5225:
5194:
5191:
5190:
5162:
5159:
5158:
5155:
5154:
5153:
5152:
5140:
5139:
5138:
5130:
5129:
5118:
5093:
5089:
5075:
5074:
5072:
5069:
5068:
5028:
5025:
5024:
5002:
5001:
4999:
4996:
4995:
4979:
4976:
4975:
4945:
4944:
4942:
4939:
4938:
4922:
4919:
4918:
4915:universal cover
4892:
4891:
4889:
4886:
4885:
4857:
4856:
4854:
4851:
4850:
4822:
4819:
4818:
4792:
4787:
4786:
4784:
4781:
4780:
4764:
4755:
4750:
4749:
4741:
4738:
4737:
4711:
4706:
4705:
4703:
4700:
4699:
4663:
4659:
4653:
4649:
4647:
4636:
4632:
4626:
4622:
4620:
4609:
4605:
4599:
4595:
4593:
4584:
4579:
4578:
4571:
4567:
4565:
4562:
4561:
4528:
4525:
4524:
4516:standard metric
4494:
4490:
4488:
4485:
4484:
4461:
4456:
4455:
4453:
4450:
4449:
4417:
4413:
4401:
4397:
4379:
4375:
4369:
4365:
4347:
4342:
4341:
4323:
4319:
4317:
4314:
4313:
4294:
4291:
4290:
4264:
4261:
4260:
4244:
4241:
4240:
4210:
4206:
4201:
4198:
4197:
4174:
4173:
4158:
4154:
4133:
4129:
4120:
4119:
4104:
4100:
4076:
4072:
4066:
4062:
4060:
4057:
4056:
4033:
4029:
4027:
4024:
4023:
4007:
4004:
4003:
3987:
3984:
3983:
3963:
3959:
3957:
3954:
3953:
3952:and the metric
3922:
3919:
3918:
3890:
3887:
3886:
3864:
3861:
3860:
3829:
3826:
3825:
3798:
3794:
3783:
3780:
3779:
3763:
3760:
3759:
3743:
3740:
3739:
3719:
3715:
3713:
3710:
3709:
3690:
3687:
3686:
3650:
3647:
3646:
3618:
3615:
3614:
3587:
3582:
3581:
3579:
3576:
3575:
3558:
3554:
3552:
3549:
3548:
3531:
3528:
3527:
3519:
3513:
3505:Euclidean space
3484:
3480:
3471:
3466:
3465:
3460:
3457:
3456:
3432:
3431:
3426:
3421:
3416:
3410:
3409:
3404:
3399:
3394:
3388:
3387:
3382:
3377:
3372:
3366:
3365:
3360:
3355:
3350:
3340:
3339:
3327:
3319:
3310:
3307:
3306:
3297:Kronecker delta
3276:
3272:
3270:
3267:
3266:
3246:
3242:
3233:
3225:
3219:
3216:
3215:
3192:
3188:
3182:
3178:
3157:
3153:
3147:
3143:
3128:
3124:
3122:
3119:
3118:
3095:
3091:
3085:
3081:
3075:
3054:
3050:
3046:
3041:
3035:
3031:
3025:
3009:
3005:
3001:
2996:
2990:
2986:
2980:
2975:
2971:
2965:
2961:
2959:
2956:
2955:
2935:
2931:
2929:
2926:
2925:
2902:
2897:
2896:
2894:
2891:
2890:
2873:
2869:
2854:
2850:
2848:
2845:
2844:
2841:
2839:Euclidean space
2836:
2814:
2810:
2801:
2797:
2795:
2792:
2791:
2774:
2771:
2770:
2748:
2745:
2744:
2728:
2725:
2724:
2699:
2695:
2690:
2687:
2686:
2669:
2666:
2665:
2636:
2633:
2632:
2616:
2613:
2612:
2609:
2567:
2564:
2563:
2547:
2544:
2543:
2521:
2518:
2517:
2482:
2479:
2478:
2452:
2448:
2434:
2431:
2430:
2408:
2405:
2404:
2372:
2368:
2347:
2343:
2322:
2318:
2294:
2290:
2288:
2285:
2284:
2261:
2257:
2249:
2246:
2245:
2213:
2210:
2209:
2178:
2175:
2174:
2146:
2143:
2142:
2132:
2107:
2103:
2101:
2098:
2097:
2078:
2075:
2074:
2035:
2031:
2011:
2008:
2007:
1988:
1985:
1984:
1942:
1939:
1938:
1931:
1925:
1904:
1901:
1900:
1865:
1861:
1859:
1856:
1855:
1835:
1832:
1831:
1800:
1797:
1796:
1780:
1762:
1758:
1756:
1753:
1752:
1732:
1729:
1728:
1725:
1700:
1696:
1684:
1680:
1667:
1663:
1651:
1639:
1636:
1635:
1612:
1608:
1590:
1586:
1578:
1575:
1574:
1548:
1545:
1544:
1520:
1516:
1514:
1511:
1510:
1494:
1491:
1490:
1468:
1465:
1464:
1448:
1430:
1426:
1424:
1421:
1420:
1403:
1399:
1397:
1394:
1393:
1367:
1354:
1350:
1346:
1341:
1338:
1337:
1328:
1315:
1311:
1307:
1302:
1299:
1298:
1297:
1293:
1287:
1283:
1274:
1269:
1268:
1259:
1255:
1253:
1250:
1249:
1230:
1227:
1226:
1204:
1201:
1200:
1180:
1176:
1174:
1171:
1170:
1145:
1139:
1138:
1137:
1128:
1124:
1120:
1115:
1100:
1094:
1093:
1092:
1083:
1079:
1075:
1070:
1069:
1065:
1063:
1060:
1059:
1040:
1037:
1036:
1015:
1010:
1009:
991:
987:
972:
968:
963:
960:
959:
956:
912:
909:
908:
888:
885:
884:
868:
865:
864:
830:
826:
824:
815:
811:
803:
800:
799:
783:
771:
767:
758:
754:
746:
743:
742:
726:
714:
710:
698:
694:
685:
681:
679:
676:
675:
655:
652:
651:
635:
632:
631:
615:
612:
611:
581:
577:
575:
572:
571:
552:
549:
548:
532:
529:
528:
508:
504:
502:
499:
498:
482:
479:
478:
462:
459:
458:
434:
430:
428:
425:
424:
402:
399:
398:
395:smooth manifold
378:
375:
374:
363:
358:
334:Albert Einstein
314:introduced the
302:smooth manifold
256:
139:is a choice of
137:smooth manifold
106:smooth surfaces
83:
80:
79:
75:Euclidean space
71:geometric space
49:sitting inside
35:
32:Riemann surface
28:
23:
22:
15:
12:
11:
5:
20810:
20800:
20799:
20794:
20789:
20772:
20771:
20769:
20768:
20763:
20758:
20753:
20747:
20745:
20741:
20740:
20738:
20737:
20735:Sub-Riemannian
20732:
20727:
20721:
20719:
20715:
20714:
20712:
20711:
20706:
20701:
20696:
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20686:
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20675:
20673:
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20655:
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20598:
20593:
20583:
20578:
20573:
20568:
20567:
20566:
20561:
20556:
20551:
20540:
20538:
20537:Basic concepts
20534:
20533:
20521:
20520:
20513:
20506:
20498:
20489:
20488:
20486:
20485:
20480:
20475:
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20463:
20453:
20448:
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20433:
20428:
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20413:
20412:
20407:
20402:
20397:
20392:
20387:
20381:
20379:
20375:
20374:
20371:
20370:
20368:
20367:
20362:
20357:
20352:
20347:
20342:
20337:
20332:
20327:
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20316:
20314:
20308:
20307:
20305:
20304:
20299:
20294:
20289:
20284:
20279:
20274:
20264:
20259:
20254:
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20239:
20234:
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20224:
20219:
20213:
20211:
20205:
20204:
20202:
20201:
20196:
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20190:
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20035:
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20020:
20015:
20001:
19996:
19991:
19986:
19981:
19979:Parallelizable
19976:
19971:
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19954:
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19919:
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19909:
19899:
19889:
19883:
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19875:
19874:
19872:
19871:
19866:
19861:
19859:Lie derivative
19856:
19854:Integral curve
19851:
19846:
19841:
19840:
19839:
19829:
19824:
19823:
19822:
19815:Diffeomorphism
19812:
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19804:
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19660:Basic concepts
19657:
19656:
19644:
19643:
19636:
19629:
19621:
19615:
19614:
19594:
19593:External links
19591:
19589:
19588:
19560:(3): 293–329.
19540:
19528:10.4171/dm/187
19506:Documenta Math
19501:
19462:
19441:
19419:
19391:
19369:
19353:Springer, Cham
19340:
19334:
19314:
19277:
19255:
19214:
19192:
19172:
19150:
19126:Ebin, David G.
19118:
19096:
19067:
19065:
19062:
19059:
19058:
19046:
19031:
19014:
18999:
18987:
18981:, p. 72;
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18774:
18771:
18768:
18763:
18759:
18755:
18739:
18737:, p. 369.
18727:
18715:
18713:, Section 4.4.
18699:
18687:
18675:
18663:
18648:
18646:, p. 210.
18636:
18634:, p. 207.
18624:
18612:
18610:, p. 200.
18600:
18598:, p. 201.
18588:
18576:
18564:
18562:, p. 131.
18552:
18550:, p. 137.
18540:
18538:, p. 156.
18528:
18526:, p. 105.
18511:
18499:
18497:, p. 103.
18487:
18475:
18473:, p. 100.
18463:
18451:
18439:
18437:, p. 276.
18435:Burtscher 2015
18427:
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17885:
17865:
17848:, is always a
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16625:
16620:
16605:
16604:
16562:external links
16551:
16549:
16542:
16536:
16533:
16522:
16497:product spaces
16403:is said to be
16385:Main article:
16382:
16379:
16319:
16269:
16226:isometry group
16190:is said to be
16175:
16172:
16147:Berger spheres
16134:associated to
16116:
16106:left-invariant
16078:
16067:
16066:
16055:
16052:
16049:
16046:
16043:
16036:
16033:
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16020:
16017:
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15998:
15994:
15989:
15985:
15982:
15977:
15973:
15969:
15966:
15963:
15960:
15957:
15954:
15949:
15945:
15932:is defined by
15922:
15913:, such as the
15902:
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15897:
15894:
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15013:Main article:
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14005:
13984:
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13934:Main article:
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11007:Main article:
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10956:is called the
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10243:shows that if
10237:
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10222:
10219:
10216:
10213:
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10207:
10204:
10201:
10198:
10195:
10192:
10189:
10184:
10180:
10176:
10173:
10170:
10167:
10162:
10158:
10154:
10151:
10148:
10145:
10142:
10119:
10114:
10110:
10106:
10103:
10100:
10082:
10079:
10066:
10046:
10041:
10037:
10033:
10030:
10027:
9999:
9994:
9990:
9986:
9983:
9980:
9956:
9953:
9950:
9947:
9944:
9923:
9919:
9916:
9913:
9910:
9907:
9902:
9898:
9877:
9855:
9851:
9836:
9835:
9822:
9802:
9797:
9793:
9789:
9786:
9783:
9757:
9754:
9749:
9744:
9741:
9738:
9735:
9732:
9706:
9701:
9696:
9693:
9690:
9687:
9684:
9681:
9670:
9669:
9658:
9655:
9652:
9648:
9645:
9642:
9639:
9635:
9632:
9628:
9623:
9618:
9614:
9608:
9603:
9600:
9597:
9594:
9591:
9568:
9565:
9562:
9559:
9556:
9553:
9533:
9513:
9504:The length of
9491:
9488:
9485:
9482:
9479:
9476:
9473:
9470:
9450:
9447:
9444:
9441:
9438:
9435:
9432:
9412:
9409:
9406:
9378:
9375:
9372:
9369:
9366:
9363:
9360:
9357:
9354:
9334:
9331:
9328:
9325:
9322:
9319:
9316:
9313:
9310:
9305:
9301:
9297:
9294:
9291:
9288:
9285:
9282:
9258:
9255:
9252:
9232:
9229:
9224:
9220:
9216:
9213:
9193:
9190:
9187:
9165:
9161:
9157:
9154:
9151:
9148:
9145:
9142:
9139:
9136:
9133:
9130:
9110:
9090:
9085:
9082:
9060:
9040:
9037:
9034:
9029:
9026:
9004:
8984:
8981:
8978:
8958:
8955:
8952:
8949:
8929:
8926:
8923:
8920:
8917:
8914:
8894:
8891:
8888:
8885:
8882:
8827:
8824:
8821:
8818:
8815:
8812:
8809:
8804:
8800:
8779:
8776:
8773:
8749:
8744:
8740:
8736:
8733:
8730:
8717:
8716:
8704:
8684:
8679:
8675:
8671:
8668:
8665:
8645:
8640:
8636:
8632:
8629:
8626:
8616:
8611:
8598:
8578:
8573:
8569:
8565:
8562:
8559:
8531:
8526:
8522:
8518:
8515:
8512:
8499:
8498:
8487:
8484:
8481:
8478:
8475:
8472:
8469:
8466:
8463:
8460:
8457:
8454:
8451:
8448:
8445:
8437:
8434:
8431:
8428:
8425:
8422:
8419:
8416:
8413:
8410:
8407:
8404:
8401:
8398:
8393:
8389:
8365:
8362:
8359:
8356:
8353:
8350:
8347:
8344:
8341:
8338:
8333:
8329:
8307:
8304:
8301:
8298:
8295:
8283:
8282:
8271:
8268:
8265:
8259:
8256:
8253:
8250:
8246:
8242:
8239:
8236:
8232:
8229:
8225:
8220:
8215:
8211:
8207:
8204:
8201:
8198:
8195:
8183:is defined as
8172:
8169:
8166:
8163:
8160:
8157:
8154:
8151:
8148:
8128:
8125:
8122:
8119:
8099:
8096:
8093:
8090:
8087:
8065:
8062:
8059:
8056:
8052:
8048:
8045:
8042:
8038:
8035:
8031:
8028:
8025:
8005:
8000:
7997:
7994:
7991:
7987:
7983:
7980:
7977:
7974:
7970:
7967:
7946:
7943:
7940:
7937:
7934:
7931:
7928:
7925:
7922:
7906:
7903:
7885:
7882:
7860:
7855:
7833:
7820:such that the
7809:
7787:
7782:
7777:
7774:
7771:
7768:
7748:
7745:
7742:
7739:
7736:
7733:
7709:
7689:
7671:
7670:
7657:
7635:
7631:
7608:
7604:
7600:
7597:
7592:
7588:
7584:
7581:
7578:
7558:
7536:
7529:
7526:
7519:
7514:
7510:
7500:, the product
7487:
7483:
7460:
7453:
7450:
7424:
7420:
7392:
7389:
7386:
7381:
7375:
7370:
7366:
7343:
7338:
7310:
7298:
7297:
7286:
7277:
7271:
7266:
7262:
7258:
7253:
7246:
7243:
7225:
7224:
7211:
7204:
7201:
7194:
7189:
7185:
7179:
7176:
7173:
7169:
7165:
7162:
7139:
7119:
7096:
7093:
7090:
7068:
7064:
7060:
7057:
7052:
7048:
7044:
7041:
7038:
7012:
7009:
7006:
7002:
6996:
6992:
6988:
6963:
6958:
6953:
6950:
6945:
6941:
6937:
6932:
6928:
6924:
6919:
6915:
6911:
6906:
6902:
6881:
6878:
6873:
6869:
6854:locally finite
6839:
6836:
6833:
6829:
6825:
6820:
6816:
6812:
6807:
6803:
6799:
6796:
6776:
6763:
6762:
6759:
6754:
6730:
6727:
6715:
6712:
6690:
6686:
6680:
6676:
6672:
6669:
6666:
6661:
6657:
6651:
6647:
6626:
6604:
6600:
6596:
6593:
6590:
6585:
6581:
6560:
6557:
6535:
6531:
6527:
6524:
6521:
6516:
6512:
6499:
6496:
6481:
6476:
6453:
6431:
6427:
6424:
6421:
6417:
6401:is called the
6388:
6384:
6361:
6357:
6334:
6330:
6326:
6323:
6320:
6315:
6311:
6307:
6302:
6298:
6276:
6264:
6263:
6262:
6261:
6250:
6245:
6237:
6233:
6229:
6227:
6224:
6223:
6220:
6217:
6213:
6209:
6205:
6204:
6202:
6197:
6194:
6189:
6186:
6179:
6176:
6169:
6147:
6143:
6139:
6133:
6129:
6125:
6119:
6116:
6109:
6106:
6097:
6094:
6090:
6086:
6080:
6077:
6052:
6049:
6046:
6043:
6040:
6037:
6034:
6031:
6028:
6025:
6022:
5999:
5996:
5973:
5970:
5967:
5964:
5961:
5941:
5919:
5915:
5894:
5891:
5888:
5885:
5882:
5862:
5840:
5836:
5815:
5812:
5809:
5806:
5786:
5783:
5780:
5777:
5774:
5771:
5768:
5765:
5762:
5759:
5756:
5736:
5716:
5713:
5710:
5707:
5704:
5684:
5664:
5661:
5658:
5655:
5652:
5641:
5640:
5639:
5628:
5625:
5620:
5616:
5612:
5607:
5603:
5599:
5594:
5590:
5586:
5583:
5578:
5574:
5570:
5565:
5561:
5557:
5552:
5548:
5544:
5541:
5538:
5533:
5529:
5525:
5520:
5516:
5512:
5509:
5506:
5501:
5497:
5493:
5488:
5484:
5480:
5477:
5472:
5469:
5466:
5459:
5456:
5430:
5427:
5422:
5418:
5414:
5411:
5406:
5402:
5398:
5395:
5392:
5389:
5386:
5383:
5378:
5375:
5372:
5369:
5366:
5362:
5337:
5334:
5331:
5328:
5305:
5302:
5279:
5259:
5239:
5236:
5233:
5210:
5207:
5204:
5201:
5198:
5178:
5175:
5172:
5169:
5166:
5142:
5141:
5132:
5131:
5123:
5122:
5121:
5120:
5119:
5117:
5114:
5101:
5096:
5092:
5088:
5082:
5079:
5044:
5041:
5038:
5035:
5032:
5009:
5006:
4983:
4972:
4971:
4952:
4949:
4926:
4899:
4896:
4873:
4870:
4864:
4861:
4838:
4835:
4832:
4829:
4826:
4815:
4801:
4798:
4795:
4790:
4767:
4763:
4758:
4753:
4748:
4745:
4729:
4728:
4714:
4709:
4695:
4694:
4693:
4692:
4680:
4676:
4673:
4666:
4662:
4656:
4652:
4646:
4639:
4635:
4629:
4625:
4619:
4612:
4608:
4602:
4598:
4592:
4587:
4582:
4577:
4574:
4570:
4544:
4541:
4538:
4535:
4532:
4520:
4519:
4510:is called the
4497:
4493:
4470:
4467:
4464:
4459:
4445:
4444:
4443:
4442:
4431:
4428:
4425:
4420:
4416:
4410:
4407:
4404:
4400:
4396:
4393:
4390:
4387:
4382:
4378:
4372:
4368:
4364:
4361:
4356:
4353:
4350:
4345:
4340:
4337:
4334:
4331:
4326:
4322:
4298:
4271:
4268:
4248:
4224:
4221:
4218:
4213:
4209:
4205:
4194:
4193:
4182:
4177:
4172:
4169:
4166:
4161:
4157:
4153:
4150:
4147:
4144:
4141:
4136:
4132:
4128:
4123:
4116:
4113:
4110:
4107:
4103:
4099:
4096:
4093:
4090:
4087:
4084:
4079:
4075:
4069:
4065:
4041:
4036:
4032:
4011:
3991:
3971:
3966:
3962:
3941:
3938:
3935:
3932:
3929:
3926:
3906:
3903:
3900:
3897:
3894:
3874:
3871:
3868:
3845:
3842:
3839:
3836:
3833:
3809:
3806:
3801:
3797:
3793:
3790:
3787:
3767:
3747:
3727:
3722:
3718:
3694:
3666:
3663:
3660:
3657:
3654:
3634:
3631:
3628:
3625:
3622:
3596:
3593:
3590:
3585:
3561:
3557:
3535:
3515:Main article:
3512:
3509:
3492:
3483:
3479:
3474:
3469:
3464:
3453:
3452:
3441:
3436:
3430:
3427:
3425:
3422:
3420:
3417:
3415:
3412:
3411:
3408:
3405:
3403:
3400:
3398:
3395:
3393:
3390:
3389:
3386:
3383:
3381:
3378:
3376:
3373:
3371:
3368:
3367:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3345:
3343:
3338:
3335:
3325:
3322:
3318:
3314:
3300:
3299:
3282:
3279:
3275:
3252:
3249:
3245:
3241:
3231:
3228:
3224:
3209:
3208:
3195:
3191:
3185:
3181:
3177:
3174:
3171:
3168:
3165:
3160:
3156:
3150:
3146:
3142:
3139:
3136:
3127:
3112:
3111:
3098:
3094:
3088:
3084:
3078:
3074:
3070:
3066:
3057:
3053:
3049:
3045:
3038:
3034:
3028:
3024:
3020:
3012:
3008:
3004:
3000:
2993:
2989:
2983:
2979:
2974:
2964:
2934:
2910:
2905:
2900:
2876:
2872:
2868:
2865:
2862:
2857:
2853:
2840:
2837:
2835:
2832:
2817:
2813:
2809:
2804:
2800:
2778:
2752:
2732:
2702:
2698:
2694:
2673:
2652:
2649:
2646:
2643:
2640:
2620:
2608:
2605:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2551:
2531:
2528:
2525:
2514:local isometry
2501:
2498:
2495:
2492:
2489:
2486:
2463:
2460:
2455:
2451:
2447:
2444:
2441:
2438:
2418:
2415:
2412:
2401:
2400:
2389:
2386:
2383:
2380:
2375:
2371:
2367:
2364:
2361:
2358:
2355:
2350:
2346:
2342:
2339:
2334:
2331:
2328:
2325:
2321:
2317:
2314:
2311:
2308:
2305:
2302:
2297:
2293:
2280:, that is, if
2269:
2264:
2260:
2256:
2253:
2229:
2226:
2223:
2220:
2217:
2207:diffeomorphism
2194:
2191:
2188:
2185:
2182:
2162:
2159:
2156:
2153:
2150:
2131:
2128:
2115:
2110:
2106:
2085:
2082:
2067:
2066:
2055:
2052:
2049:
2046:
2043:
2038:
2034:
2030:
2027:
2024:
2021:
2018:
2015:
1992:
1977:tangent bundle
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1927:Main article:
1924:
1921:
1908:
1871:
1868:
1864:
1839:
1819:
1816:
1813:
1810:
1807:
1804:
1783:
1779:
1776:
1773:
1768:
1765:
1761:
1736:
1724:
1721:
1720:
1719:
1708:
1703:
1699:
1695:
1692:
1687:
1683:
1679:
1673:
1670:
1666:
1660:
1657:
1654:
1650:
1646:
1643:
1620:
1615:
1611:
1607:
1604:
1601:
1598:
1593:
1589:
1585:
1582:
1568:tensor algebra
1552:
1523:
1519:
1498:
1478:
1475:
1472:
1451:
1447:
1444:
1441:
1436:
1433:
1429:
1406:
1402:
1390:
1389:
1376:
1370:
1365:
1357:
1353:
1349:
1345:
1340:
1336:
1331:
1326:
1318:
1314:
1310:
1306:
1301:
1296:
1290:
1286:
1282:
1277:
1272:
1265:
1262:
1258:
1234:
1214:
1211:
1208:
1188:
1183:
1179:
1167:
1166:
1154:
1148:
1142:
1131:
1127:
1123:
1119:
1114:
1111:
1108:
1103:
1097:
1086:
1082:
1078:
1074:
1068:
1055:, the vectors
1044:
1018:
1013:
1008:
1005:
1002:
999:
994:
990:
986:
983:
980:
975:
971:
967:
955:
952:
928:
925:
922:
919:
916:
892:
872:
850:
847:
844:
841:
838:
833:
829:
823:
818:
814:
810:
807:
786:
782:
779:
774:
770:
766:
761:
757:
753:
750:
729:
725:
722:
717:
713:
709:
706:
701:
697:
693:
688:
684:
674:inner product
659:
639:
619:
589:
584:
580:
556:
536:
516:
511:
507:
486:
466:
442:
437:
433:
412:
409:
406:
382:
362:
359:
357:
354:
291:local isometry
255:
252:
87:
26:
9:
6:
4:
3:
2:
20809:
20798:
20795:
20793:
20790:
20788:
20785:
20784:
20782:
20767:
20764:
20762:
20759:
20757:
20754:
20752:
20749:
20748:
20746:
20742:
20736:
20733:
20731:
20728:
20726:
20723:
20722:
20720:
20716:
20710:
20709:Schur's lemma
20707:
20705:
20702:
20700:
20697:
20695:
20692:
20690:
20687:
20685:
20684:Gauss's lemma
20682:
20680:
20677:
20676:
20674:
20670:
20664:
20661:
20659:
20656:
20654:
20651:
20649:
20646:
20645:
20643:
20639:
20633:
20630:
20628:
20625:
20623:
20620:
20618:
20615:
20611:
20607:
20604:
20602:
20599:
20597:
20594:
20592:
20589:
20588:
20587:
20586:Metric tensor
20584:
20582:
20581:Inner product
20579:
20577:
20574:
20572:
20569:
20565:
20562:
20560:
20557:
20555:
20552:
20550:
20547:
20546:
20545:
20542:
20541:
20539:
20535:
20530:
20526:
20519:
20514:
20512:
20507:
20505:
20500:
20499:
20496:
20484:
20481:
20479:
20478:Supermanifold
20476:
20474:
20471:
20469:
20466:
20462:
20459:
20458:
20457:
20454:
20452:
20449:
20447:
20444:
20442:
20439:
20437:
20434:
20432:
20429:
20427:
20424:
20423:
20421:
20417:
20411:
20408:
20406:
20403:
20401:
20398:
20396:
20393:
20391:
20388:
20386:
20383:
20382:
20380:
20376:
20366:
20363:
20361:
20358:
20356:
20353:
20351:
20348:
20346:
20343:
20341:
20338:
20336:
20333:
20331:
20328:
20326:
20323:
20321:
20318:
20317:
20315:
20313:
20309:
20303:
20300:
20298:
20295:
20293:
20290:
20288:
20285:
20283:
20280:
20278:
20275:
20273:
20269:
20265:
20263:
20260:
20258:
20255:
20253:
20249:
20245:
20243:
20240:
20238:
20235:
20233:
20230:
20228:
20225:
20223:
20220:
20218:
20215:
20214:
20212:
20210:
20206:
20200:
20199:Wedge product
20197:
20195:
20192:
20188:
20185:
20184:
20183:
20180:
20178:
20175:
20171:
20168:
20167:
20166:
20163:
20161:
20158:
20156:
20153:
20151:
20148:
20144:
20143:Vector-valued
20141:
20140:
20139:
20136:
20134:
20131:
20127:
20124:
20123:
20122:
20119:
20117:
20114:
20112:
20109:
20108:
20106:
20102:
20096:
20093:
20091:
20088:
20086:
20083:
20079:
20076:
20075:
20074:
20073:Tangent space
20071:
20069:
20066:
20064:
20061:
20059:
20056:
20055:
20053:
20049:
20046:
20044:
20040:
20034:
20031:
20029:
20025:
20021:
20019:
20016:
20014:
20010:
20006:
20002:
20000:
19997:
19995:
19992:
19990:
19987:
19985:
19982:
19980:
19977:
19975:
19972:
19970:
19967:
19963:
19960:
19959:
19958:
19955:
19953:
19950:
19948:
19945:
19943:
19940:
19938:
19935:
19933:
19930:
19928:
19925:
19923:
19920:
19918:
19915:
19913:
19910:
19908:
19904:
19900:
19898:
19894:
19890:
19888:
19885:
19884:
19882:
19876:
19870:
19867:
19865:
19862:
19860:
19857:
19855:
19852:
19850:
19847:
19845:
19842:
19838:
19837:in Lie theory
19835:
19834:
19833:
19830:
19828:
19825:
19821:
19818:
19817:
19816:
19813:
19811:
19808:
19807:
19805:
19803:
19799:
19793:
19790:
19788:
19785:
19783:
19780:
19778:
19775:
19773:
19770:
19768:
19765:
19763:
19760:
19758:
19755:
19753:
19750:
19749:
19747:
19744:
19740:Main results
19738:
19732:
19729:
19727:
19724:
19722:
19721:Tangent space
19719:
19717:
19714:
19712:
19709:
19707:
19704:
19702:
19699:
19697:
19694:
19690:
19687:
19685:
19682:
19681:
19680:
19677:
19673:
19670:
19669:
19668:
19665:
19664:
19662:
19658:
19653:
19649:
19642:
19637:
19635:
19630:
19628:
19623:
19622:
19619:
19612:
19608:
19607:
19602:
19597:
19596:
19585:
19581:
19577:
19573:
19568:
19563:
19559:
19555:
19554:
19549:
19545:
19541:
19537:
19533:
19529:
19525:
19520:
19515:
19511:
19507:
19502:
19498:
19494:
19490:
19486:
19481:
19476:
19472:
19468:
19463:
19459:
19455:
19451:
19447:
19442:
19438:
19434:
19430:
19426:
19422:
19416:
19412:
19408:
19404:
19400:
19396:
19392:
19388:
19384:
19380:
19376:
19372:
19366:
19362:
19358:
19354:
19350:
19346:
19341:
19337:
19331:
19327:
19323:
19319:
19315:
19311:
19307:
19303:
19299:
19296:
19292:
19291:
19286:
19282:
19278:
19274:
19270:
19266:
19262:
19258:
19256:0-8176-3898-9
19252:
19248:
19244:
19241:
19237:
19233:
19229:
19225:
19224:
19219:
19218:Gromov, Misha
19215:
19211:
19207:
19203:
19199:
19195:
19189:
19185:
19181:
19177:
19173:
19169:
19165:
19161:
19157:
19153:
19147:
19143:
19139:
19135:
19131:
19127:
19123:
19122:Cheeger, Jeff
19119:
19115:
19111:
19107:
19103:
19099:
19097:3-540-15279-2
19093:
19089:
19085:
19081:
19077:
19073:
19069:
19068:
19055:
19050:
19043:
19038:
19036:
19029:, Chapter 10.
19028:
19027:Petersen 2016
19023:
19021:
19019:
19012:, Section 7C.
19011:
19006:
19004:
18996:
18991:
18984:
18980:
18975:
18968:
18963:
18945:
18937:
18933:
18927:
18923:
18919:
18914:
18911:
18908:
18904:
18898:
18894:
18890:
18885:
18882:
18879:
18875:
18871:
18868:
18865:
18860:
18856:
18850:
18846:
18842:
18837:
18833:
18826:
18818:
18814:
18810:
18807:
18804:
18799:
18795:
18788:
18780:
18776:
18772:
18769:
18766:
18761:
18757:
18743:
18736:
18735:Petersen 2016
18731:
18724:
18723:Petersen 2016
18719:
18712:
18711:Petersen 2016
18708:
18703:
18696:
18691:
18684:
18679:
18672:
18667:
18660:
18655:
18653:
18645:
18640:
18633:
18628:
18621:
18616:
18609:
18604:
18597:
18592:
18585:
18580:
18573:
18572:do Carmo 1992
18568:
18561:
18556:
18549:
18544:
18537:
18532:
18525:
18520:
18518:
18516:
18508:
18503:
18496:
18491:
18484:
18479:
18472:
18467:
18460:
18455:
18448:
18443:
18436:
18431:
18425:, p. 39.
18424:
18419:
18413:, p. 11.
18412:
18407:
18401:, p. 20.
18400:
18395:
18393:
18386:, p. 16.
18385:
18380:
18374:, p. 15.
18373:
18368:
18366:
18358:
18353:
18347:, p. 31.
18346:
18341:
18335:, p. 30.
18334:
18329:
18327:
18320:, p. 12.
18319:
18314:
18312:
18305:, p. 26.
18304:
18299:
18293:, p. 13.
18292:
18287:
18285:
18278:, p. 38.
18277:
18276:do Carmo 1992
18272:
18270:
18263:, p. 37.
18262:
18261:do Carmo 1992
18257:
18255:
18247:
18246:do Carmo 1992
18242:
18238:
18222:
18219:
18217:
18214:
18212:
18209:
18207:
18204:
18202:
18201:Metric tensor
18199:
18197:
18194:
18192:
18189:
18187:
18184:
18182:
18179:
18177:
18174:
18173:
18166:
18152:
18143:
18139:
18123:
18119:
18095:
18092:
18089:
18078:
18074:
18048:
18042:
18039:
18017:
18013:
17989:
17986:
17983:
17958:
17954:
17933:
17925:
17908:
17904:
17883:
17863:
17855:
17854:
17853:
17851:
17847:
17844:, called the
17829:
17825:
17798:
17795:
17786:
17783:
17780:
17777:
17772:
17768:
17733:
17727:
17724:
17714:
17708:
17705:
17699:
17693:
17682:
17676:
17672:
17666:
17662:
17658:
17652:
17649:
17646:
17638:
17634:
17626:
17625:
17611:
17603:
17597:
17593:
17589:
17583:
17577:
17574:
17571:
17568:
17565:
17545:
17539:
17533:
17530:
17525:
17521:
17517:
17514:
17511:
17508:
17489:
17483:
17477:
17474:
17471:
17468:
17448:
17440:
17424:
17418:
17415:
17393:
17389:
17367:
17364:
17356:
17340:
17320:
17317:
17309:
17308:vector fields
17305:
17301:
17282:
17276:
17273:
17250:
17247:
17244:
17233:
17218:
17215:
17212:
17209:
17206:
17203:
17200:
17177:
17174:
17171:
17165:
17159:
17156:
17153:
17145:
17141:
17120:
17117:
17112:
17108:
17087:
17067:
17064:
17061:
17058:
17050:
17049:Hilbert space
17027:
17024:
17021:
17014:
17011:
17000:
16999:
16978:
16958:
16938:
16933:
16929:
16906:
16902:
16881:
16873:
16869:
16855:
16852:
16847:
16843:
16814:
16809:
16805:
16801:
16798:
16793:
16789:
16785:
16780:
16776:
16755:
16752:
16749:
16729:
16718:
16715:
16712:
16709:
16706:
16703:
16700:
16680:
16672:
16668:
16667:
16666:
16658:
16656:
16652:
16648:
16644:
16628:
16623:
16601:
16598:
16590:
16580:
16576:
16570:
16569:
16563:
16559:
16555:
16550:
16541:
16540:
16532:
16530:
16526:
16518:
16514:
16510:
16509:Marcel Berger
16506:
16502:
16498:
16494:
16489:
16487:
16483:
16479:
16475:
16471:
16466:
16464:
16460:
16456:
16452:
16448:
16444:
16440:
16436:
16428:
16424:
16408:
16407:
16400:
16396:
16388:
16378:
16350:
16335:
16331:
16327:
16322:
16310:
16306:
16302:
16285:
16281:
16277:
16272:
16257:
16250:
16242:
16229:
16227:
16223:
16222:group actions
16195:
16194:
16187:
16183:
16171:
16169:
16165:
16161:
16156:
16152:
16148:
16143:
16141:
16133:
16125:
16119:
16115:
16109:
16107:
16102:
16092:
16088:
16081:
16077:
16053:
16044:
16034:
16031:
16027:
16022:
16018:
16015:
16009:
15999:
15996:
15992:
15987:
15983:
15975:
15971:
15967:
15961:
15958:
15955:
15947:
15943:
15935:
15934:
15933:
15925:
15921:
15916:
15912:
15893:
15891:
15887:
15881:
15877:
15873:
15869:
15865:
15859:
15857:
15853:
15849:
15845:
15836:
15832:
15828:
15816:
15808:
15801:
15795:
15790:
15782:
15778:
15773:
15770:
15754:
15753:
15748:
15744:
15740:
15739:
15734:
15730:
15726:
15725:
15719:
15713:
15685:
15672:
15667:
15663:
15659:
15656:
15653:
15648:
15643:
15639:
15630:
15627:
15622:
15619:
15609:
15604:
15600:
15596:
15593:
15590:
15587:
15582:
15577:
15573:
15569:
15559:
15558:
15557:
15555:
15551:
15542:
15538:
15534:
15529:
15523:
15519:
15515:
15510:
15506:
15501:
15482:
15474:
15471:
15467:
15461:
15458:
15454:
15450:
15445:
15442:
15438:
15432:
15429:
15425:
15418:
15415:
15410:
15407:
15404:
15401:
15397:
15389:
15388:
15387:
15385:
15384:metric tensor
15381:
15373:
15366:
15365:
15353:
15343:
15341:
15337:
15321:
15313:
15294:
15274:
15271:
15268:
15265:
15262:
15259:
15252:
15251:
15250:
15249:
15245:
15242:is called an
15229:
15221:
15217:
15201:
15198:
15195:
15185:
15175:
15161:
15135:
15129:
15126:
15123:
15117:
15111:
15105:
15102:
15099:
15093:
15090:
15087:
15081:
15078:
15075:
15068:
15067:
15066:
15064:
15063:
15046:
15016:
15006:
14989:
14986:
14983:
14972:
14953:
14950:
14947:
14921:
14913:
14910:
14907:
14896:
14893:
14888:
14878:
14870:
14867:
14862:
14852:
14844:
14841:
14835:
14832:
14829:
14823:
14816:
14815:
14814:
14797:
14778:
14765:
14759:
14746:
14740:
14727:
14724:
14716:
14715:
14698:
14669:
14664:
14654:
14652:
14648:
14623:
14619:
14615:
14612:
14607:
14603:
14599:
14588:
14565:
14561:
14557:
14552:
14548:
14544:
14539:
14535:
14531:
14520:
14506:
14490:
14486:
14482:
14479:
14474:
14470:
14466:
14444:
14440:
14436:
14430:
14426:
14422:
14417:
14413:
14409:
14406:
14401:
14397:
14393:
14390:
14385:
14381:
14377:
14369:
14350:
14347:
14344:
14338:
14333:
14319:
14314:
14300:
14280:
14260:
14240:
14220:
14215:
14211:
14190:
14185:
14181:
14177:
14174:
14154:
14151:
14145:
14139:
14119:
14116:
14110:
14104:
14084:
14075:
14072:
14069:
14063:
14060:
14040:
14037:
14034:
14029:
14025:
14016:
14003:
13982:
13962:
13953:
13951:
13942:
13937:
13912:
13905:
13890:
13882:
13879:
13861:
13857:
13848:
13827:
13823:
13819:
13816:
13805:
13804:
13803:
13786:
13783:
13780:
13769:
13765:
13763:
13758:
13719:
13716:
13713:
13707:
13704:
13681:
13678:
13675:
13669:
13666:
13644:
13611:
13608:
13605:
13596:
13591:
13577:
13559:
13526:
13520:
13509:
13508:
13491:
13442:
13439:
13436:
13430:
13427:
13404:
13401:
13398:
13392:
13389:
13363:
13360:
13357:
13343:
13327:
13322:
13309:
13308:great circles
13291:
13287:
13278:
13261:
13246:
13245:
13239:
13225:
13205:
13196:
13193:
13190:
13184:
13181:
13172:
13158:
13155:
13149:
13142:
13139:
13118:
13115:
13109:
13103:
13095:
13079:
13076:
13070:
13063:
13060:
13039:
13036:
13030:
13024:
13004:
12984:
12981:
12975:
12968:
12965:
12944:
12941:
12935:
12929:
12909:
12889:
12883:
12880:
12877:
12857:
12852:
12848:
12844:
12841:
12821:
12818:
12815:
12806:
12804:
12788:
12768:
12748:
12745:
12741:
12738:
12732:
12728:
12707:
12686:
12683:
12677:
12673:
12664:
12651:
12629:
12620:
12617:
12614:
12608:
12605:
12585:
12556:
12553:
12544:
12543:great circles
12526:
12522:
12499:
12478:
12469:
12459:
12428:
12417:
12410:
12407:
12398:
12392:
12386:
12381:
12377:
12356:
12330:
12319:
12305:
12302:
12297:
12293:
12288:
12285:
12282:
12278:
12275:
12271:
12265:
12262:
12254:
12250:
12242:
12228:
12225:
12220:
12216:
12211:
12208:
12205:
12200:
12196:
12191:
12188:
12182:
12179:
12176:
12173:
12170:
12162:
12158:
12150:
12149:
12148:
12147:, such that:
12146:
12133:
12124:, called the
12108:
12089:
12076:
12071:
12067:
12046:
12037:
12034:
12031:
12025:
12022:
12002:
11973:
11971:
11958:
11951:extension of
11949:is called an
11930:
11902:
11896:
11886:
11879:
11873:
11867:
11847:
11821:
11798:
11778:
11755:
11749:
11736:
11733:
11710:
11704:
11698:
11692:
11689:
11683:
11674:
11671:
11661:
11660:
11659:
11634:
11631:
11628:
11622:
11619:
11599:
11579:
11556:
11523:
11520:
11517:
11511:
11508:
11488:
11480:
11474:
11470:
11466:
11460:
11454:
11434:
11431:
11422:
11419:
11416:
11410:
11407:
11399:
11386:
11364:
11355:
11352:
11349:
11343:
11340:
11326:
11312:
11303:
11298:
11296:
11277:
11274:
11271:
11245:
11239:
11236:
11233:
11227:
11224:
11219:
11211:
11208:
11203:
11191:
11190:
11189:
11187:
11164:A connection
11145:
11140:
11132:
11129:
11123:
11120:
11114:
11111:
11108:
11103:
11092:
11089:
11076:
11073:
11070:
11064:
11054:
11047:
11046:
11045:
11043:
11020:A connection
11018:
11016:
11010:
11000:
10998:
10985:
10965:
10943:
10938:
10909:
10904:
10895:
10892:
10889:
10883:
10877:
10874:
10871:
10868:
10863:
10850:
10836:
10833:
10826:
10822:
10810:
10806:
10802:
10799:
10792:
10788:
10776:
10772:
10768:
10765:
10758:
10754:
10748:
10744:
10740:
10735:
10731:
10725:
10721:
10692:
10680:
10676:
10673:
10665:
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10486:vector fields
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6029:
6026:
6023:
5997:
5994:
5968:
5965:
5962:
5952:in the chart
5939:
5917:
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5889:
5886:
5883:
5873:in the chart
5860:
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5127:
5113:
5099:
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5090:
5086:
5077:
5066:
5065:
5060:
5059:
5055:is called an
5042:
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5004:
4981:
4969:
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4947:
4924:
4916:
4897:
4894:
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3336:
3323:
3320:
3316:
3305:
3304:
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3298:
3280:
3277:
3273:
3250:
3247:
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3229:
3226:
3222:
3214:
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3193:
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3117:
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3115:
3096:
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3086:
3082:
3076:
3072:
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3064:
3055:
3051:
3036:
3032:
3026:
3022:
3018:
3010:
3006:
2991:
2987:
2981:
2977:
2972:
2962:
2954:
2953:
2952:
2932:
2924:
2908:
2903:
2874:
2870:
2866:
2863:
2860:
2855:
2851:
2831:
2815:
2811:
2807:
2802:
2798:
2789:
2776:
2766:
2750:
2730:
2722:
2718:
2700:
2696:
2692:
2685:
2671:
2663:has a unique
2647:
2644:
2641:
2618:
2604:
2587:
2581:
2575:
2572:
2569:
2549:
2529:
2526:
2523:
2515:
2499:
2496:
2490:
2487:
2484:
2475:
2461:
2458:
2453:
2449:
2445:
2442:
2439:
2436:
2416:
2413:
2410:
2381:
2373:
2369:
2365:
2362:
2356:
2348:
2344:
2340:
2329:
2323:
2319:
2315:
2309:
2306:
2303:
2295:
2291:
2283:
2282:
2281:
2267:
2262:
2258:
2254:
2251:
2243:
2240:is called an
2227:
2221:
2218:
2215:
2208:
2189:
2186:
2183:
2157:
2154:
2151:
2139:
2137:
2127:
2113:
2108:
2104:
2083:
2080:
2072:
2050:
2047:
2044:
2036:
2032:
2022:
2019:
2016:
2006:
2005:
2004:
1990:
1983:. Namely, if
1982:
1978:
1959:
1956:
1953:
1944:
1936:
1930:
1920:
1906:
1898:
1893:
1891:
1887:
1869:
1866:
1862:
1853:
1837:
1817:
1811:
1808:
1805:
1774:
1771:
1766:
1763:
1759:
1750:
1734:
1706:
1701:
1697:
1693:
1690:
1685:
1681:
1677:
1671:
1668:
1664:
1658:
1655:
1652:
1648:
1644:
1641:
1634:
1633:
1632:
1613:
1609:
1605:
1602:
1599:
1596:
1591:
1587:
1583:
1573:
1569:
1564:
1550:
1542:
1539:
1521:
1517:
1496:
1476:
1473:
1470:
1442:
1439:
1434:
1431:
1427:
1404:
1400:
1374:
1368:
1363:
1355:
1351:
1334:
1329:
1324:
1316:
1312:
1294:
1288:
1284:
1280:
1275:
1263:
1260:
1256:
1248:
1247:
1246:
1232:
1212:
1209:
1206:
1186:
1181:
1177:
1152:
1146:
1129:
1125:
1112:
1109:
1106:
1101:
1084:
1080:
1066:
1058:
1057:
1056:
1042:
1034:
1016:
1003:
1000:
992:
988:
984:
981:
978:
973:
969:
951:
949:
944:
942:
941:metric tensor
923:
920:
917:
906:
890:
870:
845:
842:
839:
831:
827:
821:
816:
808:
777:
772:
768:
764:
759:
751:
720:
715:
711:
707:
704:
699:
695:
691:
686:
682:
673:
657:
637:
617:
610:
605:
603:
602:inner product
587:
582:
578:
568:
554:
534:
514:
509:
505:
497:. Vectors in
484:
464:
456:
455:tangent space
440:
435:
431:
410:
407:
404:
396:
380:
367:
353:
351:
347:
343:
339:
335:
331:
329:
325:
321:
317:
313:
309:
307:
303:
299:
294:
292:
287:
285:
281:
277:
273:
265:
260:
251:
249:
245:
241:
237:
233:
229:
225:
221:
217:
213:
209:
205:
201:
197:
195:
191:
187:
183:
182:group actions
179:
175:
171:
166:
162:
158:
157:ambient space
153:
150:
146:
145:tangent space
142:
141:inner product
138:
134:
130:
125:
123:
119:
115:
111:
107:
103:
99:
85:
76:
72:
68:
64:
56:
55:tangent plane
52:
48:
44:
39:
33:
19:
20744:Applications
20672:Main results
20621:
20405:Moving frame
20400:Morse theory
20390:Gauge theory
20182:Tensor field
20111:Closed/Exact
20090:Vector field
20058:Distribution
20012:
19999:Hypercomplex
19994:Quaternionic
19731:Vector field
19700:
19689:Smooth atlas
19604:
19557:
19551:
19544:Milnor, John
19519:math/0409303
19509:
19505:
19470:
19466:
19449:
19445:
19398:
19344:
19321:
19318:Lee, John M.
19289:
19222:
19179:
19129:
19075:
19049:
18990:
18974:
18962:
18742:
18730:
18718:
18702:
18690:
18678:
18666:
18661:, Chapter 2.
18639:
18627:
18615:
18603:
18591:
18579:
18567:
18555:
18543:
18531:
18502:
18490:
18478:
18466:
18454:
18442:
18430:
18418:
18406:
18379:
18352:
18340:
18298:
18241:
18144:
18140:
18076:
18075:
18072:
17850:pseudometric
17845:
17759:
17381:
16871:
16670:
16664:
16608:
16593:
16584:
16573:Please help
16565:
16492:
16490:
16478:Cayley plane
16467:
16454:
16435:identity map
16427:differential
16405:
16398:
16394:
16390:
16351:
16333:
16329:
16325:
16320:
16308:
16304:
16283:
16279:
16275:
16270:
16230:
16192:
16185:
16181:
16177:
16144:
16140:bi-invariant
16139:
16117:
16113:
16110:
16105:
16100:
16090:
16086:
16079:
16075:
16068:
15923:
15919:
15904:
15879:
15868:Möbius strip
15864:Klein bottle
15860:
15844:group theory
15834:
15830:
15793:
15774:
15750:
15746:
15742:
15736:
15722:
15720:
15705:
15540:
15536:
15532:
15521:
15517:
15513:
15508:
15504:
15502:is given by
15497:
15362:
15360:
15311:
15309:
15247:
15243:
15187:
15153:
15060:
15018:
14936:
14712:
14670:
14666:
14650:
14646:
14644:
14315:
13995:
13954:
13947:
13903:are compact,
13767:
13766:
13759:
13545:. The plane
13505:
13483:
13174:Every curve
13173:
13093:
12807:
12802:
12801:is called a
12643:
12557:
12550:
12125:
11974:
11950:
11770:
11378:
11332:
11301:
11299:
11260:
11186:torsion-free
11185:
11163:
11041:
11019:
11012:
10957:
10925:
10593:
10509:
10450:
10442:
10421:
10418:
10238:
10087:
10084:
10012:metric space
9969:
9840:
9771:
9769:
9720:
9671:
9503:
9394:
9390:
9270:
8872:
8867:
8863:
8859:
8855:
8850:
8846:
8842:
8840:
8720:
8544:metric space
8501:
8500:
8286:
8284:
7910:
7908:
7675:
7299:
7226:
7109:
6978:
6766:
6738:
6733:
6732:
6501:
6402:
6265:
5156:
5062:
5056:
4973:
4515:
4512:round metric
4511:
4283:
4195:
3917:is given by
3858:
3821:
3612:
3511:Submanifolds
3504:
3454:
3301:
3210:
3113:
2951:is given by
2922:
2842:
2768:
2716:
2611:An oriented
2610:
2513:
2476:
2402:
2241:
2140:
2135:
2133:
2068:
1932:
1894:
1851:
1748:
1726:
1565:
1391:
1168:
957:
948:metric space
945:
904:
608:
606:
569:
372:
332:
326:defined the
310:
306:Hermann Weyl
295:
288:
269:
224:gauge theory
218:(especially
198:
154:
132:
128:
127:Formally, a
126:
66:
60:
20350:Levi-Civita
20340:Generalized
20312:Connections
20262:Lie algebra
20194:Volume form
20095:Vector flow
20068:Pushforward
20063:Lie bracket
19962:Lie algebra
19927:G-structure
19716:Pushforward
19696:Submanifold
19512:: 217–245.
19452:: 273–296.
18983:Milnor 1976
17355:volume form
17133:The metric
16661:Definitions
16579:introducing
16493:irreducible
16459:Élie Cartan
16423:fixed point
16301:coset space
16256:Lie algebra
16193:homogeneous
16132:Lie algebra
15852:lens spaces
15825:, with the
14813:defined by
14717:is the map
13880:converges),
11295:Lie bracket
11040:is said to
10433:Connections
6746:paracompact
4237:pushforward
2715:called the
1031:are smooth
798:defined by
453:called the
324:Levi-Civita
312:Élie Cartan
236:cartography
161:submanifold
131:(or just a
114:paraboloids
43:dot product
20781:Categories
20704:Ricci flow
20653:Hyperbolic
20473:Stratifold
20431:Diffeology
20227:Associated
20028:Symplectic
20013:Riemannian
19942:Hyperbolic
19869:Submersion
19777:Hopf–Rinow
19711:Submersion
19706:Smooth map
19584:0341.53030
19480:1910.06430
19437:1216.53003
19387:1417.53001
19310:0119.37502
19273:0953.53002
19210:0752.53001
19168:1142.53003
19114:0613.53001
19010:Besse 1987
18228:References
17441:, denoted
16247:, fix any
16155:collapsing
16130:, and the
15815:eigenvalue
15809:possesses
15775:Using the
12808:For every
11860:such that
11539:. The set
11447:such that
10662:such that
9423:such that
9121:such that
8546:, and the
6404:flat torus
5149:flat torus
4284:Examples:
3885:, the map
3503:is called
2769:volume of
2562:such that
2130:Isometries
1890:measurable
1749:continuous
1572:dual basis
1543:matrix at
1419:functions
907:, denoted
356:Definition
320:connection
174:Lie groups
110:ellipsoids
20648:Hermitian
20601:Signature
20564:Sectional
20544:Curvature
20355:Principal
20330:Ehresmann
20287:Subbundle
20277:Principal
20252:Fibration
20232:Cotangent
20104:Covectors
19957:Lie group
19937:Hermitian
19880:manifolds
19849:Immersion
19844:Foliation
19782:Noether's
19767:Frobenius
19762:De Rham's
19757:Darboux's
19648:Manifolds
19611:EMS Press
19497:204578276
19458:1076-9803
19236:S. Semmes
18912:−
18883:−
18869:…
18808:…
18789:⋅
18770:…
18683:Wolf 2011
18671:Wolf 2011
18659:Wolf 2011
18043:
17802:∞
17790:→
17784:×
17728:μ
17663:∫
17590:∈
17569:∈
17558:Then for
17534:
17518:∈
17478:
17472:∈
17419:
17341:μ
17304:Lie group
17277:
17216:∈
17181:⟩
17169:⟨
17062:∈
17032:⟩
17028:⋅
17022:⋅
17018:⟨
16818:→
16802:×
16753:∈
16722:→
16713:×
16587:July 2024
16406:symmetric
16032:−
15997:−
15911:Lie group
15729:connected
15657:⋯
15628:κ
15591:⋯
15451:−
15419:κ
15370:if every
15338:with the
15295:λ
15272:λ
15115:↦
15106:
15027:∇
14901:∇
14897:−
14885:∇
14875:∇
14871:−
14859:∇
14849:∇
14785:→
14766:×
14747:×
14679:∇
14620:θ
14562:θ
14487:θ
14441:θ
14339:∖
14281:γ
14241:γ
14178:∈
14140:γ
14105:γ
14082:→
14061:γ
14004:γ
13983:γ
13597:∖
13530:∞
13524:∞
13521:−
13349:∖
13226:γ
13203:→
13182:γ
13140:γ
13104:γ
13061:γ
13025:γ
12966:γ
12930:γ
12887:→
12878:γ
12845:∈
12819:∈
12789:γ
12739:γ
12708:γ
12684:γ
12652:γ
12627:→
12606:γ
12566:∇
12552:Geodesics
12452:Geodesics
12432:~
12408:γ
12403:∇
12334:~
12134:γ
12109:γ
12096:→
12090:γ
12044:→
12023:γ
11983:∇
11934:~
11897:γ
11890:~
11848:γ
11825:~
11799:γ
11750:γ
11737:∈
11641:→
11600:γ
11580:γ
11557:γ
11512:∈
11475:γ
11467:∈
11429:→
11387:γ
11362:→
11341:γ
11278:⋅
11272:⋅
11216:∇
11212:−
11200:∇
11172:∇
11137:∇
11100:∇
11028:∇
10935:∇
10901:∇
10860:∇
10818:∇
10784:∇
10717:∇
10685:∞
10677:∈
10641:∇
10637:↦
10563:→
10544:×
10522:∇
10355:→
10349:×
10215:∈
10144:
9918:→
9912:×
9748:λ
9743:≥
9737:γ
9695:⊂
9686:∂
9647:‖
9631:γ
9627:‖
9622:δ
9613:∫
9607:λ
9602:≥
9596:γ
9561:δ
9532:γ
9512:γ
9484:∂
9481:∈
9475:δ
9469:γ
9443:∉
9437:δ
9431:γ
9405:δ
9374:→
9353:γ
9318:⊂
9257:‖
9254:⋅
9251:‖
9215:∈
9189:∈
9160:‖
9153:‖
9150:λ
9147:≥
9109:λ
9084:¯
9033:⊂
9028:¯
8980:∋
8951:∉
8775:≠
8465:γ
8444:γ
8436:γ
8427:γ
8361:∞
8349:→
8343:×
8249:γ
8245:‖
8228:γ
8224:‖
8210:∫
8200:γ
8168:→
8147:γ
8124:γ
8055:γ
8051:‖
8034:γ
8030:‖
8027:↦
7990:γ
7982:∈
7966:γ
7942:→
7921:γ
7800:for some
7776:→
7680:to embed
7634:α
7607:α
7599:⊆
7591:α
7587:τ
7580:
7535:α
7528:~
7518:⋅
7513:α
7509:τ
7486:α
7459:α
7452:~
7423:α
7419:φ
7374:∗
7369:α
7365:φ
7270:∗
7265:α
7261:φ
7252:α
7245:~
7210:α
7203:~
7193:⋅
7188:α
7184:τ
7175:∈
7172:α
7168:∑
7092:∈
7089:α
7067:α
7059:⊆
7051:α
7047:τ
7040:
7008:∈
7005:α
6995:α
6991:τ
6952:⊆
6944:α
6931:α
6927:φ
6923:→
6918:α
6910::
6905:α
6901:φ
6877:⊆
6872:α
6835:∈
6832:α
6819:α
6815:φ
6806:α
6742:Hausdorff
6668:…
6592:…
6523:…
6426:×
6423:⋯
6420:×
6325:×
6322:⋯
6319:×
6178:~
6108:~
6089:∑
6079:~
6027:×
5998:~
5808:×
5761:×
5458:~
5413:⊕
5397:≅
5388:×
5330:×
5304:~
5235:×
5095:∗
5081:~
5040:→
5008:~
4951:~
4898:~
4869:→
4863:~
4762:→
4576:∈
4557:ellipsoid
4389:⋯
4339:∈
4068:∗
4035:∗
3965:∗
3902:→
3870:⊆
3800:∗
3721:∗
3662:→
3424:⋯
3407:⋮
3402:⋱
3397:⋮
3392:⋮
3380:⋯
3358:⋯
3274:δ
3244:δ
3167:⋯
3073:∑
3048:∂
3044:∂
3023:∑
3003:∂
2999:∂
2978:∑
2864:…
2799:∫
2579:→
2527:∈
2516:if every
2494:→
2446:∈
2414:∈
2263:∗
2225:→
2136:isometric
2109:∗
2051:⋅
2029:↦
1963:⟩
1960:⋅
1951:⟨
1948:↦
1937:given by
1886:Lipschitz
1778:→
1691:⊗
1649:∑
1600:…
1538:symmetric
1474:×
1446:→
1348:∂
1344:∂
1309:∂
1305:∂
1210:∈
1122:∂
1118:∂
1110:…
1077:∂
1073:∂
1007:→
982:…
813:‖
806:‖
781:→
756:‖
752:⋅
749:‖
724:→
708:×
570:However,
408:∈
350:spacetime
270:In 1827,
165:John Nash
143:for each
118:manifolds
20663:Kenmotsu
20576:Geodesic
20529:Glossary
20451:Orbifold
20446:K-theory
20436:Diffiety
20160:Pullback
19974:Oriented
19952:Kenmotsu
19932:Hadamard
19878:Types of
19827:Geodesic
19652:Glossary
19546:(1976).
19397:(2011).
19320:(2018).
19287:(1963).
19232:P. Pansu
19220:(1999).
19178:(1992).
19128:(2008).
19074:(1987).
18979:Lee 2018
18967:Lee 2018
18644:Lee 2018
18632:Lee 2018
18620:Lee 2018
18608:Lee 2018
18596:Lee 2018
18584:Lee 2018
18560:Lee 2018
18548:Lee 2018
18536:Lee 2018
18524:Lee 2018
18507:Lee 2018
18495:Lee 2018
18483:Lee 2018
18471:Lee 2018
18459:Lee 2018
18447:Lee 2018
18423:Lee 2018
18411:Lee 2018
18399:Lee 2018
18384:Lee 2018
18372:Lee 2018
18357:Lee 2018
18345:Lee 2018
18333:Lee 2018
18318:Lee 2018
18303:Lee 2018
18291:Lee 2018
18169:See also
17300:see here
17193:for all
16996:Examples
16525:geometry
16447:parallel
16166:with an
16160:Einstein
15876:cylinder
15854:and the
15800:subgroup
15544:, where
15526:and the
13847:complete
13768:Theorem:
13242:Examples
13143:′
13064:′
12969:′
12803:geodesic
12761:for all
12742:′
12687:′
12458:Geodesic
12411:′
12279:′
11501:for all
10284:complete
10088:diameter
10081:Diameter
10014:and the
9634:′
9461:clearly
9204:and any
9178:for any
8790:implies
8502:Theorem:
8231:′
8037:′
7969:′
7822:pullback
7437:. While
7408:pullback
7081:for all
6859:so that
6734:Theorem:
5905:and let
5116:Products
4884:, where
3707:pullback
2834:Examples
2403:for all
2242:isometry
1979:and the
1199:for any
20730:Hilbert
20725:Finsler
20395:History
20378:Related
20292:Tangent
20270:)
20250:)
20217:Adjoint
20209:Bundles
20187:density
20085:Torsion
20051:Vectors
20043:Tensors
20026:)
20011:)
20007:,
20005:Pseudo−
19984:Poisson
19917:Finsler
19912:Fibered
19907:Contact
19905:)
19897:Complex
19895:)
19864:Section
19576:0425012
19429:2742530
19379:3469435
19302:0152974
19265:1699320
19228:M. Katz
19202:1138207
19160:2394158
19106:0867684
19064:Sources
18077:Theorem
17896:, then
16647:Fréchet
16575:improve
16433:is the
16295:, then
16254:of the
15518:κg
15220:gravity
14969:is the
13849:(every
12369:, then
11920:, then
11293:is the
9273:denote
7406:is its
6637:, then
5747:, then
4913:is the
4309:-sphere
4235:is the
3546:-sphere
3295:is the
2765:compact
2721:measure
254:History
216:physics
135:) on a
98:-sphere
20658:Kähler
20554:Scalar
20549:tensor
20360:Vector
20345:Koszul
20325:Cartan
20320:Affine
20302:Vector
20297:Tensor
20282:Spinor
20272:Normal
20268:Stable
20222:Affine
20126:bundle
20078:bundle
20024:Almost
19947:Kähler
19903:Almost
19893:Almost
19887:Closed
19787:Sard's
19743:(list)
19582:
19574:
19534:
19495:
19456:
19435:
19427:
19417:
19385:
19377:
19367:
19332:
19308:
19300:
19271:
19263:
19253:
19234:, and
19208:
19200:
19190:
19166:
19158:
19148:
19112:
19104:
19094:
18079:: Let
16653:, and
16651:Banach
16527:, and
16476:, and
16122:, the
15874:, the
15870:, the
15866:, the
15813:as an
15763:, and
15716:κ
15708:κ
15541:κ
15376:κ
15368:κ
15154:where
15059:. The
14937:where
14711:. The
14233:along
12700:along
12598:. Let
11261:where
10922:holds.
9243:where
7569:since
7410:along
7227:where
6287:-torus
6160:where
4555:. The
4196:where
3778:, and
3705:. The
3681:or an
3677:be an
3265:where
2767:, the
2607:Volume
1852:smooth
1392:These
246:, and
234:, and
210:, and
133:metric
104:, and
77:, the
47:sphere
20559:Ricci
20468:Sheaf
20242:Fiber
20018:Rizza
19989:Prime
19820:Local
19810:Curve
19672:Atlas
19536:69260
19532:S2CID
19514:arXiv
19493:S2CID
19475:arXiv
18233:Notes
17353:be a
17100:with
17047:is a
16951:. If
16560:, or
16421:as a
15909:be a
15872:torus
14647:along
14097:with
12720:. If
11811:. If
10508:. An
10010:is a
9389:from
9015:with
8542:is a
7300:Here
6857:atlas
5145:torus
5067:) if
4734:graph
2684:-form
903:is a
393:be a
69:is a
20335:Form
20237:Dual
20170:flow
20033:Tame
20009:Sub−
19922:Flat
19802:Maps
19454:ISSN
19415:ISBN
19365:ISBN
19330:ISBN
19251:ISBN
19188:ISBN
19146:ISBN
19092:ISBN
18040:Diff
17531:Diff
17475:Diff
17416:Diff
17380:The
17333:Let
17274:Diff
17234:Let
16445:are
16441:and
16373:and
16200:and
15905:Let
15796:+ 1)
15731:and
15539:– 1)
15516:– 1)
14132:and
13770:Let
13760:The
13131:and
13052:and
12957:and
12834:and
11771:Let
11726:for
10451:Let
10239:The
10141:diam
10085:The
9408:>
9290:>
8969:Let
8940:and
8823:>
7577:supp
7356:and
7037:supp
6979:Let
6767:Let
6744:and
6502:Let
5826:Let
5695:and
5270:and
5189:and
5157:Let
5061:(or
4732:The
4288:The
3613:Let
3525:The
2843:Let
2429:and
2173:and
1935:dual
373:Let
222:and
188:and
176:and
112:and
65:, a
41:The
20257:Jet
19580:Zbl
19562:doi
19524:doi
19485:doi
19471:148
19433:Zbl
19407:doi
19383:Zbl
19357:doi
19306:Zbl
19269:Zbl
19243:doi
19206:Zbl
19164:Zbl
19138:doi
19110:Zbl
19084:doi
18145:If
17926:If
17856:If
17357:on
17310:on
17001:If
16874:on
16673:on
16429:at
16413:of
16341:of
16328:):
16291:of
16278:):
16258:of
16243:of
16216:to
16204:in
16126:of
16098:to
15882:× ℝ
15749:or
15530:is
15512:= (
15386:as
15246:if
15065:is
15039:on
14691:on
14017:if
13845:is
12578:on
12320:If
11995:on
11333:If
11188:if
11184:is
11044:if
10594:on
10488:on
10443:An
10422:any
10376:If
10282:is
10172:sup
10130:is
10018:on
9833:.
9579:So
9544:to
9393:to
9281:sup
8862:to
8849:to
8550:on
8415:inf
8376:by
7909:An
7873:is
7824:by
7313:can
7280:can
7150:by
7130:on
6752:.
6571:If
6063:is
5643:If
5319:on
4917:of
4817:If
4514:or
4259:by
4239:of
4052:is
3824:of
3738:of
3685:of
3486:can
3329:can
3235:can
3130:can
2967:can
2937:can
2790:is
2763:is
2723:on
2244:if
1850:is
1747:is
1245:by
1035:on
958:If
630:on
547:at
477:at
457:of
226:),
61:In
20783::
20248:Co
19609:,
19603:,
19578:.
19572:MR
19570:.
19558:21
19556:.
19550:.
19530:.
19522:.
19510:10
19508:.
19491:.
19483:.
19469:.
19450:21
19448:.
19431:.
19425:MR
19423:.
19413:.
19405:.
19381:.
19375:MR
19373:.
19363:.
19355:.
19347:.
19328:.
19324:.
19304:.
19298:MR
19283:;
19267:.
19261:MR
19259:.
19249:.
19230:,
19204:.
19198:MR
19196:.
19186:.
19162:.
19156:MR
19154:.
19144:.
19136:.
19124:;
19108:.
19102:MR
19100:.
19090:.
19082:.
19034:^
19017:^
19002:^
18651:^
18514:^
18391:^
18364:^
18325:^
18310:^
18283:^
18268:^
18253:^
17624:,
16870:A
16669:A
16657:.
16649:,
16564:,
16556:,
16519:,
16515:,
16397:,
16332:→
16318:ad
16282:→
16268:ad
16184:,
16170:.
16101:xy
16089:→
16073:,
15892:.
15858:.
15833:/
15792:O(
15765:–1
15759:,
15721:A
15522:jk
15509:jk
15342:.
15162:tr
15103:tr
14651:on
14313:.
13757:.
13697:,
13420:,
12805:.
12781:,
11972:.
11658::
11325:.
11300:A
11297:.
11017:.
10999:.
10708:,
10077:.
8609:.
7107:.
6852:a
5143:A
3856:.
3507:.
2830:.
2126:.
1563:.
1281::=
943:.
670:a
607:A
567:.
322:.
308:.
250:.
242:,
230:,
206:,
196:.
100:,
20608:/
20531:)
20527:(
20517:e
20510:t
20503:v
20266:(
20246:(
20022:(
20003:(
19901:(
19891:(
19654:)
19650:(
19640:e
19633:t
19626:v
19586:.
19564::
19538:.
19526::
19516::
19499:.
19487::
19477::
19460:.
19439:.
19409::
19389:.
19359::
19338:.
19312:.
19275:.
19245::
19212:.
19170:.
19140::
19116:.
19086::
19056:.
19044:.
18985:.
18946:.
18943:)
18938:n
18934:y
18928:n
18924:x
18920:,
18915:1
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5770:(
5767:,
5764:V
5758:U
5755:(
5735:N
5715:)
5712:y
5709:,
5706:V
5703:(
5683:M
5663:)
5660:x
5657:,
5654:U
5651:(
5627:.
5624:)
5619:2
5615:v
5611:,
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5598:(
5593:q
5589:h
5585:+
5582:)
5577:1
5573:v
5569:,
5564:1
5560:u
5556:(
5551:p
5547:g
5543:=
5540:)
5537:)
5532:2
5528:v
5524:,
5519:1
5515:v
5511:(
5508:,
5505:)
5500:2
5496:u
5492:,
5487:1
5483:u
5479:(
5476:(
5471:q
5468:,
5465:p
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5429:,
5426:N
5421:q
5417:T
5410:M
5405:p
5401:T
5394:)
5391:N
5385:M
5382:(
5377:)
5374:q
5371:,
5368:p
5365:(
5361:T
5336:,
5333:N
5327:M
5301:g
5278:h
5258:g
5238:N
5232:M
5209:)
5206:h
5203:,
5200:N
5197:(
5177:)
5174:g
5171:,
5168:M
5165:(
5100:g
5091:i
5087:=
5078:g
5043:M
5037:N
5034::
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5005:g
4982:N
4948:M
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4895:M
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4837:)
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4828:M
4825:(
4800:1
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4766:R
4757:n
4752:R
4747::
4744:f
4727:.
4713:3
4708:R
4679:}
4675:1
4672:=
4665:2
4661:c
4655:2
4651:z
4645:+
4638:2
4634:b
4628:2
4624:y
4618:+
4611:2
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4469:1
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4458:R
4430:}
4427:1
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4360::
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4330:=
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3928:(
3925:i
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3899:N
3896::
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3867:N
3844:)
3841:g
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3832:(
3808:)
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3789:N
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3766:N
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3726:g
3717:i
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3656::
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3633:)
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3621:(
3609:.
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3440:.
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3429:1
3419:0
3414:0
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3337:=
3334:)
3324:j
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3230:j
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3180:x
3176:d
3173:(
3170:+
3164:+
3159:2
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3141:d
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3135:=
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3093:b
3087:i
3083:a
3077:i
3069:=
3065:)
3056:j
3052:x
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2307:,
2304:u
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2252:g
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2222:M
2219::
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2190:h
2187:,
2184:N
2181:(
2161:)
2158:g
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2114:M
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2045:v
2042:(
2037:p
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2020:,
2017:p
2014:(
1991:g
1957:,
1954:v
1945:v
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1619:}
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381:M
266:.
86:n
34:.
20:)
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