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1179:. In Riemannian geometry, all geodesics are locally distance-minimizing paths, but the converse is not true. In fact, only paths that are both locally distance minimizing and parameterized proportionately to arc-length are geodesics. Another equivalent way of defining geodesics on a Riemannian manifold, is to define them as the minima of the following
2740:
of the curve has no components in the direction of the surface (and therefore it is perpendicular to the tangent plane of the surface at each point of the curve). So, the motion is completely determined by the bending of the surface. This is also the idea of general relativity where particles move on
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For example, when a ribbon is wound as a ring around a cone, the ribbon would not lie on the cone's surface but stick out, so that circle is not a geodesic on the cone. If the ribbon is adjusted so that all its parts touch the cone's surface, it would give an approximation to a geodesic.
351:|. Equivalently, a different quantity may be used, termed the energy of the curve; minimizing the energy leads to the same equations for a geodesic (here "constant velocity" is a consequence of minimization). Intuitively, one can understand this second formulation by noting that an
327:. This has some minor technical problems because there is an infinite-dimensional space of different ways to parameterize the shortest path. It is simpler to restrict the set of curves to those that are parameterized "with constant speed" 1, meaning that the distance from
1802:
4084:
A ribbon "test" is a way of finding a geodesic on a physical surface. The idea is to fit a bit of paper around a straight line (a ribbon) onto a curved surface as closely as possible without stretching or squishing the ribbon (without changing its internal geometry).
1978:
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1997:
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are constant real numbers. Thus apart from specifying a certain class of embedded curves, the geodesic equation also determines a preferred class of parameterizations on each of the curves. Accordingly, solutions of
358:
It is possible that several different curves between two points minimize the distance, as is the case for two diametrically opposite points on a sphere. In such a case, any of these curves is a geodesic.
2734:
2265:
3983:
1665:
3871:, Chapter 6, Addendum I). The torsion itself does not, in fact, affect the family of geodesics, since the geodesic equation depends only on the symmetric part of the connection. More precisely, if
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from the unit interval on the real number line to itself gives the shortest path between 0 and 1, but is not a geodesic because the velocity of the corresponding motion of a point is not constant.
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1591:, because they turn out to be affinely parameterized, and the inequality is an equality. The usefulness of this approach is that the problem of seeking minimizers of
365:
In general, geodesics are not the same as "shortest curves" between two points, though the two concepts are closely related. The difference is that geodesics are only
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5236:— Introduction to geodesics including two ways of derivation of the equation of geodesic with applications in geometry (geodesic on a sphere and on a
1988:
2564:{\displaystyle {\frac {d^{2}\gamma ^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {d\gamma ^{\mu }}{dt}}{\frac {d\gamma ^{\nu }}{dt}}=0\ ,}
2132:{\displaystyle \delta ^{2}E(\gamma )(\varphi ,\psi )=\left.{\frac {\partial ^{2}}{\partial s\,\partial t}}\right|_{s=t=0}E(\gamma +t\varphi +s\psi ).}
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stretched between two points will contract its width, and in so doing will minimize its energy. The resulting shape of the band is a geodesic.
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805:
This generalizes the notion of geodesic for
Riemannian manifolds. However, in metric geometry the geodesic considered is often equipped with
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have the same geodesics, with the same affine parameterizations. Furthermore, there is a unique connection having the same geodesics as
2676:
for the coordinates. It has a unique solution, given an initial position and an initial velocity. Therefore, from the point of view of
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5438:
5288:
444:
This article presents the mathematical formalism involved in defining, finding, and proving the existence of geodesics, in the case of
5686:
3631:
6792:
5881:
3630: \ {0}) it is enough that the connection be equivariant under positive rescalings: it need not be linear. That is, (cf.
4823:
1797:{\displaystyle {\frac {d^{2}x^{\lambda }}{dt^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{dt}}{\frac {dx^{\nu }}{dt}}=0,}
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Efficient solvers for the minimal geodesic problem on surfaces have been proposed by
Mitchell, Kimmel, Crane, and others.
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is formed by the geodesics joining each pair out of three points on a given surface. On the sphere, the geodesics are
441:
deals with the paths that objects may take when they are not free, and their movement is constrained in various ways.
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In general, a metric space may have no geodesics, except constant curves. At the other extreme, any two points in a
6857:
6079:
2636:
1973:{\displaystyle \delta E(\gamma )(\varphi )=\left.{\frac {\partial }{\partial t}}\right|_{t=0}E(\gamma +t\varphi ).}
1810:
369:
the shortest distance between points, and are parameterized with "constant speed". Going the "long way round" on a
1319:{\displaystyle E(\gamma )={\frac {1}{2}}\int _{a}^{b}g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))\,dt.}
7084:
6062:
5671:
2178:
1159:
of the length taken over all continuous, piecewise continuously differentiable curves γ : →
549:
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If an insect is placed on a surface and continually walks "forward", by definition it will trace out a geodesic.
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3132:
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2154:
540:
behave in a more complicated way than on a sphere; in particular, they are not closed in general (see figure).
3874:
2934:
2933:, by noticing that the geodesic equation is a second-order ODE. Existence and uniqueness then follow from the
1121:{\displaystyle L(\gamma )=\int _{a}^{b}{\sqrt {g_{\gamma (t)}({\dot {\gamma }}(t),{\dot {\gamma }}(t))}}\,dt.}
6708:
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6274:
5681:
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is equal to a constant a.e.; the path should be travelled at constant speed. It happens that minimizers of
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on the tangent bundle. For the resulting vector field to be a spray (on the deleted tangent bundle T
537:
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200:
geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's
32:
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under the influence of gravity alone. In particular, the path taken by a falling rock, an orbiting
373:
between two points on a sphere is a geodesic but not the shortest path between the points. The map
164:
4239:
3436:
7008:
6695:
6612:
6582:
5928:
5898:
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4698:
4612: – Analyzes the topology of a manifold by studying differentiable functions on that manifold
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Geodesic triangles in spaces of positive (top), negative (middle) and zero (bottom) curvature.
220:). The term has since been generalized to more abstract mathematical spaces; for example, in
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6483:
6428:
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A particular case of a non-linear connection arising in this manner is that associated to a
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6249:
6187:
6035:
5739:
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5701:
5676:
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5355:
5205:
4787:
4589: – Gives equivalent statements about the geodesic completeness of Riemannian manifolds
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4384:
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4159:
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5200:
Gravitation and
Cosmology: Principles and Applications of the General Theory of Relativity
8:
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4601: – Line along which a quadratic form applied to any two points' displacement is zero
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2150:
1843:
973:
957:
445:
410:
294:
240:
156:
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3796:) is invariant under affine reparameterizations; that is, parameterizations of the form
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6329:
6284:
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5856:
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2315:
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575:
489:
418:
271:
256:
244:
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5417:
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is then given by the inverse of the (pseudo-)Riemannian metric, evaluated against the
2142:
In an appropriate sense, zeros of the second variation along a geodesic γ arise along
38:
7028:
6797:
6772:
6587:
6498:
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6244:
6239:
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5871:
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3622:
More generally, the same construction allows one to construct a vector field for any
3472:
2977:
2754:
2408:
2195:
2158:
1345:
can be arbitrarily re-parameterized (without changing their length), while minima of
1184:
264:
248:
201:
19:
This article is about geodesics in general. For geodesics in general relativity, see
4545:); without GSP reconstruction often results in self-intersections within the surface
293:
A locally shortest path between two given points in a curved space, assumed to be a
7043:
6718:
6685:
6670:
6552:
6421:
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6196:
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5360:
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are generally not very regular, because arbitrary reparameterizations are allowed.
1180:
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63:
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205:
168:
243:
or submanifold, geodesics are characterised by the property of having vanishing
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148:
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42:
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to the space), and then minimizing this length between the points using the
182:
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2143:
795:{\displaystyle d(\gamma (t_{1}),\gamma (t_{2}))=v\left|t_{1}-t_{2}\right|.}
628:
571:
497:
370:
352:
221:
213:
209:
4819:
3258:
On a (pseudo-)Riemannian manifold, the geodesic flow is identified with a
915:{\displaystyle d(\gamma (t_{1}),\gamma (t_{2}))=\left|t_{1}-t_{2}\right|.}
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6167:
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for the solutions of ODEs with prescribed initial conditions. γ depends
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Mathematically the ribbon test can be formulated as finding a mapping
3463:
More precisely, an affine connection gives rise to a splitting of the
3427:
implies invariance of a kinematic measure on the unit tangent bundle.
3419:
remains unit speed throughout, so the geodesic flow is tangent to the
6802:
6753:
6206:
6171:
5876:
5763:
964:, although this minimizing sequence need not converge to a geodesic.
558:
430:
422:
279:
4859:
4755:
2146:. Jacobi fields are thus regarded as variations through geodesics.
460:
discusses the special case of general relativity in greater detail.
437:
are all geodesics in curved spacetime. More generally, the topic of
6832:
6370:
6365:
6355:
5567:
4064:
Geodesics without a particular parameterization are described by a
2395:
603:
298:
53:
3270:. In particular the flow preserves the (pseudo-)Riemannian metric
2332:. More precisely, in order to define the covariant derivative of
990:
of a continuously differentiable curve γ : →
581:
6526:
6488:
2753:
for geodesics states that geodesics on a smooth manifold with an
1156:
599:
196:, though many of the underlying principles can be applied to any
188:
24:
4902:
A Comprehensive introduction to differential geometry (Volume 2)
6852:
6444:
5962:
4692:
493:
2210:
along the curve preserves the tangent vector to the curve, so
114:
72:
5237:
3632:
Ehresmann connection#Vector bundles and covariant derivatives
2199:
607:
306:
193:
144:
2043:
1913:
102:
87:
78:
6413:
4735:
2729:{\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}
2260:{\displaystyle \nabla _{\dot {\gamma }}{\dot {\gamma }}=0}
69:
16:
Straight path on a curved surface or a
Riemannian manifold
4987:
Adler, Ronald; Bazin, Maurice; Schiffer, Menahem (1975),
4573: – Study of curves from a differential point of view
3978:{\displaystyle D(X,Y)=\nabla _{X}Y-{\bar {\nabla }}_{X}Y}
3526:
The geodesic spray is the unique horizontal vector field
1987:
of the first variation are precisely the geodesics. The
132:
120:
93:
4839:
3700:{\displaystyle H_{\lambda X}=d(S_{\lambda })_{X}H_{X}\,}
3634:) it is enough that the horizontal distribution satisfy
3246:. A closed orbit of the geodesic flow corresponds to a
362:
A contiguous segment of a geodesic is again a geodesic.
4566:
Pages displaying short descriptions of redirect targets
2622:{\displaystyle \gamma ^{\mu }=x^{\mu }\circ \gamma (t)}
4347:{\displaystyle g_{N}-f^{*}(g_{S})=O(\varepsilon ^{2})}
3863:
its family of affinely parameterized geodesics, up to
2929:
The proof of this theorem follows from the theory of
4557:
Introduction to the mathematics of general relativity
4438:
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4387:
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4262:
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The most familiar examples are the straight lines in
379:
135:
129:
123:
117:
96:
90:
66:
3906:
are two connections such that the difference tensor
3441:
The geodesic flow defines a family of curves in the
111:
84:
5119:
4736:Mitchell, J.; Mount, D.; Papadimitriou, C. (1987).
1478:{\displaystyle L(\gamma )^{2}\leq 2(b-a)E(\gamma )}
108:
81:
5283:
5244:) and optics (light beam in inhomogeneous medium).
5197:
5075:, vol. 1 (New ed.), Wiley-Interscience,
4986:
4564: – Formula in classical differential geometry
4444:
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4373:
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794:
398:
4840:Crane, K.; Weischedel, C.; Wardetzky, M. (2017).
2680:, geodesics can be thought of as trajectories of
192:, the science of measuring the size and shape of
7071:
3771:{\displaystyle S_{\lambda }:X\mapsto \lambda X.}
3076:{\displaystyle G^{t}(V)={\dot {\gamma }}_{V}(t)}
2741:geodesics and the bending is caused by gravity.
1866:can be applied to examine the energy functional
5070:
4779:Proceedings of the National Academy of Sciences
4521:mapping images on surfaces, for rendering; see
3718: \ {0} and λ > 0. Here
1595:is a more robust variational problem. Indeed,
1341:is a bigger set since paths that are minima of
5089:
5071:Kobayashi, Shoshichi; Nomizu, Katsumi (1996),
5013:
3378:{\displaystyle g(G^{t}(V),G^{t}(V))=g(V,V).\,}
147:representing in some sense the shortest path (
6429:
5552:
5269:
4768:
4537:geodesic shortest path (GSP) correction over
2661:{\displaystyle \Gamma _{\mu \nu }^{\lambda }}
2404:) is independent of the choice of extension.
1874:of energy is defined in local coordinates by
1835:{\displaystyle \Gamma _{\mu \nu }^{\lambda }}
167:. It is a generalization of the notion of a "
4662:The path is a local maximum of the interval
598:, a geodesic is a curve which is everywhere
409:Geodesics are commonly seen in the study of
251:, a geodesic is defined to be a curve whose
4515:horizontal distances on or near Earth; see
4495:Geodesics serve as the basis to calculate:
3445:. The derivatives of these curves define a
2744:
75:
6436:
6422:
5559:
5545:
5439:Fundamental theorem of Riemannian geometry
5276:
5262:
5046:Riemannian Geometry and Geometric Analysis
4842:"The Heat Method for Distance Computation"
925:If the last equality is satisfied for all
270:Geodesics are of particular importance in
4973:Learn how and when to remove this message
4809:
4799:
3696:
3566:
3374:
3173:{\displaystyle {\dot {\gamma }}_{V}(0)=V}
2849:
2803:
2064:
1857:
1306:
1108:
550:Gauss–Bonnet theorem § For triangles
6793:Covariance and contravariance of vectors
5566:
5192:
4936:This article includes a list of general
4641:
4639:
4618: – Surface homeomorphic to a sphere
4071:
3899:{\displaystyle \nabla ,{\bar {\nabla }}}
2757:exist, and are unique. More precisely:
2157:. They are solutions of the associated
2149:By applying variational techniques from
580:
557:
479:
467:
247:. More generally, in the presence of an
37:
4771:"Computing Geodesic Paths on Manifolds"
3129:denotes the geodesic with initial data
1659:are then given in local coordinates by
960:are joined by a minimizing sequence of
7072:
5173:
4896:
4624: – Recreational geodesics problem
3868:
2902:{\displaystyle {\dot {\gamma }}(0)=V,}
2751:local existence and uniqueness theorem
2283:at each point along the curve, where
967:
23:. For the study of Earth's shape, see
6417:
5540:
5257:
5150:
4653:, the definition is more complicated.
4636:
4531:(e.g., when painting car parts); see
4216:"doesn't change the distances around
1851:
452:discusses the more general case of a
5073:Foundations of Differential Geometry
5043:
4922:
4595: – Concept in geometry/topology
4459:
3453:of the tangent bundle, known as the
2684:in a manifold. Indeed, the equation
2398:. However, the resulting value of (
2212:
1526:{\displaystyle g(\gamma ',\gamma ')}
512:of the great circle passing through
508:on a sphere is given by the shorter
159:. The term also has meaning in any
27:. For the application on Earth, see
4769:Kimmel, R.; Sethian, J. A. (1998).
4236:by much"; that is, at the distance
3437:Spray (mathematics) § Geodesic
2960:as for example for an open disc in
2629:are the coordinates of the curve γ(
2172:
496:, the images of geodesics are the
47:(marked by 7 colors and 4 patterns)
13:
6656:Tensors in curvilinear coordinates
5170:. Note especially pages 7 and 10.
4989:Introduction to General Relativity
4942:it lacks sufficient corresponding
4918:
4904:, Houston, TX: Publish or Perish,
4509:geodesic structures – for example
4048:
4022:
3999:
3957:
3938:
3887:
3878:
3619:associated to the tangent bundle.
2692:
2641:
2478:
2312:is the derivative with respect to
2223:
2065:
2058:
2048:
1921:
1917:
1815:
1714:
589:
562:A geodesic triangle on the sphere.
14:
7096:
5227:
4577:Differential geometry of surfaces
3570:{\displaystyle \pi _{*}W_{v}=v\,}
3430:
3231:{\displaystyle G^{t}(V)=\exp(tV)}
2793:) there exists a unique geodesic
2672:of the connection ∇. This is an
2390:to a continuously differentiable
4927:
4829:from the original on 2022-10-09.
4463:
4034:{\displaystyle {\bar {\nabla }}}
2983:
2383:{\displaystyle {\dot {\gamma }}}
2361:it is necessary first to extend
2354:{\displaystyle {\dot {\gamma }}}
2305:{\displaystyle {\dot {\gamma }}}
474:geodesic on a triaxial ellipsoid
62:
4738:"The Discrete Geodesic Problem"
4455:
4126:{\displaystyle f:N(\ell )\to S}
3786:Affine and projective geodesics
2931:ordinary differential equations
2179:Geodesics in general relativity
500:. The shortest path from point
288:
276:geodesics in general relativity
259:along it. Applying this to the
5599:Differentiable/Smooth manifold
4874:
4833:
4762:
4729:
4722:Merriam-Webster.com Dictionary
4709:
4680:
4656:
4622:The spider and the fly problem
4539:Poisson surface reconstruction
4341:
4328:
4319:
4306:
4117:
4114:
4108:
4079:
4061:, but with vanishing torsion.
4025:
3960:
3931:
3919:
3890:
3809:
3756:
3677:
3663:
3516:{\displaystyle TTM=H\oplus V.}
3368:
3356:
3347:
3344:
3338:
3322:
3316:
3303:
3262:on the cotangent bundle. The
3225:
3216:
3204:
3198:
3161:
3155:
3070:
3064:
3039:
3033:
2887:
2881:
2853:{\displaystyle \gamma (0)=p\,}
2840:
2834:
2674:ordinary differential equation
2616:
2610:
2155:geodesics as Hamiltonian flows
2123:
2099:
2035:
2023:
2020:
2014:
1964:
1949:
1905:
1899:
1896:
1890:
1635:
1629:
1578:
1572:
1549:
1543:
1520:
1498:
1472:
1466:
1460:
1448:
1433:
1426:
1303:
1300:
1294:
1276:
1270:
1255:
1250:
1244:
1205:
1199:
1103:
1100:
1094:
1076:
1070:
1055:
1050:
1044:
1013:
1007:
870:
867:
854:
845:
832:
826:
747:
744:
731:
722:
709:
703:
536:shortest paths between them.
383:
297:, can be defined by using the
267:recovers the previous notion.
1:
6709:Exterior covariant derivative
6641:Tensor (intrinsic definition)
5025:, London: Benjamin-Cummings,
4673:
3827:{\displaystyle t\mapsto at+b}
3013:defined in the following way
2935:Picard–Lindelöf theorem
1655:of motion for the functional
1488:with equality if and only if
809:, i.e. in the above identity
606:minimizer. More precisely, a
458:geodesic (general relativity)
21:Geodesic (general relativity)
6734:Raising and lowering indices
5366:Raising and lowering indices
5248:Totally geodesic submanifold
4666:rather than a local minimum.
4249:{\displaystyle \varepsilon }
3852:) are called geodesics with
1862:Techniques of the classical
1846:of the metric. This is the
543:
255:remain parallel if they are
7:
6972:Gluon field strength tensor
6443:
6305:Classification of manifolds
5181:Encyclopedia of Mathematics
4549:
3848:
3792:
3733:along the scalar homothety
3412:{\displaystyle \gamma _{V}}
3122:{\displaystyle \gamma _{V}}
2400:
2273:
945:, the geodesic is called a
463:
10:
7101:
6783:Cartan formalism (physics)
6603:Penrose graphical notation
5387:Pseudo-Riemannian manifold
5157:Cambridge University Press
5099:Classical Theory of Fields
4991:(2nd ed.), New York:
4647:pseudo-Riemannian manifold
4176:in a plane into a surface
3605:pushforward (differential)
3434:
2176:
2163:(pseudo-)Riemannian metric
1641:{\displaystyle L(\gamma )}
1599:is a "convex function" of
1584:{\displaystyle L(\gamma )}
1555:{\displaystyle E(\gamma )}
547:
454:pseudo-Riemannian manifold
399:{\displaystyle t\to t^{2}}
343:) along the curve equals |
151:) between two points in a
18:
6989:
6929:
6878:
6871:
6763:
6694:
6631:
6575:
6522:
6469:
6462:
6455:Glossary of tensor theory
6451:
6381:over commutative algebras
6338:
6297:
6230:
6127:
6023:
5970:
5961:
5797:
5720:
5659:
5579:
5516:Geometrization conjecture
5503:
5477:
5431:
5400:
5296:
4847:Communications of the ACM
4743:SIAM Journal on Computing
2807:{\displaystyle \gamma \,}
1411:Cauchy–Schwarz inequality
1376:curve (more generally, a
538:Geodesics on an ellipsoid
155:, or more generally in a
33:Geodesic (disambiguation)
7039:Gregorio Ricci-Curbastro
6911:Riemann curvature tensor
6618:Van der Waerden notation
6097:Riemann curvature tensor
5023:Foundations of mechanics
4629:
4587:Hopf–Rinow theorem
4499:geodesic airframes; see
3859:An affine connection is
2745:Existence and uniqueness
1653:Euler–Lagrange equations
1349:cannot. For a piecewise
807:natural parameterization
658:there is a neighborhood
7009:Elwin Bruno Christoffel
6942:Angular momentum tensor
6613:Tetrad (index notation)
6583:Abstract index notation
5174:Volkov, Yu.A. (2001) ,
5129:Wheeler, John Archibald
4957:more precise citations.
4801:10.1073/pnas.95.15.8431
4699:Oxford University Press
4054:{\displaystyle \nabla }
4005:{\displaystyle \nabla }
1612:{\displaystyle \gamma }
1402:{\displaystyle W^{1,2}}
439:sub-Riemannian geometry
425:describe the motion of
278:describe the motion of
224:, one might consider a
161:differentiable manifold
7085:Geodesic (mathematics)
6823:Levi-Civita connection
5889:Manifold with boundary
5604:Differential structure
5526:Uniformization theorem
5459:Nash embedding theorem
5392:Riemannian volume form
5351:Levi-Civita connection
4446:
4426:
4402:
4375:
4348:
4270:
4250:
4230:
4210:
4190:
4170:
4150:
4127:
4055:
4035:
4006:
3979:
3900:
3828:
3772:
3701:
3571:
3517:
3413:
3379:
3284:
3232:
3174:
3123:
3077:
2903:
2854:
2808:
2730:
2662:
2623:
2565:
2384:
2355:
2326:
2306:
2261:
2153:, one can also regard
2133:
1974:
1864:calculus of variations
1858:Calculus of variations
1836:
1798:
1642:
1613:
1585:
1556:
1527:
1479:
1403:
1370:
1320:
1122:
916:
796:
586:
563:
485:
477:
450:Levi-Civita connection
400:
325:calculus of variations
261:Levi-Civita connection
49:
31:. For other uses, see
7080:Differential geometry
7049:Jan Arnoldus Schouten
7004:Augustin-Louis Cauchy
6484:Differential geometry
5250:at the Manifold Atlas
5206:John Wiley & Sons
5151:Ortín, Tomás (2004),
5044:Jost, Jürgen (2002),
4695:UK English Dictionary
4533:Shortest path problem
4447:
4427:
4403:
4401:{\displaystyle g_{S}}
4376:
4374:{\displaystyle g_{N}}
4349:
4271:
4251:
4231:
4229:{\displaystyle \ell }
4211:
4191:
4171:
4169:{\displaystyle \ell }
4151:
4128:
4072:Computational methods
4066:projective connection
4056:
4036:
4007:
3980:
3901:
3829:
3773:
3702:
3607:along the projection
3572:
3518:
3465:double tangent bundle
3435:Further information:
3414:
3380:
3285:
3233:
3175:
3124:
3078:
2978:geodesically complete
2904:
2855:
2809:
2731:
2663:
2624:
2566:
2385:
2356:
2327:
2307:
2262:
2134:
1975:
1837:
1799:
1643:
1614:
1586:
1557:
1528:
1480:
1404:
1371:
1369:{\displaystyle C^{1}}
1321:
1143:) between two points
1123:
917:
797:
584:
561:
483:
471:
401:
218:great-circle distance
41:
7024:Carl Friedrich Gauss
6957:stress–energy tensor
6952:Cauchy stress tensor
6704:Covariant derivative
6666:Antisymmetric tensor
6598:Multi-index notation
6036:Covariant derivative
5587:Topological manifold
5449:Gauss–Bonnet theorem
5356:Covariant derivative
5101:, Oxford: Pergamon,
5048:, Berlin, New York:
4571:Differentiable curve
4436:
4416:
4385:
4358:
4280:
4260:
4240:
4220:
4200:
4196:so that the mapping
4180:
4160:
4140:
4096:
4045:
4016:
3996:
3913:
3875:
3803:
3737:
3641:
3624:Ehresmann connection
3537:
3486:
3396:
3388:In particular, when
3297:
3274:
3185:
3133:
3106:
3020:
2866:
2828:
2797:
2688:
2637:
2578:
2430:
2421:summation convention
2365:
2336:
2316:
2287:
2219:
1998:
1881:
1811:
1666:
1623:
1603:
1566:
1537:
1492:
1420:
1380:
1353:
1193:
1001:
820:
697:
627:of the reals to the
446:Riemannian manifolds
433:, or the shape of a
377:
6901:Nonmetricity tensor
6756:(2nd-order tensors)
6724:Hodge star operator
6714:Exterior derivative
6563:Transport phenomena
6548:Continuum mechanics
6504:Multilinear algebra
6070:Exterior derivative
5672:Atiyah–Singer index
5621:Riemannian manifold
5521:Poincaré conjecture
5382:Riemannian manifold
5370:Musical isomorphism
5285:Riemannian geometry
5234:Geodesics Revisited
5153:Gravity and strings
5019:Marsden, Jerrold E.
4792:1998PNAS...95.8431K
4651:Lorentzian manifold
4562:Clairaut's relation
3425:Liouville's theorem
3421:unit tangent bundle
2769:and for any vector
2738:acceleration vector
2678:classical mechanics
2670:Christoffel symbols
2657:
2494:
2415:, we can write the
2151:classical mechanics
1844:Christoffel symbols
1831:
1730:
1333:are also minima of
1235:
1033:
974:Riemannian manifold
968:Riemannian geometry
958:length metric space
947:minimizing geodesic
813: = 1 and
554:Toponogov's theorem
413:and more generally
411:Riemannian geometry
295:Riemannian manifold
241:Riemannian manifold
157:Riemannian manifold
7034:Tullio Levi-Civita
6977:Metric tensor (GR)
6891:Levi-Civita symbol
6744:Tensor contraction
6558:General relativity
6494:Euclidean geometry
6376:Secondary calculus
6330:Singularity theory
6285:Parallel transport
6053:De Rham cohomology
5692:Generalized Stokes
5511:General relativity
5454:Hopf–Rinow theorem
5401:Types of manifolds
5377:Parallel transport
5121:Misner, Charles W.
4725:. Merriam-Webster.
4475:. You can help by
4442:
4422:
4398:
4371:
4344:
4266:
4246:
4226:
4206:
4186:
4166:
4146:
4123:
4051:
4031:
4002:
3975:
3896:
3824:
3768:
3697:
3567:
3513:
3409:
3392:is a unit vector,
3375:
3280:
3268:canonical one-form
3228:
3170:
3119:
3073:
2968:extends to all of
2956:may not be all of
2899:
2850:
2804:
2726:
2658:
2640:
2619:
2561:
2477:
2380:
2351:
2322:
2302:
2257:
2208:parallel transport
2198:∇ is defined as a
2159:Hamilton equations
2129:
1970:
1832:
1814:
1794:
1713:
1638:
1609:
1581:
1552:
1523:
1475:
1399:
1366:
1316:
1221:
1155:is defined as the
1118:
1019:
912:
792:
670:such that for any
648:such that for any
587:
576:spherical triangle
564:
490:Euclidean geometry
486:
478:
419:general relativity
396:
272:general relativity
245:geodesic curvature
180:and the adjective
50:
45:with 28 geodesics
7067:
7066:
7029:Hermann Grassmann
6985:
6984:
6937:Moment of inertia
6798:Differential form
6773:Affine connection
6588:Einstein notation
6571:
6570:
6499:Exterior calculus
6479:Coordinate system
6411:
6410:
6293:
6292:
6058:Differential form
5712:Whitney embedding
5646:Differential form
5534:
5533:
5215:978-0-471-92567-5
5166:978-0-521-82475-0
5144:978-0-7167-0344-0
5137:, W. H. Freeman,
5108:978-0-08-018176-9
5059:978-3-540-42627-1
5032:978-0-8053-0102-1
5015:Abraham, Ralph H.
5002:978-0-07-000423-8
4983:
4982:
4975:
4911:978-0-914098-71-3
4786:(15): 8431–8435.
4543:digital dentistry
4505:geodetic airframe
4501:geodesic airframe
4493:
4492:
4445:{\displaystyle S}
4425:{\displaystyle N}
4269:{\displaystyle l}
4209:{\displaystyle f}
4189:{\displaystyle S}
4149:{\displaystyle N}
4028:
3963:
3893:
3283:{\displaystyle g}
3146:
3055:
2878:
2755:affine connection
2717:
2704:
2557:
2547:
2520:
2472:
2417:geodesic equation
2409:local coordinates
2377:
2348:
2325:{\displaystyle t}
2299:
2281:
2280:
2248:
2235:
2196:affine connection
2072:
1928:
1848:geodesic equation
1783:
1756:
1708:
1291:
1267:
1219:
1185:energy functional
1106:
1091:
1067:
962:rectifiable paths
623:from an interval
568:geodesic triangle
532:, then there are
265:Riemannian metric
249:affine connection
48:
7092:
7044:Bernhard Riemann
6876:
6875:
6719:Exterior product
6686:Two-point tensor
6671:Symmetric tensor
6553:Electromagnetism
6467:
6466:
6438:
6431:
6424:
6415:
6414:
6403:Stratified space
6361:Fréchet manifold
6075:Interior product
5968:
5967:
5665:
5561:
5554:
5547:
5538:
5537:
5278:
5271:
5264:
5255:
5254:
5218:
5203:
5194:Weinberg, Steven
5188:
5169:
5147:
5111:
5085:
5062:
5035:
5005:
4978:
4971:
4967:
4964:
4958:
4953:this article by
4944:inline citations
4931:
4930:
4923:
4914:
4889:
4878:
4872:
4871:
4837:
4831:
4830:
4828:
4813:
4803:
4775:
4766:
4760:
4759:
4733:
4727:
4726:
4713:
4707:
4706:
4701:. Archived from
4684:
4667:
4660:
4654:
4643:
4593:Intrinsic metric
4567:
4488:
4485:
4467:
4460:
4451:
4449:
4448:
4443:
4431:
4429:
4428:
4423:
4407:
4405:
4404:
4399:
4397:
4396:
4380:
4378:
4377:
4372:
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4369:
4353:
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4339:
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4317:
4305:
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4275:
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4267:
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4227:
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4207:
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4021:
4011:
4009:
4008:
4003:
3984:
3982:
3981:
3976:
3971:
3970:
3965:
3964:
3956:
3946:
3945:
3905:
3903:
3902:
3897:
3895:
3894:
3886:
3854:affine parameter
3833:
3831:
3830:
3825:
3780:Finsler manifold
3777:
3775:
3774:
3769:
3749:
3748:
3706:
3704:
3703:
3698:
3695:
3694:
3685:
3684:
3675:
3674:
3656:
3655:
3610:
3591:
3576:
3574:
3573:
3568:
3559:
3558:
3549:
3548:
3522:
3520:
3519:
3514:
3477:vertical bundles
3418:
3416:
3415:
3410:
3408:
3407:
3384:
3382:
3381:
3376:
3337:
3336:
3315:
3314:
3289:
3287:
3286:
3281:
3260:Hamiltonian flow
3237:
3235:
3234:
3229:
3197:
3196:
3179:
3177:
3176:
3171:
3154:
3153:
3148:
3147:
3139:
3128:
3126:
3125:
3120:
3118:
3117:
3082:
3080:
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3074:
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2975:
2971:
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2906:
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2900:
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2879:
2871:
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2857:
2856:
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2805:
2735:
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2705:
2697:
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2659:
2656:
2651:
2628:
2626:
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2620:
2603:
2602:
2590:
2589:
2570:
2568:
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2562:
2555:
2548:
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2538:
2537:
2536:
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2511:
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2496:
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2389:
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2275:
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2264:
2263:
2258:
2250:
2249:
2241:
2238:
2237:
2236:
2228:
2213:
2173:Affine geodesics
2138:
2136:
2135:
2130:
2095:
2094:
2077:
2073:
2071:
2056:
2055:
2046:
2010:
2009:
1989:second variation
1979:
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1933:
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1234:
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890:
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866:
865:
844:
843:
801:
799:
798:
793:
788:
784:
783:
782:
770:
769:
743:
742:
721:
720:
689:
657:
647:
622:
574:arcs, forming a
530:antipodal points
405:
403:
402:
397:
395:
394:
142:
141:
138:
137:
134:
131:
126:
125:
122:
119:
116:
113:
110:
105:
104:
99:
98:
95:
92:
89:
86:
83:
80:
77:
74:
71:
68:
46:
7100:
7099:
7095:
7094:
7093:
7091:
7090:
7089:
7070:
7069:
7068:
7063:
7014:Albert Einstein
6981:
6962:Einstein tensor
6925:
6906:Ricci curvature
6886:Kronecker delta
6872:Notable tensors
6867:
6788:Connection form
6765:
6759:
6690:
6676:Tensor operator
6633:
6627:
6567:
6543:Computer vision
6536:
6518:
6514:Tensor calculus
6458:
6447:
6442:
6412:
6407:
6346:Banach manifold
6339:Generalizations
6334:
6289:
6226:
6123:
6085:Ricci curvature
6041:Cotangent space
6019:
5957:
5799:
5793:
5752:Exponential map
5716:
5661:
5655:
5575:
5565:
5535:
5530:
5499:
5478:Generalizations
5473:
5427:
5396:
5331:Exponential map
5292:
5282:
5242:brachistochrone
5230:
5216:
5176:"Geodesic line"
5167:
5145:
5109:
5095:Lifshitz, E. M.
5083:
5065:See section 1.4
5060:
5050:Springer-Verlag
5038:See section 2.7
5033:
5003:
4979:
4968:
4962:
4959:
4949:Please help to
4948:
4932:
4928:
4921:
4919:Further reading
4912:
4898:Spivak, Michael
4893:
4892:
4883:(Nov 2, 2017),
4881:Michael Stevens
4879:
4875:
4860:10.1145/3131280
4838:
4834:
4826:
4773:
4767:
4763:
4756:10.1137/0216045
4734:
4730:
4715:
4714:
4710:
4686:
4685:
4681:
4676:
4671:
4670:
4661:
4657:
4644:
4637:
4632:
4627:
4582:Geodesic circle
4565:
4552:
4529:motion planning
4517:Earth geodesics
4489:
4483:
4480:
4473:needs expansion
4458:
4437:
4434:
4433:
4417:
4414:
4413:
4392:
4388:
4386:
4383:
4382:
4365:
4361:
4359:
4356:
4355:
4335:
4331:
4313:
4309:
4300:
4296:
4287:
4283:
4281:
4278:
4277:
4261:
4258:
4257:
4241:
4238:
4237:
4221:
4218:
4217:
4201:
4198:
4197:
4181:
4178:
4177:
4161:
4158:
4157:
4141:
4138:
4137:
4097:
4094:
4093:
4082:
4074:
4046:
4043:
4042:
4020:
4019:
4017:
4014:
4013:
3997:
3994:
3993:
3966:
3955:
3954:
3953:
3941:
3937:
3914:
3911:
3910:
3885:
3884:
3876:
3873:
3872:
3804:
3801:
3800:
3788:
3744:
3740:
3738:
3735:
3734:
3728:
3690:
3686:
3680:
3676:
3670:
3666:
3648:
3644:
3642:
3639:
3638:
3608:
3595: : TT
3594:
3589:
3554:
3550:
3544:
3540:
3538:
3535:
3534:
3487:
3484:
3483:
3439:
3433:
3403:
3399:
3397:
3394:
3393:
3332:
3328:
3310:
3306:
3298:
3295:
3294:
3275:
3272:
3271:
3248:closed geodesic
3240:exponential map
3192:
3188:
3186:
3183:
3182:
3149:
3138:
3137:
3136:
3134:
3131:
3130:
3113:
3109:
3107:
3104:
3103:
3058:
3047:
3046:
3045:
3027:
3023:
3021:
3018:
3017:
2986:
2973:
2972:if and only if
2969:
2965:
2870:
2869:
2867:
2864:
2863:
2829:
2826:
2825:
2798:
2795:
2794:
2778:
2747:
2736:means that the
2709:
2708:
2696:
2695:
2691:
2689:
2686:
2685:
2652:
2644:
2638:
2635:
2634:
2598:
2594:
2585:
2581:
2579:
2576:
2575:
2539:
2532:
2528:
2524:
2522:
2512:
2505:
2501:
2497:
2495:
2489:
2481:
2465:
2461:
2457:
2450:
2446:
2440:
2436:
2435:
2433:
2431:
2428:
2427:
2369:
2368:
2366:
2363:
2362:
2340:
2339:
2337:
2334:
2333:
2317:
2314:
2313:
2291:
2290:
2288:
2285:
2284:
2240:
2239:
2227:
2226:
2222:
2220:
2217:
2216:
2189:smooth manifold
2181:
2175:
2078:
2057:
2051:
2047:
2045:
2042:
2041:
2005:
2001:
1999:
1996:
1995:
1985:critical points
1934:
1920:
1915:
1912:
1911:
1882:
1879:
1878:
1872:first variation
1860:
1826:
1818:
1812:
1809:
1808:
1775:
1768:
1764:
1760:
1758:
1748:
1741:
1737:
1733:
1731:
1725:
1717:
1701:
1697:
1693:
1686:
1682:
1676:
1672:
1671:
1669:
1667:
1664:
1663:
1624:
1621:
1620:
1604:
1601:
1600:
1567:
1564:
1563:
1538:
1535:
1534:
1512:
1501:
1493:
1490:
1489:
1436:
1432:
1421:
1418:
1417:
1387:
1383:
1381:
1378:
1377:
1360:
1356:
1354:
1351:
1350:
1283:
1282:
1259:
1258:
1240:
1236:
1230:
1225:
1211:
1194:
1191:
1190:
1083:
1082:
1059:
1058:
1040:
1036:
1034:
1028:
1023:
1002:
999:
998:
970:
939:
932:
926:
898:
894:
885:
881:
880:
876:
861:
857:
839:
835:
821:
818:
817:
778:
774:
765:
761:
760:
756:
738:
734:
716:
712:
698:
695:
694:
684:
677:
671:
649:
642:
610:
596:metric geometry
592:
590:Metric geometry
556:
546:
534:infinitely many
466:
435:planetary orbit
427:point particles
421:, geodesics in
415:metric geometry
390:
386:
378:
375:
374:
291:
253:tangent vectors
206:spherical Earth
128:
107:
101:
65:
61:
36:
17:
12:
11:
5:
7098:
7088:
7087:
7082:
7065:
7064:
7062:
7061:
7056:
7054:Woldemar Voigt
7051:
7046:
7041:
7036:
7031:
7026:
7021:
7019:Leonhard Euler
7016:
7011:
7006:
7001:
6995:
6993:
6991:Mathematicians
6987:
6986:
6983:
6982:
6980:
6979:
6974:
6969:
6964:
6959:
6954:
6949:
6944:
6939:
6933:
6931:
6927:
6926:
6924:
6923:
6918:
6916:Torsion tensor
6913:
6908:
6903:
6898:
6893:
6888:
6882:
6880:
6873:
6869:
6868:
6866:
6865:
6860:
6855:
6850:
6845:
6840:
6835:
6830:
6825:
6820:
6815:
6810:
6805:
6800:
6795:
6790:
6785:
6780:
6775:
6769:
6767:
6761:
6760:
6758:
6757:
6751:
6749:Tensor product
6746:
6741:
6739:Symmetrization
6736:
6731:
6729:Lie derivative
6726:
6721:
6716:
6711:
6706:
6700:
6698:
6692:
6691:
6689:
6688:
6683:
6678:
6673:
6668:
6663:
6658:
6653:
6651:Tensor density
6648:
6643:
6637:
6635:
6629:
6628:
6626:
6625:
6623:Voigt notation
6620:
6615:
6610:
6608:Ricci calculus
6605:
6600:
6595:
6593:Index notation
6590:
6585:
6579:
6577:
6573:
6572:
6569:
6568:
6566:
6565:
6560:
6555:
6550:
6545:
6539:
6537:
6535:
6534:
6529:
6523:
6520:
6519:
6517:
6516:
6511:
6509:Tensor algebra
6506:
6501:
6496:
6491:
6489:Dyadic algebra
6486:
6481:
6475:
6473:
6464:
6460:
6459:
6452:
6449:
6448:
6441:
6440:
6433:
6426:
6418:
6409:
6408:
6406:
6405:
6400:
6395:
6390:
6385:
6384:
6383:
6373:
6368:
6363:
6358:
6353:
6348:
6342:
6340:
6336:
6335:
6333:
6332:
6327:
6322:
6317:
6312:
6307:
6301:
6299:
6295:
6294:
6291:
6290:
6288:
6287:
6282:
6277:
6272:
6267:
6262:
6257:
6252:
6247:
6242:
6236:
6234:
6228:
6227:
6225:
6224:
6219:
6214:
6209:
6204:
6199:
6194:
6184:
6179:
6174:
6164:
6159:
6154:
6149:
6144:
6139:
6133:
6131:
6125:
6124:
6122:
6121:
6116:
6111:
6110:
6109:
6099:
6094:
6093:
6092:
6082:
6077:
6072:
6067:
6066:
6065:
6055:
6050:
6049:
6048:
6038:
6033:
6027:
6025:
6021:
6020:
6018:
6017:
6012:
6007:
6002:
6001:
6000:
5990:
5985:
5980:
5974:
5972:
5965:
5959:
5958:
5956:
5955:
5950:
5940:
5935:
5921:
5916:
5911:
5906:
5901:
5899:Parallelizable
5896:
5891:
5886:
5885:
5884:
5874:
5869:
5864:
5859:
5854:
5849:
5844:
5839:
5834:
5829:
5819:
5809:
5803:
5801:
5795:
5794:
5792:
5791:
5786:
5781:
5779:Lie derivative
5776:
5774:Integral curve
5771:
5766:
5761:
5760:
5759:
5749:
5744:
5743:
5742:
5735:Diffeomorphism
5732:
5726:
5724:
5718:
5717:
5715:
5714:
5709:
5704:
5699:
5694:
5689:
5684:
5679:
5674:
5668:
5666:
5657:
5656:
5654:
5653:
5648:
5643:
5638:
5633:
5628:
5623:
5618:
5613:
5612:
5611:
5606:
5596:
5595:
5594:
5583:
5581:
5580:Basic concepts
5577:
5576:
5564:
5563:
5556:
5549:
5541:
5532:
5531:
5529:
5528:
5523:
5518:
5513:
5507:
5505:
5501:
5500:
5498:
5497:
5495:Sub-Riemannian
5492:
5487:
5481:
5479:
5475:
5474:
5472:
5471:
5466:
5461:
5456:
5451:
5446:
5441:
5435:
5433:
5429:
5428:
5426:
5425:
5420:
5415:
5410:
5404:
5402:
5398:
5397:
5395:
5394:
5389:
5384:
5379:
5374:
5373:
5372:
5363:
5358:
5353:
5343:
5338:
5333:
5328:
5327:
5326:
5321:
5316:
5311:
5300:
5298:
5297:Basic concepts
5294:
5293:
5281:
5280:
5273:
5266:
5258:
5252:
5251:
5245:
5240:), mechanics (
5229:
5228:External links
5226:
5225:
5224:
5214:
5190:
5171:
5165:
5148:
5143:
5117:
5114:See section 87
5107:
5087:
5081:
5068:
5058:
5041:
5031:
5011:
5001:
4981:
4980:
4935:
4933:
4926:
4920:
4917:
4916:
4915:
4910:
4891:
4890:
4873:
4832:
4761:
4750:(4): 647–668.
4728:
4708:
4705:on 2020-03-16.
4678:
4677:
4675:
4672:
4669:
4668:
4655:
4634:
4633:
4631:
4628:
4626:
4625:
4619:
4613:
4607:
4602:
4599:Isotropic line
4596:
4590:
4584:
4579:
4574:
4568:
4559:
4553:
4551:
4548:
4547:
4546:
4535:
4525:
4519:
4513:
4511:geodesic domes
4507:
4491:
4490:
4470:
4468:
4457:
4454:
4441:
4421:
4395:
4391:
4368:
4364:
4343:
4338:
4334:
4330:
4327:
4324:
4321:
4316:
4312:
4308:
4303:
4299:
4295:
4290:
4286:
4265:
4245:
4225:
4205:
4185:
4165:
4145:
4122:
4119:
4116:
4113:
4110:
4107:
4104:
4101:
4081:
4078:
4073:
4070:
4050:
4027:
4024:
4001:
3990:skew-symmetric
3986:
3985:
3974:
3969:
3962:
3959:
3952:
3949:
3944:
3940:
3936:
3933:
3930:
3927:
3924:
3921:
3918:
3892:
3889:
3883:
3880:
3835:
3834:
3823:
3820:
3817:
3814:
3811:
3808:
3787:
3784:
3767:
3764:
3761:
3758:
3755:
3752:
3747:
3743:
3726:
3714: ∈ T
3708:
3707:
3693:
3689:
3683:
3679:
3673:
3669:
3665:
3662:
3659:
3654:
3651:
3647:
3611: : T
3599: → T
3592:
3584: ∈ T
3580:at each point
3578:
3577:
3565:
3562:
3557:
3553:
3547:
3543:
3524:
3523:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3491:
3443:tangent bundle
3432:
3431:Geodesic spray
3429:
3406:
3402:
3386:
3385:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3335:
3331:
3327:
3324:
3321:
3318:
3313:
3309:
3305:
3302:
3279:
3242:of the vector
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3195:
3191:
3169:
3166:
3163:
3160:
3157:
3152:
3145:
3142:
3116:
3112:
3084:
3083:
3072:
3069:
3066:
3061:
3054:
3051:
3044:
3041:
3038:
3035:
3030:
3026:
3009:of a manifold
3004:tangent bundle
2985:
2982:
2927:
2926:
2911:
2910:
2909:
2898:
2895:
2892:
2889:
2886:
2883:
2877:
2874:
2861:
2848:
2845:
2842:
2839:
2836:
2833:
2802:
2776:
2761:For any point
2746:
2743:
2725:
2722:
2716:
2713:
2703:
2700:
2694:
2682:free particles
2655:
2650:
2647:
2643:
2618:
2615:
2612:
2609:
2606:
2601:
2597:
2593:
2588:
2584:
2572:
2571:
2560:
2554:
2551:
2545:
2542:
2535:
2531:
2527:
2518:
2515:
2508:
2504:
2500:
2492:
2487:
2484:
2480:
2476:
2468:
2464:
2460:
2453:
2449:
2443:
2439:
2376:
2373:
2347:
2344:
2321:
2298:
2295:
2279:
2278:
2269:
2267:
2256:
2253:
2247:
2244:
2234:
2231:
2225:
2174:
2171:
2140:
2139:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2107:
2104:
2101:
2098:
2093:
2090:
2087:
2084:
2081:
2076:
2070:
2067:
2063:
2060:
2054:
2050:
2044:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2008:
2004:
1991:is defined by
1981:
1980:
1969:
1966:
1963:
1960:
1957:
1954:
1951:
1948:
1943:
1940:
1937:
1932:
1926:
1923:
1919:
1914:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1859:
1856:
1829:
1824:
1821:
1817:
1805:
1804:
1793:
1790:
1787:
1781:
1778:
1771:
1767:
1763:
1754:
1751:
1744:
1740:
1736:
1728:
1723:
1720:
1716:
1712:
1704:
1700:
1696:
1689:
1685:
1679:
1675:
1637:
1634:
1631:
1628:
1608:
1580:
1577:
1574:
1571:
1562:also minimize
1551:
1548:
1545:
1542:
1522:
1518:
1515:
1511:
1507:
1504:
1500:
1497:
1486:
1485:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1447:
1444:
1439:
1435:
1431:
1428:
1425:
1396:
1393:
1390:
1386:
1363:
1359:
1329:All minima of
1327:
1326:
1315:
1312:
1309:
1305:
1302:
1299:
1296:
1290:
1287:
1281:
1278:
1275:
1272:
1266:
1263:
1257:
1252:
1249:
1246:
1243:
1239:
1233:
1228:
1224:
1218:
1215:
1210:
1207:
1204:
1201:
1198:
1175:) =
1167:) =
1129:
1128:
1117:
1114:
1111:
1105:
1102:
1099:
1096:
1090:
1087:
1081:
1078:
1075:
1072:
1066:
1063:
1057:
1052:
1049:
1046:
1043:
1039:
1031:
1026:
1022:
1018:
1015:
1012:
1009:
1006:
994:is defined by
969:
966:
937:
930:
923:
922:
911:
907:
901:
897:
893:
888:
884:
879:
875:
872:
869:
864:
860:
856:
853:
850:
847:
842:
838:
834:
831:
828:
825:
803:
802:
791:
787:
781:
777:
773:
768:
764:
759:
755:
752:
749:
746:
741:
737:
733:
730:
727:
724:
719:
715:
711:
708:
705:
702:
682:
675:
638:if there is a
591:
588:
545:
542:
465:
462:
448:. The article
393:
389:
385:
382:
290:
287:
283:test particles
29:Earth geodesic
15:
9:
6:
4:
3:
2:
7097:
7086:
7083:
7081:
7078:
7077:
7075:
7060:
7057:
7055:
7052:
7050:
7047:
7045:
7042:
7040:
7037:
7035:
7032:
7030:
7027:
7025:
7022:
7020:
7017:
7015:
7012:
7010:
7007:
7005:
7002:
7000:
6997:
6996:
6994:
6992:
6988:
6978:
6975:
6973:
6970:
6968:
6965:
6963:
6960:
6958:
6955:
6953:
6950:
6948:
6945:
6943:
6940:
6938:
6935:
6934:
6932:
6928:
6922:
6919:
6917:
6914:
6912:
6909:
6907:
6904:
6902:
6899:
6897:
6896:Metric tensor
6894:
6892:
6889:
6887:
6884:
6883:
6881:
6877:
6874:
6870:
6864:
6861:
6859:
6856:
6854:
6851:
6849:
6846:
6844:
6841:
6839:
6836:
6834:
6831:
6829:
6826:
6824:
6821:
6819:
6816:
6814:
6811:
6809:
6808:Exterior form
6806:
6804:
6801:
6799:
6796:
6794:
6791:
6789:
6786:
6784:
6781:
6779:
6776:
6774:
6771:
6770:
6768:
6762:
6755:
6752:
6750:
6747:
6745:
6742:
6740:
6737:
6735:
6732:
6730:
6727:
6725:
6722:
6720:
6717:
6715:
6712:
6710:
6707:
6705:
6702:
6701:
6699:
6697:
6693:
6687:
6684:
6682:
6681:Tensor bundle
6679:
6677:
6674:
6672:
6669:
6667:
6664:
6662:
6659:
6657:
6654:
6652:
6649:
6647:
6644:
6642:
6639:
6638:
6636:
6630:
6624:
6621:
6619:
6616:
6614:
6611:
6609:
6606:
6604:
6601:
6599:
6596:
6594:
6591:
6589:
6586:
6584:
6581:
6580:
6578:
6574:
6564:
6561:
6559:
6556:
6554:
6551:
6549:
6546:
6544:
6541:
6540:
6538:
6533:
6530:
6528:
6525:
6524:
6521:
6515:
6512:
6510:
6507:
6505:
6502:
6500:
6497:
6495:
6492:
6490:
6487:
6485:
6482:
6480:
6477:
6476:
6474:
6472:
6468:
6465:
6461:
6457:
6456:
6450:
6446:
6439:
6434:
6432:
6427:
6425:
6420:
6419:
6416:
6404:
6401:
6399:
6398:Supermanifold
6396:
6394:
6391:
6389:
6386:
6382:
6379:
6378:
6377:
6374:
6372:
6369:
6367:
6364:
6362:
6359:
6357:
6354:
6352:
6349:
6347:
6344:
6343:
6341:
6337:
6331:
6328:
6326:
6323:
6321:
6318:
6316:
6313:
6311:
6308:
6306:
6303:
6302:
6300:
6296:
6286:
6283:
6281:
6278:
6276:
6273:
6271:
6268:
6266:
6263:
6261:
6258:
6256:
6253:
6251:
6248:
6246:
6243:
6241:
6238:
6237:
6235:
6233:
6229:
6223:
6220:
6218:
6215:
6213:
6210:
6208:
6205:
6203:
6200:
6198:
6195:
6193:
6189:
6185:
6183:
6180:
6178:
6175:
6173:
6169:
6165:
6163:
6160:
6158:
6155:
6153:
6150:
6148:
6145:
6143:
6140:
6138:
6135:
6134:
6132:
6130:
6126:
6120:
6119:Wedge product
6117:
6115:
6112:
6108:
6105:
6104:
6103:
6100:
6098:
6095:
6091:
6088:
6087:
6086:
6083:
6081:
6078:
6076:
6073:
6071:
6068:
6064:
6063:Vector-valued
6061:
6060:
6059:
6056:
6054:
6051:
6047:
6044:
6043:
6042:
6039:
6037:
6034:
6032:
6029:
6028:
6026:
6022:
6016:
6013:
6011:
6008:
6006:
6003:
5999:
5996:
5995:
5994:
5993:Tangent space
5991:
5989:
5986:
5984:
5981:
5979:
5976:
5975:
5973:
5969:
5966:
5964:
5960:
5954:
5951:
5949:
5945:
5941:
5939:
5936:
5934:
5930:
5926:
5922:
5920:
5917:
5915:
5912:
5910:
5907:
5905:
5902:
5900:
5897:
5895:
5892:
5890:
5887:
5883:
5880:
5879:
5878:
5875:
5873:
5870:
5868:
5865:
5863:
5860:
5858:
5855:
5853:
5850:
5848:
5845:
5843:
5840:
5838:
5835:
5833:
5830:
5828:
5824:
5820:
5818:
5814:
5810:
5808:
5805:
5804:
5802:
5796:
5790:
5787:
5785:
5782:
5780:
5777:
5775:
5772:
5770:
5767:
5765:
5762:
5758:
5757:in Lie theory
5755:
5754:
5753:
5750:
5748:
5745:
5741:
5738:
5737:
5736:
5733:
5731:
5728:
5727:
5725:
5723:
5719:
5713:
5710:
5708:
5705:
5703:
5700:
5698:
5695:
5693:
5690:
5688:
5685:
5683:
5680:
5678:
5675:
5673:
5670:
5669:
5667:
5664:
5660:Main results
5658:
5652:
5649:
5647:
5644:
5642:
5641:Tangent space
5639:
5637:
5634:
5632:
5629:
5627:
5624:
5622:
5619:
5617:
5614:
5610:
5607:
5605:
5602:
5601:
5600:
5597:
5593:
5590:
5589:
5588:
5585:
5584:
5582:
5578:
5573:
5569:
5562:
5557:
5555:
5550:
5548:
5543:
5542:
5539:
5527:
5524:
5522:
5519:
5517:
5514:
5512:
5509:
5508:
5506:
5502:
5496:
5493:
5491:
5488:
5486:
5483:
5482:
5480:
5476:
5470:
5469:Schur's lemma
5467:
5465:
5462:
5460:
5457:
5455:
5452:
5450:
5447:
5445:
5444:Gauss's lemma
5442:
5440:
5437:
5436:
5434:
5430:
5424:
5421:
5419:
5416:
5414:
5411:
5409:
5406:
5405:
5403:
5399:
5393:
5390:
5388:
5385:
5383:
5380:
5378:
5375:
5371:
5367:
5364:
5362:
5359:
5357:
5354:
5352:
5349:
5348:
5347:
5346:Metric tensor
5344:
5342:
5341:Inner product
5339:
5337:
5334:
5332:
5329:
5325:
5322:
5320:
5317:
5315:
5312:
5310:
5307:
5306:
5305:
5302:
5301:
5299:
5295:
5290:
5286:
5279:
5274:
5272:
5267:
5265:
5260:
5259:
5256:
5249:
5246:
5243:
5239:
5235:
5232:
5231:
5222:
5221:See chapter 3
5217:
5211:
5207:
5202:
5201:
5195:
5191:
5187:
5183:
5182:
5177:
5172:
5168:
5162:
5158:
5154:
5149:
5146:
5140:
5136:
5135:
5130:
5126:
5122:
5118:
5115:
5110:
5104:
5100:
5096:
5092:
5091:Landau, L. D.
5088:
5084:
5082:0-471-15733-3
5078:
5074:
5069:
5066:
5061:
5055:
5051:
5047:
5042:
5039:
5034:
5028:
5024:
5020:
5016:
5012:
5009:
5008:See chapter 2
5004:
4998:
4994:
4990:
4985:
4984:
4977:
4974:
4966:
4956:
4952:
4946:
4945:
4939:
4934:
4925:
4924:
4913:
4907:
4903:
4899:
4895:
4894:
4887:
4886:
4882:
4877:
4869:
4865:
4861:
4857:
4854:(11): 90–99.
4853:
4849:
4848:
4843:
4836:
4825:
4821:
4817:
4812:
4807:
4802:
4797:
4793:
4789:
4785:
4781:
4780:
4772:
4765:
4757:
4753:
4749:
4745:
4744:
4739:
4732:
4724:
4723:
4718:
4712:
4704:
4700:
4696:
4694:
4689:
4683:
4679:
4665:
4659:
4652:
4648:
4642:
4640:
4635:
4623:
4620:
4617:
4614:
4611:
4608:
4606:
4603:
4600:
4597:
4594:
4591:
4588:
4585:
4583:
4580:
4578:
4575:
4572:
4569:
4563:
4560:
4558:
4555:
4554:
4544:
4540:
4536:
4534:
4530:
4526:
4524:
4520:
4518:
4514:
4512:
4508:
4506:
4502:
4498:
4497:
4496:
4487:
4478:
4474:
4471:This section
4469:
4466:
4462:
4461:
4453:
4439:
4419:
4411:
4393:
4389:
4366:
4362:
4336:
4332:
4325:
4322:
4314:
4310:
4301:
4297:
4293:
4288:
4284:
4263:
4243:
4223:
4203:
4183:
4163:
4143:
4136:
4120:
4111:
4105:
4102:
4099:
4090:
4086:
4077:
4069:
4067:
4062:
3991:
3972:
3967:
3950:
3947:
3942:
3934:
3928:
3925:
3922:
3916:
3909:
3908:
3907:
3881:
3870:
3866:
3862:
3861:determined by
3857:
3855:
3851:
3850:
3844:
3840:
3821:
3818:
3815:
3812:
3806:
3799:
3798:
3797:
3795:
3794:
3783:
3781:
3765:
3762:
3759:
3753:
3750:
3745:
3741:
3732:
3725:
3721:
3717:
3713:
3691:
3687:
3681:
3671:
3667:
3660:
3657:
3652:
3649:
3645:
3637:
3636:
3635:
3633:
3629:
3625:
3620:
3618:
3615: →
3614:
3606:
3602:
3598:
3587:
3583:
3563:
3560:
3555:
3551:
3545:
3541:
3533:
3532:
3531:
3529:
3510:
3507:
3504:
3501:
3498:
3495:
3492:
3489:
3482:
3481:
3480:
3478:
3474:
3470:
3466:
3461:
3459:
3458:
3452:
3448:
3444:
3438:
3428:
3426:
3422:
3404:
3400:
3391:
3371:
3365:
3362:
3359:
3353:
3350:
3341:
3333:
3329:
3325:
3319:
3311:
3307:
3300:
3293:
3292:
3291:
3277:
3269:
3265:
3261:
3256:
3255:
3251:
3249:
3243:
3241:
3222:
3219:
3213:
3210:
3207:
3201:
3193:
3189:
3167:
3164:
3158:
3150:
3143:
3140:
3114:
3110:
3101:
3098: ∈
3097:
3093:
3090: ∈
3089:
3067:
3059:
3052:
3049:
3042:
3036:
3028:
3024:
3016:
3015:
3014:
3012:
3008:
3005:
3001:
2997:
2993:
2992:
2984:Geodesic flow
2981:
2979:
2963:
2959:
2955:
2950:
2948:
2944:
2940:
2936:
2932:
2925:containing 0.
2924:
2920:
2919:open interval
2917:is a maximal
2916:
2912:
2896:
2893:
2890:
2884:
2875:
2872:
2862:
2846:
2843:
2837:
2831:
2824:
2823:
2821:
2817:
2800:
2792:
2788:
2784:
2783:tangent space
2780:
2772:
2768:
2764:
2760:
2759:
2758:
2756:
2752:
2742:
2739:
2723:
2720:
2714:
2711:
2701:
2698:
2683:
2679:
2675:
2671:
2653:
2648:
2645:
2632:
2613:
2607:
2604:
2599:
2595:
2591:
2586:
2582:
2558:
2552:
2549:
2543:
2540:
2533:
2529:
2525:
2516:
2513:
2506:
2502:
2498:
2490:
2485:
2482:
2474:
2466:
2462:
2458:
2451:
2447:
2441:
2437:
2426:
2425:
2424:
2422:
2418:
2414:
2410:
2405:
2403:
2402:
2397:
2393:
2374:
2371:
2345:
2342:
2319:
2296:
2293:
2277:
2270:
2268:
2254:
2251:
2245:
2242:
2232:
2229:
2215:
2214:
2211:
2209:
2205:
2201:
2197:
2193:
2190:
2186:
2180:
2170:
2168:
2164:
2160:
2156:
2152:
2147:
2145:
2144:Jacobi fields
2126:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2096:
2091:
2088:
2085:
2082:
2079:
2074:
2068:
2061:
2052:
2038:
2032:
2029:
2026:
2017:
2011:
2006:
2002:
1994:
1993:
1992:
1990:
1986:
1967:
1961:
1958:
1955:
1952:
1946:
1941:
1938:
1935:
1930:
1924:
1908:
1902:
1893:
1887:
1884:
1877:
1876:
1875:
1873:
1869:
1865:
1855:
1853:
1849:
1845:
1827:
1822:
1819:
1791:
1788:
1785:
1779:
1776:
1769:
1765:
1761:
1752:
1749:
1742:
1738:
1734:
1726:
1721:
1718:
1710:
1702:
1698:
1694:
1687:
1683:
1677:
1673:
1662:
1661:
1660:
1658:
1654:
1649:
1632:
1626:
1606:
1598:
1594:
1575:
1569:
1546:
1540:
1516:
1513:
1509:
1505:
1502:
1495:
1469:
1463:
1457:
1454:
1451:
1445:
1442:
1437:
1429:
1423:
1416:
1415:
1414:
1412:
1394:
1391:
1388:
1384:
1361:
1357:
1348:
1344:
1340:
1336:
1332:
1313:
1310:
1307:
1297:
1288:
1285:
1279:
1273:
1264:
1261:
1247:
1241:
1237:
1231:
1226:
1222:
1216:
1213:
1208:
1202:
1196:
1189:
1188:
1187:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1146:
1142:
1138:
1134:
1131:The distance
1115:
1112:
1109:
1097:
1088:
1085:
1079:
1073:
1064:
1061:
1047:
1041:
1037:
1029:
1024:
1020:
1016:
1010:
1004:
997:
996:
995:
993:
989:
986:, the length
985:
982:
981:metric tensor
978:
975:
965:
963:
959:
954:
952:
951:shortest path
948:
943:
936:
929:
909:
905:
899:
895:
891:
886:
882:
877:
873:
862:
858:
851:
848:
840:
836:
829:
823:
816:
815:
814:
812:
808:
789:
785:
779:
775:
771:
766:
762:
757:
753:
750:
739:
735:
728:
725:
717:
713:
706:
700:
693:
692:
691:
688:
681:
674:
669:
665:
661:
656:
652:
645:
641:
637:
633:
630:
626:
621:
617:
613:
609:
605:
601:
597:
583:
579:
577:
573:
569:
560:
555:
551:
541:
539:
535:
531:
527:
523:
519:
515:
511:
507:
503:
499:
498:great circles
495:
491:
482:
475:
470:
461:
459:
455:
451:
447:
442:
440:
436:
432:
428:
424:
420:
416:
412:
407:
391:
387:
380:
372:
368:
363:
360:
356:
354:
350:
346:
342:
338:
334:
330:
326:
322:
321:
316:
315:open interval
312:
308:
304:
300:
296:
286:
284:
281:
277:
273:
268:
266:
262:
258:
254:
250:
246:
242:
237:
235:
231:
227:
223:
219:
215:
211:
207:
203:
199:
195:
191:
190:
185:
184:
179:
178:
172:
170:
169:straight line
166:
162:
158:
154:
150:
146:
140:
59:
55:
44:
43:Klein quartic
40:
34:
30:
26:
22:
7059:Hermann Weyl
6863:Vector space
6848:Pseudotensor
6817:
6813:Fiber bundle
6766:abstractions
6661:Mixed tensor
6646:Tensor field
6453:
6325:Moving frame
6320:Morse theory
6310:Gauge theory
6102:Tensor field
6031:Closed/Exact
6010:Vector field
5978:Distribution
5919:Hypercomplex
5914:Quaternionic
5746:
5651:Vector field
5609:Smooth atlas
5504:Applications
5432:Main results
5335:
5220:
5204:, New York:
5199:
5179:
5152:
5133:
5113:
5098:
5072:
5064:
5045:
5037:
5022:
5007:
4988:
4969:
4960:
4941:
4901:
4884:
4876:
4851:
4845:
4835:
4783:
4777:
4764:
4747:
4741:
4731:
4720:
4711:
4703:the original
4691:
4682:
4663:
4658:
4616:Zoll surface
4610:Morse theory
4605:Jacobi field
4494:
4481:
4477:adding to it
4472:
4456:Applications
4135:neighborhood
4091:
4087:
4083:
4075:
4063:
3987:
3860:
3858:
3853:
3847:
3842:
3838:
3836:
3791:
3789:
3723:
3719:
3715:
3711:
3709:
3627:
3621:
3616:
3612:
3603:denotes the
3600:
3596:
3585:
3581:
3579:
3527:
3525:
3468:
3462:
3454:
3447:vector field
3440:
3389:
3387:
3257:
3253:
3245:
3181:
3099:
3095:
3091:
3087:
3085:
3010:
3006:
2995:
2988:
2987:
2961:
2957:
2953:
2952:In general,
2951:
2946:
2942:
2928:
2922:
2914:
2819:
2815:
2790:
2786:
2774:
2770:
2766:
2762:
2750:
2748:
2630:
2573:
2416:
2412:
2406:
2399:
2392:vector field
2282:
2271:
2206:) such that
2203:
2191:
2184:
2182:
2148:
2141:
1982:
1867:
1861:
1850:, discussed
1847:
1806:
1656:
1650:
1596:
1592:
1487:
1409:curve), the
1346:
1342:
1338:
1334:
1330:
1328:
1176:
1172:
1168:
1164:
1163:such that γ(
1160:
1152:
1148:
1144:
1140:
1136:
1132:
1130:
991:
987:
983:
976:
971:
955:
950:
946:
941:
934:
927:
924:
810:
804:
686:
679:
672:
667:
663:
659:
654:
650:
643:
635:
631:
629:metric space
624:
619:
615:
611:
593:
572:great circle
567:
565:
533:
525:
521:
517:
513:
505:
501:
487:
443:
408:
371:great circle
366:
364:
361:
357:
353:elastic band
348:
344:
340:
336:
332:
328:
318:
310:
309:(a function
292:
289:Introduction
280:free falling
274:. Timelike
269:
238:
232:/nodes of a
228:between two
222:graph theory
214:great circle
187:
181:
175:
173:
57:
51:
6999:Élie Cartan
6947:Spin tensor
6921:Weyl tensor
6879:Mathematics
6843:Multivector
6634:definitions
6532:Engineering
6471:Mathematics
6270:Levi-Civita
6260:Generalized
6232:Connections
6182:Lie algebra
6114:Volume form
6015:Vector flow
5988:Pushforward
5983:Lie bracket
5882:Lie algebra
5847:G-structure
5636:Pushforward
5616:Submanifold
5134:Gravitation
5125:Thorne, Kip
4993:McGraw-Hill
4955:introducing
4080:Ribbon test
3869:Spivak 1999
3731:pushforward
3530:satisfying
3451:total space
3264:Hamiltonian
2994:is a local
2419:(using the
2167:Hamiltonian
257:transported
198:ellipsoidal
7074:Categories
6828:Linear map
6696:Operations
6393:Stratifold
6351:Diffeology
6147:Associated
5948:Symplectic
5933:Riemannian
5862:Hyperbolic
5789:Submersion
5697:Hopf–Rinow
5631:Submersion
5626:Smooth map
5464:Ricci flow
5413:Hyperbolic
4938:references
4717:"geodesic"
4688:"geodesic"
4674:References
4649:, e.g., a
4523:UV mapping
4156:of a line
3790:Equation (
3710:for every
3473:horizontal
2822:such that
2177:See also:
548:See also:
216:(see also
208:, it is a
186:come from
165:connection
6967:EM tensor
6803:Dimension
6754:Transpose
6275:Principal
6250:Ehresmann
6207:Subbundle
6197:Principal
6172:Fibration
6152:Cotangent
6024:Covectors
5877:Lie group
5857:Hermitian
5800:manifolds
5769:Immersion
5764:Foliation
5702:Noether's
5687:Frobenius
5682:De Rham's
5677:Darboux's
5568:Manifolds
5408:Hermitian
5361:Signature
5324:Sectional
5304:Curvature
5186:EMS Press
4963:July 2014
4541:(e.g. in
4484:June 2014
4333:ε
4302:∗
4294:−
4244:ε
4224:ℓ
4164:ℓ
4118:→
4112:ℓ
4049:∇
4026:¯
4023:∇
4000:∇
3961:¯
3958:∇
3951:−
3939:∇
3891:¯
3888:∇
3879:∇
3810:↦
3760:λ
3757:↦
3746:λ
3729:) is the
3672:λ
3650:λ
3546:∗
3542:π
3505:⊕
3455:geodesic
3401:γ
3214:
3180:. Thus,
3144:˙
3141:γ
3111:γ
3053:˙
3050:γ
2989:Geodesic
2945:and
2876:˙
2873:γ
2832:γ
2801:γ
2715:˙
2712:γ
2702:˙
2699:γ
2693:∇
2654:λ
2649:ν
2646:μ
2642:Γ
2608:γ
2605:∘
2600:μ
2587:μ
2583:γ
2534:ν
2530:γ
2507:μ
2503:γ
2491:λ
2486:ν
2483:μ
2479:Γ
2452:λ
2448:γ
2375:˙
2372:γ
2346:˙
2343:γ
2297:˙
2294:γ
2246:˙
2243:γ
2233:˙
2230:γ
2224:∇
2165:taken as
2121:ψ
2112:φ
2103:γ
2066:∂
2059:∂
2049:∂
2033:ψ
2027:φ
2018:γ
2003:δ
1962:φ
1953:γ
1922:∂
1918:∂
1903:φ
1894:γ
1885:δ
1828:λ
1823:ν
1820:μ
1816:Γ
1770:ν
1743:μ
1727:λ
1722:ν
1719:μ
1715:Γ
1688:λ
1633:γ
1607:γ
1576:γ
1547:γ
1514:γ
1503:γ
1470:γ
1455:−
1443:≤
1430:γ
1289:˙
1286:γ
1265:˙
1262:γ
1242:γ
1223:∫
1203:γ
1089:˙
1086:γ
1065:˙
1062:γ
1042:γ
1021:∫
1011:γ
892:−
852:γ
830:γ
772:−
729:γ
707:γ
544:Triangles
504:to point
431:satellite
423:spacetime
384:→
174:The noun
6833:Manifold
6818:Geodesic
6576:Notation
6371:Orbifold
6366:K-theory
6356:Diffiety
6080:Pullback
5894:Oriented
5872:Kenmotsu
5852:Hadamard
5798:Types of
5747:Geodesic
5572:Glossary
5423:Kenmotsu
5336:Geodesic
5289:Glossary
5196:(1972),
5131:(1973),
5097:(1975),
5021:(1978),
4900:(1999),
4824:Archived
4550:See also
4276:we have
3250:on
2941:on both
2939:smoothly
2818:→
2814: :
2668:are the
2396:open set
2194:with an
2185:geodesic
1842:are the
1517:′
1506:′
690:we have
640:constant
636:geodesic
614: :
604:distance
492:. On a
464:Examples
313:from an
301:for the
299:equation
230:vertices
226:geodesic
204:. For a
183:geodetic
177:geodesic
58:geodesic
54:geometry
6930:Physics
6764:Related
6527:Physics
6445:Tensors
6315:History
6298:Related
6212:Tangent
6190:)
6170:)
6137:Adjoint
6129:Bundles
6107:density
6005:Torsion
5971:Vectors
5963:Tensors
5946:)
5931:)
5927:,
5925:Pseudo−
5904:Poisson
5837:Finsler
5832:Fibered
5827:Contact
5825:)
5817:Complex
5815:)
5784:Section
5490:Hilbert
5485:Finsler
4951:improve
4868:7078650
4820:9671694
4788:Bibcode
4410:metrics
3992:, then
3865:torsion
3588:; here
3449:on the
3290:, i.e.
3238:is the
3002:on the
2161:, with
1870:. The
1157:infimum
600:locally
367:locally
347:−
210:segment
202:surface
189:geodesy
163:with a
153:surface
143:) is a
25:Geodesy
6858:Vector
6853:Spinor
6838:Matrix
6632:Tensor
6280:Vector
6265:Koszul
6245:Cartan
6240:Affine
6222:Vector
6217:Tensor
6202:Spinor
6192:Normal
6188:Stable
6142:Affine
6046:bundle
5998:bundle
5944:Almost
5867:Kähler
5823:Almost
5813:Almost
5807:Closed
5707:Sard's
5663:(list)
5418:Kähler
5314:Scalar
5309:tensor
5212:
5163:
5141:
5105:
5079:
5056:
5029:
4999:
4940:, but
4908:
4866:
4818:
4808:
4693:Lexico
4645:For a
4527:robot
4354:where
3837:where
3086:where
3000:action
2964:. Any
2913:where
2633:) and
2574:where
2556:
2407:Using
2394:in an
1807:where
1413:gives
1337:, but
1181:action
1171:and γ(
552:, and
494:sphere
303:length
6778:Basis
6463:Scope
6388:Sheaf
6162:Fiber
5938:Rizza
5909:Prime
5740:Local
5730:Curve
5592:Atlas
5319:Ricci
5238:torus
4864:S2CID
4827:(PDF)
4811:21092
4774:(PDF)
4630:Notes
4256:from
4133:of a
3471:into
3457:spray
2781:(the
2423:) as
2200:curve
2187:on a
1852:below
979:with
972:In a
634:is a
608:curve
520:. If
417:. In
335:) to
307:curve
305:of a
263:of a
239:In a
234:graph
212:of a
194:Earth
145:curve
6255:Form
6157:Dual
6090:flow
5953:Tame
5929:Sub−
5842:Flat
5722:Maps
5210:ISBN
5161:ISBN
5139:ISBN
5103:ISBN
5077:ISBN
5054:ISBN
5027:ISBN
4997:ISBN
4906:ISBN
4816:PMID
4432:and
4408:are
4381:and
4012:and
3841:and
3475:and
3102:and
2991:flow
2749:The
1983:The
1651:The
1147:and
528:are
524:and
516:and
456:and
56:, a
6177:Jet
4856:doi
4806:PMC
4796:doi
4752:doi
4503:or
4479:.
4412:on
3988:is
3423:.
3211:exp
2976:is
2921:in
2860:and
2789:at
2785:to
2773:in
2765:in
2411:on
1183:or
1151:of
949:or
666:in
662:of
646:≥ 0
594:In
510:arc
317:of
171:".
149:arc
106:-,-
52:In
7076::
6168:Co
5219:.
5208:,
5184:,
5178:,
5159:,
5155:,
5127:;
5123:;
5112:.
5093:;
5063:.
5052:,
5036:.
5017:;
5006:.
4995:,
4862:.
4852:60
4850:.
4844:.
4822:.
4814:.
4804:.
4794:.
4784:95
4782:.
4776:.
4748:16
4746:.
4740:.
4719:.
4697:.
4690:.
4638:^
4452:.
4068:.
3856:.
3782:.
3479::
3467:TT
3460:.
3244:tV
3100:TM
3094:,
3007:TM
2980:.
2949:.
2202:γ(
2183:A
2169:.
1854:.
1139:,
953:.
940:∈
933:,
685:∈
678:,
653:∈
618:→
602:a
578:.
566:A
472:A
285:.
236:.
127:,-
115:iː
103:oʊ
100:,-
73:iː
70:dʒ
6437:e
6430:t
6423:v
6186:(
6166:(
5942:(
5923:(
5821:(
5811:(
5574:)
5570:(
5560:e
5553:t
5546:v
5368:/
5291:)
5287:(
5277:e
5270:t
5263:v
5223:.
5189:.
5116:.
5086:.
5067:.
5040:.
5010:.
4976:)
4970:(
4965:)
4961:(
4947:.
4888:.
4870:.
4858::
4798::
4790::
4758:.
4754::
4664:k
4486:)
4482:(
4440:S
4420:N
4394:S
4390:g
4367:N
4363:g
4342:)
4337:2
4329:(
4326:O
4323:=
4320:)
4315:S
4311:g
4307:(
4298:f
4289:N
4285:g
4264:l
4204:f
4184:S
4144:N
4121:S
4115:)
4109:(
4106:N
4103::
4100:f
3973:Y
3968:X
3948:Y
3943:X
3935:=
3932:)
3929:Y
3926:,
3923:X
3920:(
3917:D
3882:,
3867:(
3849:1
3846:(
3843:b
3839:a
3822:b
3819:+
3816:t
3813:a
3807:t
3793:1
3766:.
3763:X
3754:X
3751::
3742:S
3727:λ
3724:S
3722:(
3720:d
3716:M
3712:X
3692:X
3688:H
3682:X
3678:)
3668:S
3664:(
3661:d
3658:=
3653:X
3646:H
3628:M
3617:M
3613:M
3609:π
3601:M
3597:M
3593:∗
3590:π
3586:M
3582:v
3564:v
3561:=
3556:v
3552:W
3528:W
3511:.
3508:V
3502:H
3499:=
3496:M
3493:T
3490:T
3469:M
3405:V
3390:V
3372:.
3369:)
3366:V
3363:,
3360:V
3357:(
3354:g
3351:=
3348:)
3345:)
3342:V
3339:(
3334:t
3330:G
3326:,
3323:)
3320:V
3317:(
3312:t
3308:G
3304:(
3301:g
3278:g
3254:.
3252:M
3226:)
3223:V
3220:t
3217:(
3208:=
3205:)
3202:V
3199:(
3194:t
3190:G
3168:V
3165:=
3162:)
3159:0
3156:(
3151:V
3115:V
3096:V
3092:R
3088:t
3071:)
3068:t
3065:(
3060:V
3043:=
3040:)
3037:V
3034:(
3029:t
3025:G
3011:M
2998:-
2996:R
2974:M
2970:ℝ
2966:γ
2962:R
2958:R
2954:I
2947:V
2943:p
2923:R
2915:I
2897:,
2894:V
2891:=
2888:)
2885:0
2882:(
2847:p
2844:=
2841:)
2838:0
2835:(
2820:M
2816:I
2791:p
2787:M
2779:M
2777:p
2775:T
2771:V
2767:M
2763:p
2724:0
2721:=
2631:t
2617:)
2614:t
2611:(
2596:x
2592:=
2559:,
2553:0
2550:=
2544:t
2541:d
2526:d
2517:t
2514:d
2499:d
2475:+
2467:2
2463:t
2459:d
2442:2
2438:d
2413:M
2401:1
2320:t
2276:)
2274:1
2272:(
2255:0
2252:=
2204:t
2192:M
2127:.
2124:)
2118:s
2115:+
2109:t
2106:+
2100:(
2097:E
2092:0
2089:=
2086:t
2083:=
2080:s
2075:|
2069:t
2062:s
2053:2
2039:=
2036:)
2030:,
2024:(
2021:)
2015:(
2012:E
2007:2
1968:.
1965:)
1959:t
1956:+
1950:(
1947:E
1942:0
1939:=
1936:t
1931:|
1925:t
1909:=
1906:)
1900:(
1897:)
1891:(
1888:E
1868:E
1792:,
1789:0
1786:=
1780:t
1777:d
1766:x
1762:d
1753:t
1750:d
1739:x
1735:d
1711:+
1703:2
1699:t
1695:d
1684:x
1678:2
1674:d
1657:E
1636:)
1630:(
1627:L
1597:E
1593:E
1579:)
1573:(
1570:L
1550:)
1544:(
1541:E
1521:)
1510:,
1499:(
1496:g
1473:)
1467:(
1464:E
1461:)
1458:a
1452:b
1449:(
1446:2
1438:2
1434:)
1427:(
1424:L
1395:2
1392:,
1389:1
1385:W
1362:1
1358:C
1347:E
1343:L
1339:L
1335:L
1331:E
1314:.
1311:t
1308:d
1304:)
1301:)
1298:t
1295:(
1280:,
1277:)
1274:t
1271:(
1256:(
1251:)
1248:t
1245:(
1238:g
1232:b
1227:a
1217:2
1214:1
1209:=
1206:)
1200:(
1197:E
1177:q
1173:b
1169:p
1165:a
1161:M
1153:M
1149:q
1145:p
1141:q
1137:p
1135:(
1133:d
1116:.
1113:t
1110:d
1104:)
1101:)
1098:t
1095:(
1080:,
1077:)
1074:t
1071:(
1056:(
1051:)
1048:t
1045:(
1038:g
1030:b
1025:a
1017:=
1014:)
1008:(
1005:L
992:M
988:L
984:g
977:M
942:I
938:2
935:t
931:1
928:t
910:.
906:|
900:2
896:t
887:1
883:t
878:|
874:=
871:)
868:)
863:2
859:t
855:(
849:,
846:)
841:1
837:t
833:(
827:(
824:d
811:v
790:.
786:|
780:2
776:t
767:1
763:t
758:|
754:v
751:=
748:)
745:)
740:2
736:t
732:(
726:,
723:)
718:1
714:t
710:(
704:(
701:d
687:J
683:2
680:t
676:1
673:t
668:I
664:t
660:J
655:I
651:t
644:v
632:M
625:I
620:M
616:I
612:γ
526:B
522:A
518:B
514:A
506:B
502:A
476:.
392:2
388:t
381:t
349:t
345:s
341:t
339:(
337:f
333:s
331:(
329:f
320:R
311:f
139:/
136:k
133:ɪ
130:z
124:k
121:ɪ
118:s
112:d
109:ˈ
97:k
94:ɪ
91:s
88:ɛ
85:d
82:ˈ
79:ə
76:.
67:ˌ
64:/
60:(
35:.
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