Manifold - Knowledge

360: 43: 1273: 6593: 2722: 112: 4371:, and a Cartesian product of charts is a chart for the product manifold. Thus, an atlas for the product manifold can be constructed using atlases for its factors. If these atlases define a differential structure on the factors, the corresponding atlas defines a differential structure on the product manifold. The same is true for any other structure defined on the factors. If one of the factors has a boundary, the product manifold also has a boundary. Cartesian products may be used to construct tori and finite 2399:. It is not generally possible to describe a manifold with just one chart, because the global structure of the manifold is different from the simple structure of the charts. For example, no single flat map can represent the entire Earth without separation of adjacent features across the map's boundaries or duplication of coverage. When a manifold is constructed from multiple overlapping charts, the regions where they overlap carry information essential to understanding the global structure. 1903:, possibly with additional structure. A manifold can be constructed by giving a collection of coordinate charts, that is, a covering by open sets with homeomorphisms to a Euclidean space, and patching functions: homeomorphisms from one region of Euclidean space to another region if they correspond to the same part of the manifold in two different coordinate charts. A manifold can be given additional structure if the patching functions satisfy axioms beyond continuity. For instance, 6518:. Gluing the circles together will produce a new, closed manifold without boundary, the Klein bottle. Closing the surface does nothing to improve the lack of orientability, it merely removes the boundary. Thus, the Klein bottle is a closed surface with no distinction between inside and outside. In three-dimensional space, a Klein bottle's surface must pass through itself. Building a Klein bottle which is not self-intersecting requires four or more dimensions of space. 6685: 3618: 1791: 6490:

open, so that it could unroll to become a rectangle, but keep a grasp on the cut ends. Twist one end 180°, making the inner surface face out, and glue the ends back together seamlessly. This results in a strip with a permanent half-twist: the Möbius strip. Its boundary is no longer a pair of circles, but (topologically) a single circle; and what was once its "inside" has merged with its "outside", so that it now has only a

6476: 4391: 100: 6446:, a chart causes its piece of the manifold to itself acquire a sense of ordering, which in 3-dimensions can be viewed as either right-handed or left-handed. Overlapping charts are not required to agree in their sense of ordering, which gives manifolds an important freedom. For some manifolds, like the sphere, charts can be chosen so that overlapping regions agree on their "handedness"; these are 1678:, so neither chart alone is sufficient to cover the whole circle. It can be proved that it is not possible to cover the full circle with a single chart. For example, although it is possible to construct a circle from a single line interval by overlapping and "gluing" the ends, this does not produce a chart; a portion of the circle will be mapped to both ends at once, losing invertibility. 1259: 6556:) to mathematically merge each antipode pair into a single point. The closed surface so produced is the real projective plane, yet another non-orientable surface. It has a number of equivalent descriptions and constructions, but this route explains its name: all the points on any given line through the origin project to the same "point" on this "plane". 6568:, or "number of handles" present in a surface. A torus is a sphere with one handle, a double torus is a sphere with two handles, and so on. Indeed, it is possible to fully characterize compact, two-dimensional manifolds on the basis of genus and orientability. In higher-dimensional manifolds genus is replaced by the notion of 6533: 6262:

Orientable surfaces can be visualized, and their diffeomorphism classes enumerated, by genus. Given two orientable surfaces, one can determine if they are diffeomorphic by computing their respective genera and comparing: they are diffeomorphic if and only if the genera are equal, so the genus forms a

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All differentiable manifolds (of constant dimension) can be given the structure of a Riemannian manifold. The Euclidean space itself carries a natural structure of Riemannian manifold (the tangent spaces are naturally identified with the Euclidean space itself and carry the standard scalar product of

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between the two boundaries. Two points are identified when they are mapped onto each other. For a topological manifold, this bijection should be a homeomorphism, otherwise the result will not be a topological manifold. Similarly, for a differentiable manifold, it has to be a diffeomorphism. For other

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algebraic varieties over the real or complex numbers are manifolds. One generalizes this first by allowing singularities, secondly by allowing different fields, and thirdly by emulating the patching construction of manifolds: just as a manifold is glued together from open subsets of Euclidean space,

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of points which are mapped to each other by transition maps. Charts map equivalence classes to points of a single patch. There are usually strong demands on the consistency of the transition maps. For topological manifolds they are required to be homeomorphisms; if they are also diffeomorphisms, the

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A manifold can be constructed by gluing together pieces in a consistent manner, making them into overlapping charts. This construction is possible for any manifold and hence it is often used as a characterisation, especially for differentiable and Riemannian manifolds. It focuses on an atlas, as the

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This is much harder in higher dimensions: higher-dimensional manifolds cannot be directly visualized (though visual intuition is useful in understanding them), nor can their diffeomorphism classes be enumerated, nor can one in general determine if two different descriptions of a higher-dimensional

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However, excluded are examples like two touching circles that share a point to form a figure-8; at the shared point, a satisfactory chart cannot be created. Even with the bending allowed by topology, the vicinity of the shared point looks like a "+", not a line. A "+" is not homeomorphic to a line

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Begin with an infinite circular cylinder standing vertically, a manifold without boundary. Slice across it high and low to produce two circular boundaries, and the cylindrical strip between them. This is an orientable manifold with boundary, upon which "surgery" will be performed. Slice the strip

5401:. By definition, all manifolds are topological manifolds, so the phrase "topological manifold" is usually used to emphasize that a manifold lacks additional structure, or that only its topological properties are being considered. Formally, a topological manifold is a topological space 2549:

Charts in an atlas may overlap and a single point of a manifold may be represented in several charts. If two charts overlap, parts of them represent the same region of the manifold, just as a map of Europe and a map of Russia may both contain Moscow. Given two overlapping charts, a

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manifold looks locally like a Euclidean space in a rather weak manner: while for each individual chart it is possible to distinguish differentiable functions or measure distances and angles, merely by virtue of being a topological manifold a space does not have any

7016:(top dimensional manifold and codimension 1 boundary) and manifolds with corners (top dimensional manifold, codimension 1 boundary, codimension 2 corners). Whitney stratified spaces are a broad class of spaces, including algebraic varieties, analytic varieties, 7049:

is a space that behaves like a manifold from the point of view of homology theory. These are not all manifolds, but (in high dimension) can be analyzed by surgery theory similarly to manifolds, and failure to be a manifold is a local obstruction, as in surgery

8234:(3rd edition) Publish or Perish Inc. Encyclopedic five-volume series presenting a systematic treatment of the theory of manifolds, Riemannian geometry, classical differential geometry, and numerous other topics at the first- and second-year graduate levels. 1598: 4222: 485: 3098: 2425:, of a manifold is an invertible map between a subset of the manifold and a simple space such that both the map and its inverse preserve the desired structure. For a topological manifold, the simple space is a subset of some Euclidean space 932: 2541:(i.e. an equivalence class containing that given atlas). Unlike an ordinary atlas, the maximal atlas of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations). 2640:

An atlas can also be used to define additional structure on the manifold. The structure is first defined on each chart separately. If all transition maps are compatible with this structure, the structure transfers to the manifold.

5299:, formulated in the 1970s, provided a far-reaching extension of the Poincaré conjecture to the general three-dimensional manifolds. Four-dimensional manifolds were brought to the forefront of mathematical research in the 1980s by 3814: 6259:. This is a classification in principle: the general question of whether two smooth manifolds are diffeomorphic is not computable in general. Further, specific computations remain difficult, and there are many open questions. 7034:

is a topological space formed by gluing disks of different dimensionality together. In general the resulting space is singular, hence not a manifold. However, they are of central interest in algebraic topology, especially in

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manifold have the same dimension. Some authors require that all charts of a topological manifold map to Euclidean spaces of same dimension. In that case every topological manifold has a topological invariant, its dimension.

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After a line, a circle is the simplest example of a topological manifold. Topology ignores bending, so a small piece of a circle is treated the same as a small piece of a line. Considering, for instance, the top part of the

4738:), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using 2959: 1254:{\displaystyle {\begin{aligned}T(a)&=\chi _{\mathrm {right} }\left(\chi _{\mathrm {top} }^{-1}\left\right)\\&=\chi _{\mathrm {right} }\left(a,{\sqrt {1-a^{2}}}\right)\\&={\sqrt {1-a^{2}}}\end{aligned}}} 5469:

discussed below. In particular, the same underlying topological manifold can have several mutually incompatible classes of differentiable functions and an infinite number of ways to specify distances and angles.

4035:. When a manifold is viewed in this way, it is easy to use intuition from Euclidean spaces to define additional structure. For example, in a Euclidean space, it is always clear whether a vector at some point is 6284:, because, while they may be defined in terms of some presentation (such as the genus in terms of a triangulation), they are the same relative to all possible descriptions of a particular manifold: they are 1361: 5515:, is used. If the local charts on a manifold are compatible in a certain sense, one can define directions, tangent spaces, and differentiable functions on that manifold. In particular it is possible to use 1441: 7222: 7370: 4332:

Two manifolds with boundaries can be glued together along a boundary. If this is done the right way, the result is also a manifold. Similarly, two boundaries of a single manifold can be glued together.

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formalisms of classical mechanics. This space is, in fact, a high-dimensional manifold, whose dimension corresponds to the degrees of freedom of the system and where the points are specified by their

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In dimensions two and higher, a simple but important invariant criterion is the question of whether a manifold admits a meaningful orientation. Consider a topological manifold with charts mapping to

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is locally isomorphic to the sheaf of continuous (or differentiable, or complex-analytic, etc.) functions on Euclidean space. This definition is mostly used when discussing analytic manifolds in

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manifolds. For others, this is impossible. The latter possibility is easy to overlook, because any closed surface embedded (without self-intersection) in three-dimensional space is orientable.

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The definition of a manifold can be generalized by dropping the requirement of finite dimensionality. Thus an infinite dimensional manifold is a topological space locally homeomorphic to a

5548:. For smooth manifolds the transition maps are smooth, that is, infinitely differentiable. Analytic manifolds are smooth manifolds with the additional condition that the transition maps are 4123: 5622: 1280:

The top, bottom, left, and right charts do not form the only possible atlas. Charts need not be geometric projections, and the number of charts is a matter of choice. Consider the charts

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independently discovered them 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classical Euclidean space; these gave rise to

2534:. An atlas is not unique as all manifolds can be covered in multiple ways using different combinations of charts. Two atlases are said to be equivalent if their union is also an atlas. 6753: 6291:

One could hope to develop an arsenal of invariant criteria that would definitively classify all manifolds up to isomorphism. It is known that for manifolds of dimension 4 and higher,

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side. Similarly to the Klein Bottle below, this two dimensional surface would need to intersect itself in two dimensions, but can easily be constructed in three or more dimensions.

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As with the circle, one may define one chart that covers the whole sphere excluding one point. Thus two charts are sufficient, but the sphere cannot be covered by a single chart.

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In technical language, a manifold with boundary is a space containing both interior points and boundary points. Every interior point has a neighborhood homeomorphic to the open

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This is the standard way differentiable manifolds are defined. If the transition functions of an atlas for a topological manifold preserve the natural differential structure of

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in which the surface lies. Such a surface would, in modern terminology, be called a manifold; and in modern terms, the theorem proved that the curvature of the surface is an

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A finite cylinder may be constructed as a manifold by starting with a strip × and gluing a pair of opposite edges on the boundary by a suitable diffeomorphism. A

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because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of

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faces, and in fact, will remain true for any spherical map, even if it does not arise from any convex polytope. Thus 2 is a topological invariant of the sphere, called its

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A "stratified space" is a space that can be divided into pieces ("strata"), with each stratum a manifold, with the strata fitting together in prescribed ways (formally, a

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are likewise glued together from affine schemes, which are a generalization of algebraic varieties. Both are related to manifolds, but are constructed algebraically using

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of a continuously differentiable function is a manifold in the latter sense. He then proposes a new, more general, definition of manifold based on a 'chain of manifolds' (

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with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, thus differentiable on the manifold as a whole.

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A sphere can be treated in almost the same way as the circle. In mathematics a sphere is just the surface (not the solid interior), which can be defined as a subset of

3427: 2984: 825: 5982: 4597: = 1 face. Thus the Euler characteristic of the torus is 1 − 2 + 1 = 0. The Euler characteristic of other surfaces is a useful 4972: 4925: 3164: 3131: 4486:. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space. 2300: 7462: 5927: 2818: 1044: 820: 6716: 6008: 5076:

in a neighborhood of every point except the points (1, 0) and (−1, 0); and in a neighborhood of those points, it can be thought of as the graph of, respectively,

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It is possible to define different points of a manifold to be the same point. This can be visualized as gluing these points together in a single point, forming a

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The patchwork construction does not use any embedding, but simply views the manifold as a topological space by itself. This abstract point of view is called the

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The first construction and this construction are very similar, but represent rather different points of view. In the first construction, the manifold is seen as

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gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912, opening the road to the general concept of a

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translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a

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A single manifold can be constructed in different ways, each stressing a different aspect of the manifold, thereby leading to a slightly different viewpoint.

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a pure manifold. Since dimension is a local invariant (i.e. the map sending each point to the dimension of its neighbourhood over which a chart is defined, is

658:{\displaystyle {\begin{aligned}\chi _{\mathrm {bottom} }(x,y)&=x\\\chi _{\mathrm {left} }(x,y)&=y\\\chi _{\mathrm {right} }(x,y)&=y.\end{aligned}}} 2696:

The structure on the manifold depends on the atlas, but sometimes different atlases can be said to give rise to the same structure. Such atlases are called

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allows the definition of distance but does not require the concept of angle; it is an analytic manifold in which each tangent space is equipped with a

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Usually additional technical assumptions on the topological space are made to exclude pathological cases. It is customary to require that the space be

4256:. There is, however, no reason to expect such quotient spaces to be manifolds. Among the possible quotient spaces that are not necessarily manifolds, 1283: 1366: 4681: 1760:

plane of coordinates. This provides two charts; the four other charts are provided by a similar construction with the two other coordinate planes.

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showed in 1960 that no algorithm exists for classifying four-dimensional manifolds. Important work on higher-dimensional manifolds, including

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Different notions of manifolds have different notions of classification and invariant; in this section we focus on smooth closed manifolds.

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showed that the intrinsic definition in terms of charts was equivalent to Poincaré's definition in terms of subsets of Euclidean space.

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The manifold is constructed by specifying an atlas, which is itself defined by transition maps. A point of the manifold is therefore an

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Take two Möbius strips; each has a single loop as a boundary. Straighten out those loops into circles, and let the strips distort into

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Smooth closed manifolds have no local invariants (other than dimension), though geometric manifolds have local invariants, notably the

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A Euclidean vector space with the group operation of vector addition is an example of a non-compact Lie group. A simple example of a

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aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed through

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The description of most manifolds requires more than one chart. A specific collection of charts which covers a manifold is called an

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is a precursor to the modern notion of atlas. In particular, he considers two manifolds defined respectively as graphs of functions

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choice of such concepts. In order to discuss such properties for a manifold, one needs to specify further structure and consider

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The spherical Earth is navigated using flat maps or charts, collected in an atlas. Similarly, a manifold can be described using

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This example is historically significant, as it has motivated the terminology; it became apparent that the whole surface of the

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patches naturally provide charts, and since there is no exterior space involved it leads to an intrinsic view of the manifold.

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grew historically from constructions like this. Here is another example, applying this method to the construction of a sphere:

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The simplest kind of manifold to define is the topological manifold, which looks locally like some "ordinary" Euclidean space

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of the manifold. Generally manifolds are taken to have a constant local dimension, and the local dimension is then called the

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of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g.

8741: 6927:. The singularities correspond to fixed points of the group actions, and the actions must be compatible in a certain sense. 6548:

Begin with a sphere centered on the origin. Every line through the origin pierces the sphere in two opposite points called

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Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. His

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is glued together from affine algebraic varieties, which are zero sets of polynomials over algebraically closed fields.

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of the manifold at a certain point is the dimension of the Euclidean space that the charts at that point map to (number

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Perhaps the simplest way to construct a manifold is the one used in the example above of the circle. First, a subset of

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maps an open subset around an interior point to an open Euclidean subset, while the boundary chart with transition map

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is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as

5595: 2836:(sphere plus interior) is a 3-manifold with boundary. Its boundary is a sphere, a 2-manifold. (Do not confuse with 2716: 8826: 8435: 7099: 6846: 6621:. In addition to continuous functions and smooth functions generally, there are maps with special properties. In 6330: 4695:

Riemann was the first one to do extensive work generalizing the idea of a surface to higher dimensions. The name

4676:. For an unconstrained movement of free particles the manifold is equivalent to the Euclidean space, but various 6915:" in the topology. Roughly speaking, it is a space which locally looks like the quotients of some simple space ( 4523: 1833:, unless the dimension of the manifold is 0. Putting these freedoms together, other examples of manifolds are a 7009: 6818: 5413: 5336: 400: 363:

Figure 1: The four charts each map part of the circle to an open interval, and together cover the whole circle.

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if there is some intrinsic characteristic that differentiates them. Such criteria are commonly referred to as

2321:. However, some authors admit manifolds that are not connected, and where different points can have different 972: 712: 9472: 9404: 9038: 8445: 8277: 7255: 4840: 4452: 1593:{\displaystyle {\begin{aligned}x&={\frac {1-s^{2}}{1+s^{2}}}\\y&={\frac {2s}{1+s^{2}}}\end{aligned}}} 8208: 6975: 2382: 937: 677: 9848: 9497: 9023: 8746: 8520: 8221: 6337:. This distinction between local invariants and no local invariants is a common way to distinguish between 5824: 5734:-spheres, are specified as subspaces of a Euclidean space and inherit a metric from their embedding in it. 3509: 3347: 3221: 4561:

The same formula will hold if we project the vertices and edges of the polytope onto a sphere, creating a

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excludes spaces such as "the line with two origins" (these generalizations of manifolds are discussed in

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in the first half of 19th century led them to consider special types of complex manifolds, now known as

4217:{\displaystyle \mathbb {R} ^{n}\setminus \{0\}\to \mathbb {R} ^{n}\setminus \{0\}:x\mapsto x/\|x\|^{2}.} 4019:= 0 are not identified with any point on the first and second copy, respectively). This gives a circle. 482:. Similarly, there are charts for the bottom (red), left (blue), and right (green) parts of the circle: 9546: 9366: 9073: 9043: 8751: 8707: 8688: 8455: 8399: 7227: 6920: 6264: 5288: 5240: 4440: 4367:

The dimension of the product manifold is the sum of the dimensions of its factors. Its topology is the

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respectively, overlap in their domain: their intersection lies in the quarter of the circle where both

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In Riemannian geometry, one may ask for maps to preserve the Riemannian metric, leading to notions of

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is a generalisation of the idea of a circle (1-sphere) and sphere (2-sphere) to higher dimensions. An

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by closed subsets). There are various technical definitions, notably a Whitney stratified space (see

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The study of manifolds combines many important areas of mathematics: it generalizes concepts such as

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This function is its own inverse and thus can be used in both directions. As the transition map is a

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on the manifold. Two points are identified if one is moved onto the other by some group element. If

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maps a closed subset around a boundary point to a closed Euclidean subset. The boundary is itself a

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from the second half of the circle example. Start with two copies of the line. Use the coordinate

3093:{\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1{\text{ and }}x_{1}\geq 0\}} 927:{\displaystyle T:(0,1)\rightarrow (0,1)=\chi _{\mathrm {right} }\circ \chi _{\mathrm {top} }^{-1}} 9772: 9459: 9376: 9346: 8692: 8662: 8586: 8576: 8532: 8362: 8315: 6646: 6634: 6601: 6347:

is a source of a number of important global invariant properties. Some key criteria include the

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One method of identifying points (gluing them together) is through a right (or left) action of a

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to a Euclidean space. This means that every point has a neighbourhood for which there exists a

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The sphere is two-dimensional, so each chart will map part of the sphere to an open subset of

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are introduced in an analogous way by requiring that the transition functions of an atlas are

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Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse.

4660:. The possible states of a mechanical system are thought to be points of an abstract space, 2279: 9787: 9715: 9601: 9467: 9429: 9361: 9013: 8951: 8799: 8503: 8493: 8465: 8440: 8350: 7800: 7435: 6942: 6569: 6205: 6059: 5905: 5869: 5865: 5783: 5367: 4684:. The theory of a rotating solid body, developed in the 18th century by Leonhard Euler and 4665: 4649: 4494: 4490: 4444: 4055: 2791: 2682: 2356: 1940: 1900: 1276:

Figure 2: A circle manifold chart based on slope, covering all but one point of the circle.

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property and orientability (see below). Indeed, several branches of mathematics, such as

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Manifolds can be equipped with additional structure. One important class of manifolds are

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Bryant, J.; Ferry, S.; Mio, W.; Weinberger, S. (1996). "Topology of homology manifolds".

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For example, the unit circle in the plane can be thought of as the graph of the function

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The surface of the Earth requires (at least) two charts to include every point. Here the

9124: 7804: 6886: 6093: 5782:, in a manner which varies smoothly from point to point. This norm can be extended to a 4952: 4905: 4641:

further contributed to their theory, clarifying the geometric meaning of the process of

9843: 9797: 9654: 9507: 9321: 9257: 9093: 9048: 8945: 8816: 8620: 8308: 7861: 7843: 7790: 7759: 7724: 7415: 7395: 7375: 7154: 7134: 7079: 7059: 7005: 6622: 6344: 6307: 6303: 6073: 5647: 5627: 4932: 4885: 4626: 4610: 4475: 4436: 3535: 3489: 3469: 3373: 3300: 3247: 3201: 3177: 2964: 2846: 2833: 2821: 2771: 2364: 1932: 773: 753: 336: 238: 205: 171: 150: 8630: 6154:'Infinite dimensional manifolds': to allow for infinite dimensions, one may consider 5821:

Lie group is the circle: the group operation is simply rotation. This group, known as

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Kervaire, M. (1961). "A Manifold which does not admit any differentiable structure".

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A basic example of maps between manifolds are scalar-valued functions on a manifold,

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function between Euclidean spaces that satisfies the nondegeneracy hypothesis of the

4677: 4630: 4622: 4509: 4471: 4460: 4361: 3972: 3809:{\displaystyle S=\left\{(x,y,z)\in \mathbb {R} ^{3}\mid x^{2}+y^{2}+z^{2}=1\right\}.} 3658:

is identified, and then an atlas covering this subset is constructed. The concept of

2690: 2537:

The atlas containing all possible charts consistent with a given atlas is called the

2474: 2388: 1889: 1695: 300: 140: 7649: 4689: 4653: 9807: 9482: 9449: 9434: 9316: 9185: 9166: 8960: 8915: 8838: 8809: 8667: 8600: 8595: 8590: 8580: 8372: 8355: 8209:

Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus

8159: 7932: 7914: 7892: 7853: 7812: 7808: 7751: 7716: 7683: 6894: 6870: 6865:

over the reals. This omits the point-set axioms, allowing higher cardinalities and

6830: 6795: 6541: 6349: 6218: 6217:

is a special kind of combinatorial manifold which is defined in digital space. See

6213: 6188: 6063: 6025: 5930: 5743: 5300: 5292: 5284: 4638: 4368: 4345: 2721: 2678: 2530: 2523: 2408: 2376: 2342: 1915: 1272: 668: 404: 242: 7536: – locally convex vector spaces satisfying a very mild completeness condition 7131:. It is an algebra with respect to the pointwise addition and multiplication. Let 6890: 6163: 2824:(circle plus interior) is a 2-manifold with boundary. Its boundary is a circle, a 2454:

and interest focuses on the topological structure. This structure is preserved by

1618:. These two charts provide a second atlas for the circle, with the transition map 111: 9777: 9725: 9669: 9649: 9551: 9439: 9306: 9277: 9109: 9018: 8848: 8804: 8570: 8239: 7999: 7704: 7667: 7036: 7021: 6878: 6605: 6552:. Although there is no way to do so physically, it is possible (by considering a 6184: 6155: 6067: 5624:

in a manner which varies smoothly from point to point. Given two tangent vectors

5581:

To measure distances and angles on manifolds, the manifold must be Riemannian. A

5560: 5532: 5493: 5474: 5304: 5268: 5236: 4759: 4700: 4634: 4585:

can be sliced open by its 'parallel' and 'meridian' circles, creating a map with

4562: 4227: 2346: 2318: 1966: 1918: 1826: 1819: 1795: 226: 199: 144: 6484: 6456: 6200:

is a kind of manifold which is discretization of a manifold. It usually means a

4448: 4349: 9817: 9782: 9679: 9512: 9502: 9492: 9414: 9386: 9371: 9356: 9272: 8975: 8900: 8870: 8768: 8761: 8701: 8672: 8542: 8537: 8498: 8227: 8203: 6874: 6806: 6642: 6256: 5934: 5858: 5854: 5267:

embedded in our common 3D space, were considered by Riemann under the guise of

4498: 4414: 3962: 2674: 2455: 2334: 2325:. If a manifold has a fixed dimension, this can be emphasized by calling it a 2008: 1775: 9762: 5567:

to higher dimensions; however, rectifiable sets are not in general manifolds.

4758:

evolved into what is today formalized as a manifold. Riemannian manifolds and

4425:

Before the modern concept of a manifold there were several important results.

4268:. An example of a quotient space of a manifold that is also a manifold is the 1866:

segment, since deleting the center point from the "+" gives a space with four

9837: 9754: 9659: 9571: 9444: 9161: 8985: 8980: 8965: 8955: 8905: 8882: 8756: 8716: 8657: 8605: 8404: 8126: 8077: 7965: 7820: 7464:

for which the above conditions hold, is called a Sikorski differential space.

6597: 6448: 6311: 5818: 5586: 5524: 5406: 5332: 5272: 4479: 4051: 4040: 3997:

for the second copy. Now, glue both copies together by identifying the point

2273: 1908: 1871: 1830: 411: 7988:

Foundational Essays on Topological Manifolds. Smoothings, and Triangulations

7897: 7880: 3950:) plane, an atlas of six charts is obtained which covers the entire sphere. 2583:

to the manifold and then back to another (or perhaps the same) open ball in

1461:

is the opposite of the slope of the line through the points at coordinates (

9822: 9626: 9611: 9576: 9424: 9409: 9088: 9083: 8925: 8892: 8865: 8773: 8414: 8284: 8146:

Knots, Groups, and 3-Manifolds. Papers Dedicated to the Memory of R. H. Fox

7983: 7602: 7560:

Differential-Algebraic Systems: Analytical Aspects and Circuit Applications

6956:, a variety is in general not a manifold, though linguistically the French 6924: 6882: 6573: 6509: 6460: 6159: 5753: 5721: 5553: 5344: 5228: 4781:

In the first section of Analysis Situs, Poincaré defines a manifold as the

4602: 4265: 1958: 1829:; thus a line segment without its end points is a manifold. They are never 254: 195: 104: 5324: 3166:. This invariance allows to "define" boundary points; see next paragraph. 667:

Together, these parts cover the whole circle, and the four charts form an

190:

for short, is a topological space with the property that each point has a

9710: 9684: 9606: 9295: 9234: 8931: 8920: 8877: 8778: 8379: 8059: 7474: 6838: 6630: 6617:

Just as there are various types of manifolds, there are various types of

6463:, which must intersect itself in its 3-space representation, and (3) the 6118: 5938: 5899: 5728:

the space). Many familiar curves and surfaces, including for example all

5697: 5511:

For most applications, a special kind of topological manifold, namely, a

5328: 5222: 4661: 4443:

first studied such geometries in 1733, but sought only to disprove them.

4070: 2954:{\displaystyle \{(x_{1},x_{2},\dots ,x_{n})\vert \Sigma x_{i}^{2}<1\}} 2704: 1962: 1846: 1691: 369: 316: 132: 6455:

Some illustrative examples of non-orientable manifolds include: (1) the

5307:, who was motivated by the then recent progress in theoretical physics ( 9591: 9156: 9114: 8940: 8853: 8485: 8389: 8300: 8212:. W.A. Benjamin Inc. (reprinted by Addison-Wesley and Westview Press). 7947:. Advanced undergraduate / first-year graduate text inspired by Milnor. 7865: 7755: 7728: 7527: 7521: 7515: 7031: 6969: 6810: 6581: 6244: 5807: 5717: 4261: 4036: 2825: 2765: 234: 124: 120: 6684: 5339:. A very pervasive and flexible technique underlying much work on the 4648:

Another important source of manifolds in 19th century mathematics was

4311: 2214:

This implies also that every point has a neighborhood homeomorphic to

9566: 9517: 8970: 8935: 8640: 8527: 8287:(A film explaining and visualizing manifolds up to fourth dimension.) 7848: 6626: 6532: 6515: 6175:

is a kind of manifold which is used to represent the phase spaces in

6010:

is the dimension of the sphere. Further examples can be found in the

5798: 5713: 5410: 5312: 5248: 4782: 4464: 4337: 4028: 3923: 2961:. Every boundary point has a neighborhood homeomorphic to the "half" 2322: 2317:

of the manifold. This is, in particular, the case when manifolds are

2038: 1838: 332: 7857: 7720: 7579:

Introduction to Holomorphic Functions of Several Variables, Volume 2

7171:. Suppose also that the following conditions hold. First: for every 6310:. The most familiar invariants, which are visible for surfaces, are 5814:

which is such that the group operations are defined by smooth maps.

5271:, and rigorously classified in the beginning of the 20th century by 3851:

coordinate (coloured red in the picture on the right). The function

3847:. Consider the northern hemisphere, which is the part with positive 3617: 2473:(see below) are all differentiable, allows us to do calculus on it. 9581: 9134: 9129: 9119: 8510: 7795: 7480: 6908: 6358: 6062:

on the overlaps. These manifolds are the basic objects of study in

5516: 5276: 4778:) which served as a precursor to the modern concept of a manifold. 4502: 4257: 4243: 1834: 1790: 343: 304: 296: 265: 8291: 7483: – Straight path on a curved surface or a Riemannian manifold 6365:

were founded in order to study invariant properties of manifolds.

5810:, are differentiable manifolds that carry also the structure of a 342:

The study of manifolds requires working knowledge of calculus and

9290: 9252: 8034:. Detailed and comprehensive first-year graduate text; sequel to 7076:

be a nonempty set. Suppose that some family of real functions on

6239:

The classification of smooth closed manifolds is well understood

4774:, Henri Poincaré gave a definition of a differentiable manifold ( 4310:

Manifolds which can be constructed by identifying points include

4234:= 1, the example simplifies to the circle example given earlier. 281: 7694:; translation in Russian Math. Surveys 53 (1998), no. 1, 229–236 7688: 7671: 6475: 5416:

whose inverse is also continuous) mapping that neighbourhood to

1939:

excludes spaces which are in some sense 'too large' such as the

1822:(all in "one piece"); an example is a pair of separate circles. 1356:{\displaystyle \chi _{\mathrm {minus} }(x,y)=s={\frac {y}{1+x}}} 790:-coordinates are positive. Both map this part into the interval 9616: 9208: 8726: 6911:

is a generalization of manifold allowing for certain kinds of "

5709: 4432: 4390: 2829: 2764:

is a manifold with an edge. For example, a sheet of paper is a

1957:

to a Euclidean space means that every point has a neighborhood

1687: 1436:{\displaystyle \chi _{\mathrm {plus} }(x,y)=t={\frac {y}{1-x}}} 476:

Such functions along with the open regions they map are called

246: 230: 99: 8295: 8164:

Gesammelte mathematische Werke und wissenschaftlicher Nachlass

6564:

For two dimensional manifolds a key invariant property is the

6341:. All invariants of a smooth closed manifold are thus global. 5898:

matrices with non-zero determinant. If the matrix entries are

4272:, identified as a quotient space of the corresponding sphere. 3953:

This can be easily generalized to higher-dimensional spheres.

7217:{\displaystyle H\in C^{\infty }\left(\mathbb {R} ^{n}\right)} 6869:; and it omits finite dimension, allowing structures such as 5701: 4976:'). In this way he introduces a precursor to the notion of a 4582: 4410: 3100:. Any homeomorphism between half-balls must send points with 2747:-manifold without boundary, so the chart with transition map 2703:

These notions are made precise in general through the use of

2510:-axis and the origin. Another example of a chart is the map χ 1779: 1767: 934:

can be constructed, which takes values from the co-domain of

308: 250: 116: 8191:

Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.

7489: – Subdiscipline of statistics: statistics on manifolds 2725:

A smooth 2-manifold: The interior chart with transition map

7365:{\displaystyle H\circ \left(f_{1},\dots ,f_{n}\right)\in C} 6255:, and in high dimension (5 and above) it is algebraic, via 6162:. Similarly, Fréchet manifolds are locally homeomorphic to 5705: 5217:

has nonzero gradient at every point of the circle. By the

3264:

which have neighborhoods homeomorphic to an open subset of

1449:

is the slope of the line through the point at coordinates (

9177: 4348:

may be obtained by gluing a sphere with a hole in it to a

2458:, invertible maps that are continuous in both directions. 1770:

cannot have a plane representation consisting of a single

391:). Any point of this arc can be uniquely described by its 6295:

that can decide whether two manifolds are diffeomorphic.

4291:

is the group, the resulting quotient space is denoted by

1771: 1670:

Each chart omits a single point, either (−1, 0) for

7833: 7603:

Shigeyuki Morita; Teruko Nagase; Katsumi Nomizu (2001).

5311:), where they serve as a substitute for ordinary 'flat' 4793:. In the third section, he begins by remarking that the 27:

Topological space that locally resembles Euclidean space

7779:"Stratified vector bundles: Examples and constructions" 6247:: in low dimensions (2 and 3) it is geometric, via the 5864:

Other examples of Lie groups include special groups of

5445:. These homeomorphisms are the charts of the manifold. 5225:

of Euclidean space is locally the graph of a function.

2370: 2309:

that appears in the preceding definition is called the

8074:. Classic brief introduction to differential topology. 8017:. Detailed and comprehensive first-year graduate text. 6897:, while infinite-dimensional manifolds are studied in 5538:

Two important classes of differentiable manifolds are

5125: 5082: 5036: 4993: 4902:

depend continuously differentiably on the coordinates

264:

The concept of a manifold is central to many parts of

8232:

A Comprehensive Introduction to Differential Geometry

8100: 7438: 7418: 7398: 7378: 7310: 7258: 7230: 7177: 7157: 7137: 7102: 7082: 7062: 6761: 6727: 6698: 6559: 6423: 6390: 6128: 6096: 6076: 6035: 5990: 5947: 5908: 5878: 5827: 5762: 5670: 5650: 5630: 5598: 5422: 5378: 5171: 4955: 4935: 4908: 4888: 4843: 4814: 4526: 4126: 4093: 3861: 3824: 3705: 3676: 3635: 3581: 3558: 3538: 3512: 3492: 3472: 3435: 3396: 3376: 3350: 3323: 3303: 3270: 3250: 3224: 3204: 3180: 3139: 3106: 2987: 2967: 2869: 2849: 2794: 2774: 2650: 2589: 2560: 2483: 2431: 2337:

of a sphere and a line in three-dimensional space is

2282: 2249: 2220: 2046: 2017: 1974: 1624: 1487: 1457:) and the fixed pivot point (−1, 0); similarly, 1369: 1286: 1052: 1020: 975: 940: 828: 796: 776: 756: 715: 680: 488: 420: 208: 174: 153: 7538:

Pages displaying wikidata descriptions as a fallback

5258: 4318:(starting with a plane and a sphere, respectively). 4230:, this atlas defines a smooth manifold. In the case 4087:

can be constructed by gluing together two copies of

3621:

The chart maps the part of the sphere with positive

2832:

with interior is also a 2-manifold with boundary. A

7611:. American Mathematical Society Bookstore. p. 4601:, which can be extended to higher dimensions using 4237: 2717:

Topological manifold § Manifolds with boundary

2461:In the case of a differentiable manifold, a set of 1921:that is locally homeomorphic to a Euclidean space. 1862:(a closed loop piece and an open, infinite piece). 8224:advanced undergraduate / first-year graduate text. 8115: 7604: 7456: 7424: 7404: 7384: 7372:. Second: every function, which in every point of 7364: 7296: 7244: 7216: 7163: 7143: 7123: 7088: 7068: 7012:for topological manifolds. Basic examples include 6784: 6747: 6710: 6633:is a central example, and generalizations such as 6438: 6405: 6143: 6122:is a manifold modeled on boundaries of domains in 6105: 6082: 6050: 6002: 5976: 5921: 5890: 5845: 5774: 5688: 5656: 5636: 5616: 5437: 5393: 5209: 5157: 5111: 5068: 5022: 4966: 4941: 4919: 4894: 4870: 4829: 4680:constrain it to more complicated formations, e.g. 4553: 4489:Another, more topological example of an intrinsic 4216: 4108: 3912: 3839: 3808: 3691: 3650: 3593: 3567: 3544: 3524: 3498: 3478: 3455: 3421: 3382: 3362: 3332: 3309: 3285: 3256: 3236: 3210: 3186: 3158: 3125: 3092: 2973: 2953: 2855: 2812: 2780: 2768:with a 1-dimensional boundary. The boundary of an 2665: 2604: 2575: 2498: 2446: 2294: 2264: 2235: 2206: 2029: 1992: 1643: 1592: 1435: 1355: 1253: 1038: 1002: 969:back to the circle using the inverse, followed by 961: 926: 814: 782: 762: 742: 701: 657: 466: 214: 180: 159: 6805:In geometric topology, most commonly studied are 6066:. A one-complex-dimensional manifold is called a 4341:manifolds, other structures should be preserved. 3976:resulting manifold is a differentiable manifold. 1907:have homeomorphisms on overlapping neighborhoods 9835: 8086:. Addison-Wesley (reprinted by Westview Press) 6274:However, one can determine if two manifolds are 6225: 4474:gives a method for computing the curvature of a 3979:This can be illustrated with the transition map 2037:), or it has a neighborhood homeomorphic to the 1895:There are many different kinds of manifolds. In 467:{\displaystyle \chi _{\mathrm {top} }(x,y)=x.\,} 6852: 5789:Any Riemannian manifold is a Finsler manifold. 5531:-dimensional Euclidean space consisting of the 5519:on a differentiable manifold. Each point of an 4022: 2554:can be defined which goes from an open ball in 8182:The 1851 doctoral thesis in which "manifold" ( 5861:1 with multiplication as the group operation. 5563:generalizes the idea of a piecewise smooth or 5263:Two-dimensional manifolds, also known as a 2D 4505:in the three-dimensional Euclidean space with 4394:A finite cylinder is a manifold with boundary. 3575:is a manifold (without boundary) of dimension 3532:is a manifold (without boundary) of dimension 9193: 8316: 6459:, which is a manifold with boundary, (2) the 5617:{\displaystyle \langle \cdot ,\cdot \rangle } 5462: 1469:) and (+1, 0). The inverse mapping from 8296:Max Planck Institute for Mathematics in Bonn 6090:-dimensional complex manifold has dimension 5853:, can be also characterised as the group of 5769: 5763: 5683: 5671: 5611: 5599: 4352:along their respective circular boundaries. 4202: 4195: 4175: 4169: 4148: 4142: 3087: 3039: 2988: 2948: 2921: 2870: 387:is positive (indicated by the yellow arc in 237:. Two-dimensional manifolds are also called 8094:. Undergraduate text treating manifolds in 7124:{\displaystyle C\subseteq \mathbb {R} ^{M}} 6972:. In French an algebraic variety is called 6017: 5585:is a differentiable manifold in which each 5523:-dimensional differentiable manifold has a 5500: 9200: 9186: 8323: 8309: 8064:Topology from the Differentiable Viewpoint 7392:locally coincides with some function from 6785:{\displaystyle f\colon M\to \mathbb {C} ,} 6679: 6187:. A closely related type of manifold is a 6029:is a manifold whose charts take values in 4321: 4061: 3942:) plane and two charts projecting on the ( 2477:, for example, form a chart for the plane 1877: 1729:divides the sphere into two half spheres ( 8103: 7896: 7847: 7794: 7687: 7656:. Serié 11 (in French). Gauthier-Villars. 7238: 7200: 7151:be equipped with the topology induced by 7111: 6775: 6748:{\displaystyle f\colon M\to \mathbb {R} } 6741: 6426: 6393: 6131: 6038: 5466: 5425: 5381: 5350: 4882:), then he requires that the coordinates 4156: 4129: 4096: 3922:maps the northern hemisphere to the open 3827: 3743: 3679: 3638: 3486:is a manifold with boundary of dimension 3438: 3273: 2710: 2653: 2616:in the circle example above, is called a 2592: 2563: 2514:mentioned above, a chart for the circle. 2486: 2434: 2252: 2223: 2120: 2007:This implies that either the point is an 1977: 463: 87:Learn how and when to remove this message 9557:Covariance and contravariance of vectors 8330: 7878: 7741: 7648: 7635:The notion of a map can formalized as a 6683: 6591: 6531: 6521: 6474: 5929:-dimensional disconnected manifold. The 5361: 5287:proved the Poincaré conjecture (see the 5235:that followed shortly. During the 1930s 4765: 4389: 3616: 3169: 2720: 1914:Formally, a (topological) manifold is a 1789: 1743:), which may both be mapped on the disc 1651:(that is, one has this relation between 1271: 1003:{\displaystyle \chi _{\mathrm {right} }} 743:{\displaystyle \chi _{\mathrm {right} }} 358: 110: 98: 50:This article includes a list of general 7703: 7666: 7575: 7297:{\displaystyle f_{1},\dots ,f_{n}\in C} 6841:methods of studying manifolds, such as 5570: 4871:{\displaystyle \theta '\left(y'\right)} 4609:linked the Euler characteristic to the 2635: 235:self-crossing curves such as a figure 8 14: 9836: 8194:The 1854 Göttingen inaugural lecture ( 6833:. This leads to such functions as the 6467:, which arises naturally in geometry. 6314:(a normal invariant, also detected by 1892:that is "modeled on" Euclidean space. 1874:always preserve the number of pieces. 1782:for covering the whole Earth surface. 962:{\displaystyle \chi _{\mathrm {top} }} 822:, though differently. Thus a function 702:{\displaystyle \chi _{\mathrm {top} }} 349: 9181: 8304: 8036:Introduction to Topological Manifolds 8007:Introduction to Topological Manifolds 7596: 7556: 6983:), while a smooth manifold is called 5846:{\displaystyle \operatorname {U} (1)} 4605:. In the mid nineteenth century, the 4355: 4336:Formally, the gluing is defined by a 4116:. The transition map between them is 4031:in some Euclidean space. This is the 3913:{\displaystyle \chi (x,y,z)=(x,y),\ } 3665: 3525:{\displaystyle \operatorname {Int} M} 3363:{\displaystyle \operatorname {Int} M} 3237:{\displaystyle \operatorname {Int} M} 2756:must map to an open Euclidean subset. 119:is decomposed into charts around the 8244:(2nd ed.). New York: Springer. 7872: 7776: 7707:(1936). "Differentiable Manifolds". 6587: 5737: 5492:in the definition). All points in a 5321:analogues of the Poincaré conjecture 4981: 4420: 4054:, but instead there is an intrinsic 4043:to some surface through that point. 3456:{\displaystyle \mathbb {R} _{+}^{n}} 2371:Charts, atlases, and transition maps 147:near each point. More precisely, an 36: 8256:. Concise first-year graduate text. 8046:Algebraic Topology: An Introduction 7986:and Siebenmann, Laurence C. (1977) 7827: 7674:[On Teaching Mathematics]. 6821:, an important example of which is 6271:manifold refer to the same object. 6253:solution of the Poincaré conjecture 6058:and whose transition functions are 5689:{\displaystyle \langle u,v\rangle } 5289:Solution of the Poincaré conjecture 4744:n-fach ausgedehnte Mannigfaltigkeit 4692:, one of the founders of topology. 4645:of functions of complex variables. 4413:and surfaces as well as ideas from 2689:, the transition functions must be 107:immersed in three-dimensional space 24: 9420:Tensors in curvilinear coordinates 8237: 7530: – Manifold of dimension five 7189: 6560:Genus and the Euler characteristic 6327:curvature of a Riemannian manifold 6306:, classic algebraic topology, and 6158:which are locally homeomorphic to 6113:as a real differentiable manifold. 5828: 4405:History of manifolds and varieties 4001:on the second copy with the point 3559: 3324: 3042: 2924: 2612:. The resultant map, like the map 2544: 1718:may be covered by an atlas of six 1385: 1382: 1379: 1376: 1305: 1302: 1299: 1296: 1293: 1175: 1172: 1169: 1166: 1163: 1117: 1114: 1111: 1094: 1091: 1088: 1085: 1082: 994: 991: 988: 985: 982: 953: 950: 947: 910: 907: 904: 889: 886: 883: 880: 877: 734: 731: 728: 725: 722: 693: 690: 687: 617: 614: 611: 608: 605: 564: 561: 558: 555: 514: 511: 508: 505: 502: 499: 433: 430: 427: 225:One-dimensional manifolds include 56:it lacks sufficient corresponding 25: 9860: 8260: 7563:, World Scientific, p. 110, 7504:Mathematics of general relativity 7245:{\displaystyle n\in \mathbb {N} } 5868:, which are all subgroups of the 5535:of the curves through the point. 5259:Topology of manifolds: highlights 5158:{\textstyle x=-{\sqrt {1-y^{2}}}} 5069:{\textstyle y=-{\sqrt {1-x^{2}}}} 4719:values. He distinguishes between 4364:of manifolds is also a manifold. 4166: 4139: 3194:be a manifold with boundary. The 1993:{\displaystyle \mathbb {R} ^{n},} 1610: = 1 for all values of 8116:{\displaystyle \mathbb {R} ^{n}} 8023:Introduction to Smooth Manifolds 7654:Journal de l'École Polytechnique 6817:, one often studies solution to 6439:{\displaystyle \mathbb {R} ^{n}} 6406:{\displaystyle \mathbb {R} ^{n}} 6373: 6144:{\displaystyle \mathbb {C} ^{n}} 6051:{\displaystyle \mathbb {C} ^{n}} 5438:{\displaystyle \mathbb {R} ^{n}} 5394:{\displaystyle \mathbb {R} ^{n}} 5112:{\textstyle x={\sqrt {1-y^{2}}}} 5023:{\textstyle y={\sqrt {1-x^{2}}}} 4715:, because the variable can have 4264:are considered to be relatively 4238:Identifying points of a manifold 4109:{\displaystyle \mathbb {R} ^{n}} 3840:{\displaystyle \mathbb {R} ^{2}} 3692:{\displaystyle \mathbb {R} ^{3}} 3651:{\displaystyle \mathbb {R} ^{2}} 3286:{\displaystyle \mathbb {R} ^{n}} 2666:{\displaystyle \mathbb {R} ^{n}} 2605:{\displaystyle \mathbb {R} ^{n}} 2576:{\displaystyle \mathbb {R} ^{n}} 2499:{\displaystyle \mathbb {R} ^{2}} 2447:{\displaystyle \mathbb {R} ^{n}} 2265:{\displaystyle \mathbb {R} ^{n}} 2236:{\displaystyle \mathbb {R} ^{n}} 2049: 1690:is an example of a surface. The 1644:{\displaystyle t={\frac {1}{s}}} 41: 7881:"Abstract covariant derivative" 7783:Journal of Geometry and Physics 7509: 6930:Algebraic varieties and schemes 6503: 6470: 6333:of a manifold equipped with an 5303:and in a different setting, by 5210:{\displaystyle x^{2}+y^{2}-1=0} 4949:sont fonctions analytiques des 4770:In his very influential paper, 3604: 1811: 1807: 1803: 1799: 1785: 399:onto the first coordinate is a 8363:Differentiable/Smooth manifold 8148:. Princeton University Press. 8066:. Princeton University Press. 7990:. Princeton University Press. 7954:. Princeton University Press. 7921:. Princeton University Press. 7813:10.1016/j.geomphys.2024.105114 7770: 7735: 7697: 7660: 7642: 7629: 7607:Geometry of Differential Forms 7549: 7451: 7439: 7010:topologically stratified space 6858:Infinite dimensional manifolds 6819:partial differential equations 6771: 6737: 6288:under different descriptions. 5963: 5951: 5840: 5834: 4878:. If these manifolds overlap ( 4824: 4818: 4699:comes from Riemann's original 4184: 4151: 4011:on the first copy (the points 3901: 3889: 3883: 3865: 3735: 3717: 3416: 3397: 3036: 2991: 2918: 2873: 2807: 2795: 2788:-manifold with boundary is an 2395:, collected in a mathematical 2112: 2067: 1403: 1391: 1323: 1311: 1066: 1060: 1033: 1021: 865: 853: 850: 847: 835: 809: 797: 635: 623: 582: 570: 532: 520: 451: 439: 222:-dimensional Euclidean space. 13: 1: 9473:Exterior covariant derivative 9405:Tensor (intrinsic definition) 7908: 7008:) for smooth manifolds and a 6226:Classification and invariants 5984:dimensional manifolds, where 5792: 4748:n times extended manifoldness 4387: × , respectively. 3463:under some coordinate chart. 2000:for some nonnegative integer 9498:Raising and lowering indices 8241:An Introduction to Manifolds 8144:Neuwirth, L. P., ed. (1975) 7500: – Mathematics timeline 6853:Generalizations of manifolds 6649:. Basic results include the 4616: 4023:Intrinsic and extrinsic view 3956: 2276:to any open ball in it (for 1888:Informally, a manifold is a 1263:Such a function is called a 7: 9736:Gluon field strength tensor 9207: 9069:Classification of manifolds 8273:Encyclopedia of Mathematics 7672:"О преподавании математики" 7468: 6847:Atiyah–Singer index theorem 6843:hearing the shape of a drum 6368: 6322:(a homological invariant). 6232:Classification of manifolds 5323:, had been done earlier by 4977: 4787:continuously differentiable 3612: 1778:), and therefore one needs 10: 9865: 9547:Cartan formalism (physics) 9367:Penrose graphical notation 7998:. A detailed study of the 7917:, and Quinn, Frank (1990) 7777:Ross, Ethan (2024-04-01). 7652:(1895). "Analysis Situs". 7524: – Mathematical space 7518: – Mathematical space 6610: 6525: 6507: 6496: 6482: 6377: 6361:theory, and the theory of 6265:complete set of invariants 6229: 6179:. They are endowed with a 5796: 5775:{\displaystyle \|\cdot \|} 5741: 5574: 5552:(they can be expressed as 5504: 5365: 5354: 5283:. After nearly a century, 4830:{\displaystyle \theta (y)} 4752:n-dimensional manifoldness 4736:discontinuous manifoldness 4593: = 2 edges, and 4554:{\displaystyle V-E+F=2.\ } 4402: 4398: 4325: 4248:Group action (mathematics) 4241: 3960: 3568:{\displaystyle \partial M} 3333:{\displaystyle \partial M} 3244:, is the set of points in 2714: 2521: 2517: 2406: 2380: 2374: 2272:is homeomorphic, and even 1881: 1774:(also called "chart", see 674:The top and right charts, 410:from the upper arc to the 380: = 1, where the 167:-dimensional manifold, or 29: 9753: 9693: 9642: 9635: 9527: 9458: 9395: 9339: 9286: 9233: 9226: 9219:Glossary of tensor theory 9215: 9145:over commutative algebras 9102: 9061: 8994: 8891: 8787: 8734: 8725: 8561: 8484: 8423: 8343: 8002:of topological manifolds. 7935:and Pollack, Alan (1974) 7582:, CRC Press, p. 73, 7096:was chosen. Denote it by 6813:decompositions, while in 6655:Whitney immersion theorem 6651:Whitney embedding theorem 6202:piecewise linear manifold 5891:{\displaystyle n\times n} 5696:gives a real number. The 5253:Whitney embedding theorem 5239:and others clarified the 5219:implicit function theorem 4791:implicit function theorem 4762:are named after Riemann. 4501:showed that for a convex 4328:Quotient space (topology) 3926:by projecting it on the ( 3422:{\displaystyle (x_{n}=0)} 2622:coordinate transformation 2402: 1754:by the projection on the 1681: 1602:It can be confirmed that 1010:back to the interval. If 354: 327:, while four-dimensional 32:Manifold (disambiguation) 9803:Gregorio Ricci-Curbastro 9675:Riemann curvature tensor 9382:Van der Waerden notation 8861:Riemann curvature tensor 7543: 6919:Euclidean space) by the 6863:topological vector space 6672:; a basic result is the 6647:ramified covering spaces 6300:a rich set of invariants 6018:Other types of manifolds 5977:{\displaystyle n(n-1)/2} 5501:Differentiable manifolds 5463:differentiable manifolds 4742:, Riemann constructs an 4721:stetige Mannigfaltigkeit 4709:William Kingdon Clifford 4478:without considering the 3993:for the first copy, and 1905:differentiable manifolds 293:differentiable structure 289:differentiable manifolds 9773:Elwin Bruno Christoffel 9706:Angular momentum tensor 9377:Tetrad (index notation) 9347:Abstract index notation 7919:Topology of 4-Manifolds 7898:10.4064/cm-18-1-251-272 7885:Colloquium Mathematicum 7576:Gunning, R. C. (1990), 7557:Riaza, Ricardo (2008), 6867:non-Hausdorff manifolds 6845:and some proofs of the 6680:Scalar-valued functions 5513:differentiable manifold 5507:Differentiable manifold 5357:Categories of manifolds 4804:Poincaré's notion of a 4799:une chaîne des variétés 4732:continuous manifoldness 4674:generalized coordinates 4589: = 1 vertex, 4581:. On the other hand, a 4431:considers spaces where 4322:Gluing along boundaries 3159:{\displaystyle x_{1}=0} 3126:{\displaystyle x_{1}=0} 2383:Differentiable manifold 2349:has a fixed dimension. 1949:non-Hausdorff manifolds 1884:Categories of manifolds 1878:Mathematical definition 241:. Examples include the 143:that locally resembles 71:more precise citations. 9587:Levi-Civita connection 8653:Manifold with boundary 8368:Differential structure 8238:Tu, Loring W. (2011). 8117: 7487:Directional statistics 7458: 7426: 7406: 7386: 7366: 7298: 7246: 7218: 7165: 7145: 7125: 7090: 7070: 7014:manifold with boundary 6987:variété différentielle 6786: 6749: 6718: 6712: 6674:Nash embedding theorem 6670:Riemannian submersions 6608: 6545: 6480: 6440: 6407: 6363:characteristic classes 6298:Smooth manifolds have 6249:uniformization theorem 6197:combinatorial manifold 6145: 6107: 6084: 6052: 6004: 5978: 5923: 5892: 5847: 5776: 5690: 5658: 5638: 5618: 5439: 5395: 5297:geometrization program 5211: 5159: 5113: 5070: 5024: 4968: 4943: 4921: 4896: 4872: 4831: 4658:William Rowan Hamilton 4555: 4429:Non-Euclidean geometry 4395: 4316:real projective spaces 4218: 4110: 4065:-Sphere as a patchwork 3914: 3841: 3810: 3693: 3652: 3626: 3595: 3569: 3546: 3526: 3500: 3480: 3457: 3423: 3384: 3364: 3334: 3311: 3287: 3258: 3238: 3212: 3188: 3160: 3127: 3094: 2975: 2955: 2857: 2814: 2782: 2762:manifold with boundary 2757: 2711:Manifold with boundary 2673:(that is, if they are 2667: 2606: 2577: 2500: 2448: 2296: 2295:{\displaystyle n>0} 2266: 2237: 2208: 2031: 1994: 1872:topological operations 1825:Manifolds need not be 1818:Manifolds need not be 1815: 1659:for every point where 1645: 1594: 1437: 1357: 1277: 1255: 1040: 1004: 963: 928: 816: 784: 764: 744: 703: 659: 468: 364: 216: 182: 161: 128: 108: 18:Manifold with boundary 9813:Jan Arnoldus Schouten 9768:Augustin-Louis Cauchy 9248:Differential geometry 8118: 8083:Analysis on Manifolds 7970:Differential Topology 7937:Differential Topology 7879:Sikorski, R. (1967). 7836:Annals of Mathematics 7709:Annals of Mathematics 7534:Manifolds of mappings 7498:Timeline of manifolds 7459: 7457:{\displaystyle (M,C)} 7427: 7407: 7387: 7367: 7299: 7247: 7219: 7166: 7146: 7126: 7091: 7071: 6815:mathematical analysis 6787: 6750: 6713: 6688:3D color plot of the 6687: 6595: 6572:, and more generally 6538:real projective plane 6535: 6528:Real projective space 6522:Real projective plane 6478: 6465:real projective plane 6441: 6408: 6339:geometry and topology 6230:Further information: 6146: 6108: 6085: 6053: 6005: 5979: 5924: 5922:{\displaystyle n^{2}} 5893: 5848: 5777: 5691: 5659: 5639: 5619: 5440: 5396: 5362:Topological manifolds 5341:topology of manifolds 5251:theory. Notably, the 5245:differential geometry 5212: 5160: 5114: 5071: 5030:or else the function 5025: 4969: 4944: 4922: 4897: 4873: 4832: 4766:Poincaré's definition 4686:Joseph-Louis Lagrange 4643:analytic continuation 4599:topological invariant 4556: 4493:of a manifold is its 4403:Further information: 4393: 4270:real projective space 4219: 4118:inversion in a sphere 4111: 3961:Further information: 3915: 3842: 3811: 3694: 3653: 3625:coordinate to a disc. 3620: 3596: 3570: 3547: 3527: 3501: 3481: 3458: 3424: 3385: 3365: 3335: 3312: 3288: 3259: 3239: 3213: 3189: 3170:Boundary and interior 3161: 3128: 3095: 2976: 2956: 2858: 2815: 2813:{\displaystyle (n-1)} 2783: 2724: 2683:holomorphic functions 2668: 2618:change of coordinates 2607: 2578: 2501: 2449: 2297: 2267: 2238: 2209: 2032: 1995: 1901:topological manifolds 1897:geometry and topology 1882:Further information: 1793: 1646: 1595: 1438: 1358: 1275: 1256: 1041: 1039:{\displaystyle (0,1)} 1005: 964: 929: 817: 815:{\displaystyle (0,1)} 785: 765: 745: 704: 660: 469: 362: 321:Hamiltonian formalism 303:on a manifold allows 259:real projective plane 217: 183: 162: 114: 102: 9788:Carl Friedrich Gauss 9721:stress–energy tensor 9716:Cauchy stress tensor 9468:Covariant derivative 9430:Antisymmetric tensor 9362:Multi-index notation 8800:Covariant derivative 8351:Topological manifold 8196:Habilitationsschrift 8098: 8020:Lee, John M. (2003) 8005:Lee, John M. (2000) 7950:Hempel, John (1976) 7915:Freedman, Michael H. 7436: 7416: 7396: 7376: 7308: 7256: 7228: 7175: 7155: 7135: 7100: 7080: 7060: 6992:differential variety 6829:: the kernel of the 6825:, where one studies 6759: 6725: 6696: 6666:isometric immersions 6662:isometric embeddings 6570:Euler characteristic 6421: 6388: 6206:simplicial complexes 6126: 6094: 6074: 6033: 5988: 5945: 5906: 5876: 5870:general linear group 5825: 5760: 5668: 5664:, the inner product 5648: 5628: 5596: 5589:is equipped with an 5571:Riemannian manifolds 5467:Riemannian manifolds 5420: 5403:locally homeomorphic 5376: 5368:topological manifold 5351:Additional structure 5169: 5123: 5080: 5034: 4991: 4953: 4933: 4906: 4886: 4880:a une partie commune 4841: 4812: 4650:analytical mechanics 4607:Gauss–Bonnet theorem 4579:Euler characteristic 4524: 4495:Euler characteristic 4287:is the manifold and 4124: 4091: 4056:stable normal bundle 3859: 3822: 3703: 3674: 3633: 3579: 3556: 3536: 3510: 3490: 3470: 3433: 3394: 3374: 3348: 3321: 3301: 3268: 3248: 3222: 3202: 3178: 3137: 3104: 2985: 2965: 2867: 2847: 2792: 2772: 2687:symplectic manifolds 2648: 2636:Additional structure 2587: 2558: 2481: 2471:transition functions 2429: 2357:locally ringed space 2280: 2247: 2218: 2044: 2015: 1972: 1955:Locally homeomorphic 1899:, all manifolds are 1794:Four manifolds from 1674:or (+1, 0) for 1622: 1485: 1367: 1284: 1050: 1018: 973: 938: 826: 794: 774: 754: 713: 678: 486: 418: 329:Lorentzian manifolds 313:Symplectic manifolds 274:systems of equations 270:mathematical physics 206: 172: 151: 30:For other uses, see 9849:Geometry processing 9665:Nonmetricity tensor 9520:(2nd-order tensors) 9488:Hodge star operator 9478:Exterior derivative 9327:Transport phenomena 9312:Continuum mechanics 9268:Multilinear algebra 8834:Exterior derivative 8436:Atiyah–Singer index 8385:Riemannian manifold 8285:Dimensions-math.org 8048:. Springer-Verlag. 8026:. Springer-Verlag. 8009:. Springer-Verlag. 7972:. Springer Verlag. 7805:2024JGP...19805114R 7744:Comment. Math. Helv 7053:Differential spaces 6949:instead of atlases. 6899:functional analysis 6835:spherical harmonics 6711:{\displaystyle n=5} 6690:spherical harmonics 6499:Quasitoric manifold 6380:Orientable manifold 6177:classical mechanics 6172:symplectic manifold 6012:table of Lie groups 6003:{\displaystyle n-1} 5583:Riemannian manifold 5577:Riemannian manifold 5414:continuous function 5281:Poincaré conjecture 4457:hyperbolic geometry 3594:{\displaystyle n-1} 3452: 3059: 2941: 2838:Boundary (topology) 2626:transition function 2552:transition function 2506:minus the positive 2353:Sheaf-theoretically 2347:connected component 2189: 2165: 2147: 2030:{\displaystyle n=0} 1667:are both nonzero). 1130: 923: 350:Motivating examples 325:classical mechanics 9798:Tullio Levi-Civita 9741:Metric tensor (GR) 9655:Levi-Civita symbol 9508:Tensor contraction 9322:General relativity 9258:Euclidean geometry 9140:Secondary calculus 9094:Singularity theory 9049:Parallel transport 8817:De Rham cohomology 8456:Generalized Stokes 8166:, Sändig Reprint. 8113: 8042:Massey, William S. 7756:10.1007/BF02565940 7637:cell decomposition 7454: 7422: 7412:, also belongs to 7402: 7382: 7362: 7304:, the composition 7294: 7242: 7214: 7161: 7141: 7121: 7086: 7066: 7042:Homology manifolds 7018:semialgebraic sets 7006:Whitney conditions 6976:variété algébrique 6827:harmonic functions 6782: 6745: 6719: 6708: 6623:geometric topology 6609: 6546: 6481: 6436: 6403: 6345:Algebraic topology 6308:geometric topology 6304:point-set topology 6141: 6106:{\displaystyle 2n} 6103: 6080: 6048: 6000: 5974: 5937:of the sphere and 5919: 5902:, this will be an 5888: 5843: 5772: 5686: 5654: 5634: 5614: 5545:analytic manifolds 5435: 5391: 5207: 5155: 5109: 5066: 5020: 4967:{\displaystyle y'} 4964: 4939: 4920:{\displaystyle y'} 4917: 4892: 4868: 4827: 4806:chain of manifolds 4652:, as developed by 4631:elliptic integrals 4627:Carl Gustav Jacobi 4621:Investigations of 4611:Gaussian curvature 4551: 4484:intrinsic property 4437:parallel postulate 4396: 4375:, for example, as 4356:Cartesian products 4214: 4106: 3910: 3837: 3806: 3689: 3666:Sphere with charts 3648: 3627: 3591: 3565: 3542: 3522: 3496: 3476: 3453: 3436: 3419: 3380: 3360: 3330: 3307: 3283: 3254: 3234: 3208: 3184: 3156: 3123: 3090: 3045: 2971: 2951: 2927: 2853: 2810: 2778: 2758: 2691:symplectomorphisms 2663: 2602: 2573: 2496: 2444: 2365:algebraic geometry 2359:, whose structure 2355:, a manifold is a 2292: 2262: 2233: 2204: 2175: 2151: 2133: 2027: 1990: 1816: 1641: 1590: 1588: 1433: 1353: 1278: 1251: 1249: 1105: 1036: 1000: 959: 924: 898: 812: 780: 760: 740: 699: 655: 653: 464: 365: 337:general relativity 212: 178: 157: 129: 109: 9831: 9830: 9793:Hermann Grassmann 9749: 9748: 9701:Moment of inertia 9562:Differential form 9537:Affine connection 9352:Einstein notation 9335: 9334: 9263:Exterior calculus 9243:Coordinate system 9175: 9174: 9057: 9056: 8822:Differential form 8476:Whitney embedding 8410:Differential form 8251:978-1-4419-7399-3 8160:Riemann, Bernhard 8154:978-0-691-08170-0 8133:. Prentice Hall. 8127:Munkres, James R. 8078:Munkres, James R. 7939:. Prentice-Hall. 7933:Guillemin, Victor 7838:. Second Series. 7711:. Second Series. 7676:Uspekhi Mat. Nauk 7493:List of manifolds 7425:{\displaystyle C} 7405:{\displaystyle C} 7385:{\displaystyle M} 7164:{\displaystyle C} 7144:{\displaystyle M} 7089:{\displaystyle M} 7069:{\displaystyle M} 7047:homology manifold 6981:algebraic variety 6939:algebraic variety 6889:to be modeled on 6887:Fréchet manifolds 6881:to be modeled on 6873:to be modeled on 6871:Hilbert manifolds 6823:harmonic analysis 6796:regular functions 6794:sometimes called 6625:a basic type are 6619:maps of manifolds 6613:Maps of manifolds 6588:Maps of manifolds 6335:affine connection 6293:no program exists 6183:that defines the 6083:{\displaystyle n} 5931:orthogonal groups 5738:Finsler manifolds 5657:{\displaystyle v} 5637:{\displaystyle u} 5565:rectifiable curve 5317:Andrey Markov Jr. 5309:Yang–Mills theory 5233:topological space 5153: 5107: 5064: 5018: 4942:{\displaystyle y} 4927:and vice versa (' 4895:{\displaystyle y} 4678:conservation laws 4623:Niels Henrik Abel 4550: 4472:theorema egregium 4461:elliptic geometry 4421:Early development 4362:Cartesian product 3973:equivalence class 3909: 3545:{\displaystyle n} 3499:{\displaystyle n} 3479:{\displaystyle M} 3383:{\displaystyle M} 3310:{\displaystyle M} 3257:{\displaystyle M} 3211:{\displaystyle M} 3187:{\displaystyle M} 3069: 2974:{\displaystyle n} 2856:{\displaystyle n} 2781:{\displaystyle n} 2679:Complex manifolds 2475:Polar coordinates 2393:coordinate charts 2389:mathematical maps 1810: hyperbola, 1696:implicit equation 1639: 1584: 1540: 1431: 1351: 1245: 1210: 1014:is any number in 783:{\displaystyle y} 763:{\displaystyle x} 395:-coordinate. So, 301:Riemannian metric 215:{\displaystyle n} 181:{\displaystyle n} 160:{\displaystyle n} 141:topological space 97: 96: 89: 16:(Redirected from 9856: 9808:Bernhard Riemann 9640: 9639: 9483:Exterior product 9450:Two-point tensor 9435:Symmetric tensor 9317:Electromagnetism 9231: 9230: 9202: 9195: 9188: 9179: 9178: 9167:Stratified space 9125:Fréchet manifold 8839:Interior product 8732: 8731: 8429: 8325: 8318: 8311: 8302: 8301: 8281: 8255: 8186:) first appears. 8184:Mannigfaltigkeit 8122: 8120: 8119: 8114: 8112: 8111: 8106: 7984:Kirby, Robion C. 7903: 7902: 7900: 7876: 7870: 7869: 7851: 7831: 7825: 7824: 7798: 7774: 7768: 7767: 7739: 7733: 7732: 7701: 7695: 7693: 7691: 7682:(319): 229–234. 7664: 7658: 7657: 7646: 7640: 7633: 7627: 7626: 7610: 7600: 7594: 7592: 7573: 7553: 7539: 7463: 7461: 7460: 7455: 7431: 7429: 7428: 7423: 7411: 7409: 7408: 7403: 7391: 7389: 7388: 7383: 7371: 7369: 7368: 7363: 7355: 7351: 7350: 7349: 7331: 7330: 7303: 7301: 7300: 7295: 7287: 7286: 7268: 7267: 7252:, and arbitrary 7251: 7249: 7248: 7243: 7241: 7223: 7221: 7220: 7215: 7213: 7209: 7208: 7203: 7193: 7192: 7170: 7168: 7167: 7162: 7150: 7148: 7147: 7142: 7130: 7128: 7127: 7122: 7120: 7119: 7114: 7095: 7093: 7092: 7087: 7075: 7073: 7072: 7067: 7022:subanalytic sets 6997:Stratified space 6962:Mannigfaltigkeit 6895:general topology 6879:Banach manifolds 6831:Laplace operator 6791: 6789: 6788: 6783: 6778: 6754: 6752: 6751: 6746: 6744: 6717: 6715: 6714: 6709: 6445: 6443: 6442: 6437: 6435: 6434: 6429: 6412: 6410: 6409: 6404: 6402: 6401: 6396: 6350:simply connected 6219:digital topology 6214:digital manifold 6189:contact manifold 6156:Banach manifolds 6150: 6148: 6147: 6142: 6140: 6139: 6134: 6112: 6110: 6109: 6104: 6089: 6087: 6086: 6081: 6064:complex geometry 6057: 6055: 6054: 6049: 6047: 6046: 6041: 6026:complex manifold 6009: 6007: 6006: 6001: 5983: 5981: 5980: 5975: 5970: 5928: 5926: 5925: 5920: 5918: 5917: 5897: 5895: 5894: 5889: 5852: 5850: 5849: 5844: 5781: 5779: 5778: 5773: 5750:Finsler manifold 5744:Finsler manifold 5733: 5695: 5693: 5692: 5687: 5663: 5661: 5660: 5655: 5643: 5641: 5640: 5635: 5623: 5621: 5620: 5615: 5479:second countable 5444: 5442: 5441: 5436: 5434: 5433: 5428: 5400: 5398: 5397: 5392: 5390: 5389: 5384: 5301:Michael Freedman 5293:William Thurston 5285:Grigori Perelman 5269:Riemann surfaces 5216: 5214: 5213: 5208: 5194: 5193: 5181: 5180: 5164: 5162: 5161: 5156: 5154: 5152: 5151: 5136: 5118: 5116: 5115: 5110: 5108: 5106: 5105: 5090: 5075: 5073: 5072: 5067: 5065: 5063: 5062: 5047: 5029: 5027: 5026: 5021: 5019: 5017: 5016: 5001: 4973: 4971: 4970: 4965: 4963: 4948: 4946: 4945: 4940: 4926: 4924: 4923: 4918: 4916: 4901: 4899: 4898: 4893: 4877: 4875: 4874: 4869: 4867: 4863: 4851: 4836: 4834: 4833: 4828: 4760:Riemann surfaces 4756:Mannigfaltigkeit 4728:Mannigfaltigkeit 4713:Mannigfaltigkeit 4705:Mannigfaltigkeit 4639:Bernhard Riemann 4629:on inversion of 4560: 4558: 4557: 4552: 4548: 4467:, respectively. 4369:product topology 4346:projective plane 4223: 4221: 4220: 4215: 4210: 4209: 4194: 4165: 4164: 4159: 4138: 4137: 4132: 4115: 4113: 4112: 4107: 4105: 4104: 4099: 3919: 3917: 3916: 3911: 3907: 3854: 3846: 3844: 3843: 3838: 3836: 3835: 3830: 3815: 3813: 3812: 3807: 3802: 3798: 3791: 3790: 3778: 3777: 3765: 3764: 3752: 3751: 3746: 3698: 3696: 3695: 3690: 3688: 3687: 3682: 3657: 3655: 3654: 3649: 3647: 3646: 3641: 3600: 3598: 3597: 3592: 3574: 3572: 3571: 3566: 3551: 3549: 3548: 3543: 3531: 3529: 3528: 3523: 3505: 3503: 3502: 3497: 3485: 3483: 3482: 3477: 3462: 3460: 3459: 3454: 3451: 3446: 3441: 3428: 3426: 3425: 3420: 3409: 3408: 3389: 3387: 3386: 3381: 3369: 3367: 3366: 3361: 3339: 3337: 3336: 3331: 3316: 3314: 3313: 3308: 3292: 3290: 3289: 3284: 3282: 3281: 3276: 3263: 3261: 3260: 3255: 3243: 3241: 3240: 3235: 3217: 3215: 3214: 3209: 3193: 3191: 3190: 3185: 3165: 3163: 3162: 3157: 3149: 3148: 3132: 3130: 3129: 3124: 3116: 3115: 3099: 3097: 3096: 3091: 3080: 3079: 3070: 3067: 3058: 3053: 3035: 3034: 3016: 3015: 3003: 3002: 2980: 2978: 2977: 2972: 2960: 2958: 2957: 2952: 2940: 2935: 2917: 2916: 2898: 2897: 2885: 2884: 2862: 2860: 2859: 2854: 2819: 2817: 2816: 2811: 2787: 2785: 2784: 2779: 2755: 2746: 2742: 2733: 2672: 2670: 2669: 2664: 2662: 2661: 2656: 2611: 2609: 2608: 2603: 2601: 2600: 2595: 2582: 2580: 2579: 2574: 2572: 2571: 2566: 2524:Atlas (topology) 2505: 2503: 2502: 2497: 2495: 2494: 2489: 2453: 2451: 2450: 2445: 2443: 2442: 2437: 2419:coordinate chart 2409:Coordinate chart 2377:Atlas (topology) 2343:locally constant 2331: 2330: 2308: 2301: 2299: 2298: 2293: 2271: 2269: 2268: 2263: 2261: 2260: 2255: 2242: 2240: 2239: 2234: 2232: 2231: 2226: 2213: 2211: 2210: 2205: 2200: 2196: 2188: 2183: 2164: 2159: 2146: 2141: 2129: 2128: 2123: 2111: 2110: 2092: 2091: 2079: 2078: 2058: 2057: 2052: 2036: 2034: 2033: 2028: 2003: 1999: 1997: 1996: 1991: 1986: 1985: 1980: 1937:second countable 1925:Second countable 1916:second countable 1861: 1813: 1809: 1806: parabola, 1805: 1801: 1796:algebraic curves 1759: 1753: 1742: 1735: 1728: 1714: 1650: 1648: 1647: 1642: 1640: 1632: 1599: 1597: 1596: 1591: 1589: 1585: 1583: 1582: 1581: 1565: 1557: 1541: 1539: 1538: 1537: 1521: 1520: 1519: 1503: 1442: 1440: 1439: 1434: 1432: 1430: 1416: 1390: 1389: 1388: 1362: 1360: 1359: 1354: 1352: 1350: 1336: 1310: 1309: 1308: 1260: 1258: 1257: 1252: 1250: 1246: 1244: 1243: 1228: 1220: 1216: 1212: 1211: 1209: 1208: 1193: 1180: 1179: 1178: 1150: 1146: 1142: 1141: 1129: 1121: 1120: 1099: 1098: 1097: 1045: 1043: 1042: 1037: 1009: 1007: 1006: 1001: 999: 998: 997: 968: 966: 965: 960: 958: 957: 956: 933: 931: 930: 925: 922: 914: 913: 894: 893: 892: 821: 819: 818: 813: 789: 787: 786: 781: 769: 767: 766: 761: 749: 747: 746: 741: 739: 738: 737: 708: 706: 705: 700: 698: 697: 696: 671:for the circle. 664: 662: 661: 656: 654: 622: 621: 620: 569: 568: 567: 519: 518: 517: 473: 471: 470: 465: 438: 437: 436: 311:to be measured. 221: 219: 218: 213: 187: 185: 184: 179: 166: 164: 163: 158: 92: 85: 81: 78: 72: 67:this article by 58:inline citations 45: 44: 37: 21: 9864: 9863: 9859: 9858: 9857: 9855: 9854: 9853: 9834: 9833: 9832: 9827: 9778:Albert Einstein 9745: 9726:Einstein tensor 9689: 9670:Ricci curvature 9650:Kronecker delta 9636:Notable tensors 9631: 9552:Connection form 9529: 9523: 9454: 9440:Tensor operator 9397: 9391: 9331: 9307:Computer vision 9300: 9282: 9278:Tensor calculus 9222: 9211: 9206: 9176: 9171: 9110:Banach manifold 9103:Generalizations 9098: 9053: 8990: 8887: 8849:Ricci curvature 8805:Cotangent space 8783: 8721: 8563: 8557: 8516:Exponential map 8480: 8425: 8419: 8339: 8329: 8294:project of the 8266: 8263: 8252: 8228:Spivak, Michael 8204:Spivak, Michael 8107: 8102: 8101: 8099: 8096: 8095: 7911: 7906: 7877: 7873: 7858:10.2307/2118532 7832: 7828: 7775: 7771: 7740: 7736: 7721:10.2307/1968482 7702: 7698: 7665: 7661: 7647: 7643: 7634: 7630: 7623: 7601: 7597: 7590: 7571: 7554: 7550: 7546: 7537: 7512: 7471: 7437: 7434: 7433: 7417: 7414: 7413: 7397: 7394: 7393: 7377: 7374: 7373: 7345: 7341: 7326: 7322: 7321: 7317: 7309: 7306: 7305: 7282: 7278: 7263: 7259: 7257: 7254: 7253: 7237: 7229: 7226: 7225: 7204: 7199: 7198: 7194: 7188: 7184: 7176: 7173: 7172: 7156: 7153: 7152: 7136: 7133: 7132: 7115: 7110: 7109: 7101: 7098: 7097: 7081: 7078: 7077: 7061: 7058: 7057: 7037:homotopy theory 6954:singular points 6855: 6807:Morse functions 6774: 6760: 6757: 6756: 6740: 6726: 6723: 6722: 6697: 6694: 6693: 6682: 6643:covering spaces 6615: 6606:sphere eversion 6590: 6562: 6530: 6524: 6512: 6506: 6501: 6487: 6473: 6430: 6425: 6424: 6422: 6419: 6418: 6397: 6392: 6391: 6389: 6386: 6385: 6382: 6376: 6371: 6234: 6228: 6185:Poisson bracket 6135: 6130: 6129: 6127: 6124: 6123: 6095: 6092: 6091: 6075: 6072: 6071: 6068:Riemann surface 6042: 6037: 6036: 6034: 6031: 6030: 6020: 5989: 5986: 5985: 5966: 5946: 5943: 5942: 5935:symmetry groups 5913: 5909: 5907: 5904: 5903: 5877: 5874: 5873: 5872:, the group of 5855:complex numbers 5826: 5823: 5822: 5801: 5795: 5761: 5758: 5757: 5746: 5740: 5729: 5669: 5666: 5665: 5649: 5646: 5645: 5629: 5626: 5625: 5597: 5594: 5593: 5579: 5573: 5561:rectifiable set 5533:tangent vectors 5509: 5503: 5429: 5424: 5423: 5421: 5418: 5417: 5385: 5380: 5379: 5377: 5374: 5373: 5370: 5364: 5359: 5353: 5305:Simon Donaldson 5261: 5237:Hassler Whitney 5189: 5185: 5176: 5172: 5170: 5167: 5166: 5147: 5143: 5135: 5124: 5121: 5120: 5101: 5097: 5089: 5081: 5078: 5077: 5058: 5054: 5046: 5035: 5032: 5031: 5012: 5008: 5000: 4992: 4989: 4988: 4956: 4954: 4951: 4950: 4934: 4931: 4930: 4909: 4907: 4904: 4903: 4887: 4884: 4883: 4856: 4852: 4844: 4842: 4839: 4838: 4813: 4810: 4809: 4768: 4619: 4563:topological map 4525: 4522: 4521: 4423: 4407: 4401: 4358: 4330: 4324: 4250: 4242:Main articles: 4240: 4228:smooth function 4205: 4201: 4190: 4160: 4155: 4154: 4133: 4128: 4127: 4125: 4122: 4121: 4100: 4095: 4094: 4092: 4089: 4088: 4067: 4025: 4010: 3988: 3965: 3959: 3860: 3857: 3856: 3852: 3831: 3826: 3825: 3823: 3820: 3819: 3786: 3782: 3773: 3769: 3760: 3756: 3747: 3742: 3741: 3716: 3712: 3704: 3701: 3700: 3683: 3678: 3677: 3675: 3672: 3671: 3668: 3642: 3637: 3636: 3634: 3631: 3630: 3615: 3607: 3580: 3577: 3576: 3557: 3554: 3553: 3537: 3534: 3533: 3511: 3508: 3507: 3491: 3488: 3487: 3471: 3468: 3467: 3447: 3442: 3437: 3434: 3431: 3430: 3404: 3400: 3395: 3392: 3391: 3375: 3372: 3371: 3349: 3346: 3345: 3322: 3319: 3318: 3302: 3299: 3298: 3277: 3272: 3271: 3269: 3266: 3265: 3249: 3246: 3245: 3223: 3220: 3219: 3203: 3200: 3199: 3179: 3176: 3175: 3172: 3144: 3140: 3138: 3135: 3134: 3133:to points with 3111: 3107: 3105: 3102: 3101: 3075: 3071: 3068: and 3066: 3054: 3049: 3030: 3026: 3011: 3007: 2998: 2994: 2986: 2983: 2982: 2966: 2963: 2962: 2936: 2931: 2912: 2908: 2893: 2889: 2880: 2876: 2868: 2865: 2864: 2848: 2845: 2844: 2793: 2790: 2789: 2773: 2770: 2769: 2754: 2748: 2744: 2741: 2735: 2732: 2726: 2719: 2713: 2675:diffeomorphisms 2657: 2652: 2651: 2649: 2646: 2645: 2638: 2596: 2591: 2590: 2588: 2585: 2584: 2567: 2562: 2561: 2559: 2556: 2555: 2547: 2545:Transition maps 2526: 2520: 2513: 2490: 2485: 2484: 2482: 2479: 2478: 2438: 2433: 2432: 2430: 2427: 2426: 2411: 2405: 2385: 2379: 2373: 2328: 2327: 2311:local dimension 2306: 2281: 2278: 2277: 2256: 2251: 2250: 2248: 2245: 2244: 2227: 2222: 2221: 2219: 2216: 2215: 2184: 2179: 2160: 2155: 2142: 2137: 2124: 2119: 2118: 2106: 2102: 2087: 2083: 2074: 2070: 2066: 2062: 2053: 2048: 2047: 2045: 2042: 2041: 2016: 2013: 2012: 2001: 1981: 1976: 1975: 1973: 1970: 1969: 1967:Euclidean space 1919:Hausdorff space 1886: 1880: 1849: 1845:of points on a 1802: circles, 1788: 1755: 1744: 1737: 1730: 1723: 1701: 1684: 1631: 1623: 1620: 1619: 1587: 1586: 1577: 1573: 1566: 1558: 1556: 1549: 1543: 1542: 1533: 1529: 1522: 1515: 1511: 1504: 1502: 1495: 1488: 1486: 1483: 1482: 1420: 1415: 1375: 1374: 1370: 1368: 1365: 1364: 1340: 1335: 1292: 1291: 1287: 1285: 1282: 1281: 1248: 1247: 1239: 1235: 1227: 1218: 1217: 1204: 1200: 1192: 1185: 1181: 1162: 1161: 1157: 1148: 1147: 1131: 1122: 1110: 1109: 1104: 1100: 1081: 1080: 1076: 1069: 1053: 1051: 1048: 1047: 1019: 1016: 1015: 981: 980: 976: 974: 971: 970: 946: 945: 941: 939: 936: 935: 915: 903: 902: 876: 875: 871: 827: 824: 823: 795: 792: 791: 775: 772: 771: 755: 752: 751: 721: 720: 716: 714: 711: 710: 686: 685: 681: 679: 676: 675: 652: 651: 638: 604: 603: 599: 596: 595: 585: 554: 553: 549: 546: 545: 535: 498: 497: 493: 489: 487: 484: 483: 426: 425: 421: 419: 416: 415: 357: 352: 253:, and also the 207: 204: 203: 173: 170: 169: 152: 149: 148: 145:Euclidean space 93: 82: 76: 73: 63:Please help to 62: 46: 42: 35: 28: 23: 22: 15: 12: 11: 5: 9862: 9852: 9851: 9846: 9829: 9828: 9826: 9825: 9820: 9818:Woldemar Voigt 9815: 9810: 9805: 9800: 9795: 9790: 9785: 9783:Leonhard Euler 9780: 9775: 9770: 9765: 9759: 9757: 9755:Mathematicians 9751: 9750: 9747: 9746: 9744: 9743: 9738: 9733: 9728: 9723: 9718: 9713: 9708: 9703: 9697: 9695: 9691: 9690: 9688: 9687: 9682: 9680:Torsion tensor 9677: 9672: 9667: 9662: 9657: 9652: 9646: 9644: 9637: 9633: 9632: 9630: 9629: 9624: 9619: 9614: 9609: 9604: 9599: 9594: 9589: 9584: 9579: 9574: 9569: 9564: 9559: 9554: 9549: 9544: 9539: 9533: 9531: 9525: 9524: 9522: 9521: 9515: 9513:Tensor product 9510: 9505: 9503:Symmetrization 9500: 9495: 9493:Lie derivative 9490: 9485: 9480: 9475: 9470: 9464: 9462: 9456: 9455: 9453: 9452: 9447: 9442: 9437: 9432: 9427: 9422: 9417: 9415:Tensor density 9412: 9407: 9401: 9399: 9393: 9392: 9390: 9389: 9387:Voigt notation 9384: 9379: 9374: 9372:Ricci calculus 9369: 9364: 9359: 9357:Index notation 9354: 9349: 9343: 9341: 9337: 9336: 9333: 9332: 9330: 9329: 9324: 9319: 9314: 9309: 9303: 9301: 9299: 9298: 9293: 9287: 9284: 9283: 9281: 9280: 9275: 9273:Tensor algebra 9270: 9265: 9260: 9255: 9253:Dyadic algebra 9250: 9245: 9239: 9237: 9228: 9224: 9223: 9216: 9213: 9212: 9205: 9204: 9197: 9190: 9182: 9173: 9172: 9170: 9169: 9164: 9159: 9154: 9149: 9148: 9147: 9137: 9132: 9127: 9122: 9117: 9112: 9106: 9104: 9100: 9099: 9097: 9096: 9091: 9086: 9081: 9076: 9071: 9065: 9063: 9059: 9058: 9055: 9054: 9052: 9051: 9046: 9041: 9036: 9031: 9026: 9021: 9016: 9011: 9006: 9000: 8998: 8992: 8991: 8989: 8988: 8983: 8978: 8973: 8968: 8963: 8958: 8948: 8943: 8938: 8928: 8923: 8918: 8913: 8908: 8903: 8897: 8895: 8889: 8888: 8886: 8885: 8880: 8875: 8874: 8873: 8863: 8858: 8857: 8856: 8846: 8841: 8836: 8831: 8830: 8829: 8819: 8814: 8813: 8812: 8802: 8797: 8791: 8789: 8785: 8784: 8782: 8781: 8776: 8771: 8766: 8765: 8764: 8754: 8749: 8744: 8738: 8736: 8729: 8723: 8722: 8720: 8719: 8714: 8704: 8699: 8685: 8680: 8675: 8670: 8665: 8663:Parallelizable 8660: 8655: 8650: 8649: 8648: 8638: 8633: 8628: 8623: 8618: 8613: 8608: 8603: 8598: 8593: 8583: 8573: 8567: 8565: 8559: 8558: 8556: 8555: 8550: 8545: 8543:Lie derivative 8540: 8538:Integral curve 8535: 8530: 8525: 8524: 8523: 8513: 8508: 8507: 8506: 8499:Diffeomorphism 8496: 8490: 8488: 8482: 8481: 8479: 8478: 8473: 8468: 8463: 8458: 8453: 8448: 8443: 8438: 8432: 8430: 8421: 8420: 8418: 8417: 8412: 8407: 8402: 8397: 8392: 8387: 8382: 8377: 8376: 8375: 8370: 8360: 8359: 8358: 8347: 8345: 8344:Basic concepts 8341: 8340: 8328: 8327: 8320: 8313: 8305: 8299: 8298: 8292:manifold atlas 8288: 8282: 8262: 8261:External links 8259: 8258: 8257: 8250: 8235: 8225: 8222:Famously terse 8201: 8200: 8199: 8187: 8157: 8142: 8124: 8110: 8105: 8075: 8057: 8039: 8018: 8003: 7981: 7966:Hirsch, Morris 7963: 7948: 7930: 7910: 7907: 7905: 7904: 7871: 7842:(3): 435–467. 7826: 7769: 7734: 7715:(3): 645–680. 7696: 7678:(in Russian). 7668:Arnolʹd, V. I. 7659: 7641: 7628: 7621: 7595: 7588: 7569: 7547: 7545: 7542: 7541: 7540: 7531: 7525: 7519: 7511: 7508: 7507: 7506: 7501: 7495: 7490: 7484: 7478: 7470: 7467: 7466: 7465: 7453: 7450: 7447: 7444: 7441: 7421: 7401: 7381: 7361: 7358: 7354: 7348: 7344: 7340: 7337: 7334: 7329: 7325: 7320: 7316: 7313: 7293: 7290: 7285: 7281: 7277: 7274: 7271: 7266: 7262: 7240: 7236: 7233: 7212: 7207: 7202: 7197: 7191: 7187: 7183: 7180: 7160: 7140: 7118: 7113: 7108: 7105: 7085: 7065: 7054: 7051: 7043: 7040: 7028: 7025: 6998: 6995: 6950: 6931: 6928: 6905: 6902: 6891:Fréchet spaces 6875:Hilbert spaces 6859: 6854: 6851: 6809:, which yield 6781: 6777: 6773: 6770: 6767: 6764: 6743: 6739: 6736: 6733: 6730: 6707: 6704: 6701: 6681: 6678: 6611:Main article: 6589: 6586: 6561: 6558: 6554:quotient space 6526:Main article: 6523: 6520: 6508:Main article: 6505: 6502: 6483:Main article: 6472: 6469: 6433: 6428: 6400: 6395: 6378:Main article: 6375: 6372: 6370: 6367: 6302:, coming from 6257:surgery theory 6227: 6224: 6223: 6222: 6209: 6192: 6167: 6164:Fréchet spaces 6152: 6138: 6133: 6114: 6102: 6099: 6079: 6045: 6040: 6019: 6016: 5999: 5996: 5993: 5973: 5969: 5965: 5962: 5959: 5956: 5953: 5950: 5916: 5912: 5887: 5884: 5881: 5842: 5839: 5836: 5833: 5830: 5806:, named after 5797:Main article: 5794: 5791: 5771: 5768: 5765: 5742:Main article: 5739: 5736: 5685: 5682: 5679: 5676: 5673: 5653: 5633: 5613: 5610: 5607: 5604: 5601: 5575:Main article: 5572: 5569: 5505:Main article: 5502: 5499: 5432: 5427: 5388: 5383: 5366:Main article: 5363: 5360: 5355:Main article: 5352: 5349: 5337:Sergei Novikov 5260: 5257: 5206: 5203: 5200: 5197: 5192: 5188: 5184: 5179: 5175: 5150: 5146: 5142: 5139: 5134: 5131: 5128: 5104: 5100: 5096: 5093: 5088: 5085: 5061: 5057: 5053: 5050: 5045: 5042: 5039: 5015: 5011: 5007: 5004: 4999: 4996: 4982:transition map 4974:et inversement 4962: 4959: 4938: 4915: 4912: 4891: 4866: 4862: 4859: 4855: 4850: 4847: 4826: 4823: 4820: 4817: 4772:Analysis Situs 4767: 4764: 4690:Henri Poincaré 4682:Liouville tori 4656:, Jacobi, and 4654:Siméon Poisson 4618: 4615: 4547: 4544: 4541: 4538: 4535: 4532: 4529: 4512:(or corners), 4499:Leonhard Euler 4422: 4419: 4417:and topology. 4415:linear algebra 4400: 4397: 4357: 4354: 4326:Main article: 4323: 4320: 4254:quotient space 4239: 4236: 4213: 4208: 4204: 4200: 4197: 4193: 4189: 4186: 4183: 4180: 4177: 4174: 4171: 4168: 4163: 4158: 4153: 4150: 4147: 4144: 4141: 4136: 4131: 4103: 4098: 4066: 4060: 4048:intrinsic view 4033:extrinsic view 4024: 4021: 4006: 3984: 3963:Surgery theory 3958: 3955: 3906: 3903: 3900: 3897: 3894: 3891: 3888: 3885: 3882: 3879: 3876: 3873: 3870: 3867: 3864: 3834: 3829: 3805: 3801: 3797: 3794: 3789: 3785: 3781: 3776: 3772: 3768: 3763: 3759: 3755: 3750: 3745: 3740: 3737: 3734: 3731: 3728: 3725: 3722: 3719: 3715: 3711: 3708: 3686: 3681: 3667: 3664: 3645: 3640: 3614: 3611: 3606: 3603: 3590: 3587: 3584: 3564: 3561: 3541: 3521: 3518: 3515: 3495: 3475: 3450: 3445: 3440: 3418: 3415: 3412: 3407: 3403: 3399: 3379: 3359: 3356: 3353: 3329: 3326: 3306: 3280: 3275: 3253: 3233: 3230: 3227: 3207: 3183: 3171: 3168: 3155: 3152: 3147: 3143: 3122: 3119: 3114: 3110: 3089: 3086: 3083: 3078: 3074: 3065: 3062: 3057: 3052: 3048: 3044: 3041: 3038: 3033: 3029: 3025: 3022: 3019: 3014: 3010: 3006: 3001: 2997: 2993: 2990: 2970: 2950: 2947: 2944: 2939: 2934: 2930: 2926: 2923: 2920: 2915: 2911: 2907: 2904: 2901: 2896: 2892: 2888: 2883: 2879: 2875: 2872: 2852: 2809: 2806: 2803: 2800: 2797: 2777: 2752: 2739: 2730: 2712: 2709: 2660: 2655: 2637: 2634: 2630:transition map 2599: 2594: 2570: 2565: 2546: 2543: 2522:Main article: 2519: 2516: 2511: 2493: 2488: 2456:homeomorphisms 2441: 2436: 2421:, or simply a 2415:coordinate map 2407:Main article: 2404: 2401: 2375:Main article: 2372: 2369: 2335:disjoint union 2291: 2288: 2285: 2259: 2254: 2230: 2225: 2203: 2199: 2195: 2192: 2187: 2182: 2178: 2174: 2171: 2168: 2163: 2158: 2154: 2150: 2145: 2140: 2136: 2132: 2127: 2122: 2117: 2114: 2109: 2105: 2101: 2098: 2095: 2090: 2086: 2082: 2077: 2073: 2069: 2065: 2061: 2056: 2051: 2026: 2023: 2020: 2009:isolated point 1989: 1984: 1979: 1879: 1876: 1787: 1784: 1776:nautical chart 1716: 1715: 1683: 1680: 1638: 1635: 1630: 1627: 1580: 1576: 1572: 1569: 1564: 1561: 1555: 1552: 1550: 1548: 1545: 1544: 1536: 1532: 1528: 1525: 1518: 1514: 1510: 1507: 1501: 1498: 1496: 1494: 1491: 1490: 1481:) is given by 1429: 1426: 1423: 1419: 1414: 1411: 1408: 1405: 1402: 1399: 1396: 1393: 1387: 1384: 1381: 1378: 1373: 1349: 1346: 1343: 1339: 1334: 1331: 1328: 1325: 1322: 1319: 1316: 1313: 1307: 1304: 1301: 1298: 1295: 1290: 1266:transition map 1242: 1238: 1234: 1231: 1226: 1223: 1221: 1219: 1215: 1207: 1203: 1199: 1196: 1191: 1188: 1184: 1177: 1174: 1171: 1168: 1165: 1160: 1156: 1153: 1151: 1149: 1145: 1140: 1137: 1134: 1128: 1125: 1119: 1116: 1113: 1108: 1103: 1096: 1093: 1090: 1087: 1084: 1079: 1075: 1072: 1070: 1068: 1065: 1062: 1059: 1056: 1055: 1035: 1032: 1029: 1026: 1023: 996: 993: 990: 987: 984: 979: 955: 952: 949: 944: 921: 918: 912: 909: 906: 901: 897: 891: 888: 885: 882: 879: 874: 870: 867: 864: 861: 858: 855: 852: 849: 846: 843: 840: 837: 834: 831: 811: 808: 805: 802: 799: 779: 759: 736: 733: 730: 727: 724: 719: 695: 692: 689: 684: 650: 647: 644: 641: 639: 637: 634: 631: 628: 625: 619: 616: 613: 610: 607: 602: 598: 597: 594: 591: 588: 586: 584: 581: 578: 575: 572: 566: 563: 560: 557: 552: 548: 547: 544: 541: 538: 536: 534: 531: 528: 525: 522: 516: 513: 510: 507: 504: 501: 496: 492: 491: 462: 459: 456: 453: 450: 447: 444: 441: 435: 432: 429: 424: 356: 353: 351: 348: 299:to be done. A 211: 177: 156: 95: 94: 49: 47: 40: 26: 9: 6: 4: 3: 2: 9861: 9850: 9847: 9845: 9842: 9841: 9839: 9824: 9821: 9819: 9816: 9814: 9811: 9809: 9806: 9804: 9801: 9799: 9796: 9794: 9791: 9789: 9786: 9784: 9781: 9779: 9776: 9774: 9771: 9769: 9766: 9764: 9761: 9760: 9758: 9756: 9752: 9742: 9739: 9737: 9734: 9732: 9729: 9727: 9724: 9722: 9719: 9717: 9714: 9712: 9709: 9707: 9704: 9702: 9699: 9698: 9696: 9692: 9686: 9683: 9681: 9678: 9676: 9673: 9671: 9668: 9666: 9663: 9661: 9660:Metric tensor 9658: 9656: 9653: 9651: 9648: 9647: 9645: 9641: 9638: 9634: 9628: 9625: 9623: 9620: 9618: 9615: 9613: 9610: 9608: 9605: 9603: 9600: 9598: 9595: 9593: 9590: 9588: 9585: 9583: 9580: 9578: 9575: 9573: 9572:Exterior form 9570: 9568: 9565: 9563: 9560: 9558: 9555: 9553: 9550: 9548: 9545: 9543: 9540: 9538: 9535: 9534: 9532: 9526: 9519: 9516: 9514: 9511: 9509: 9506: 9504: 9501: 9499: 9496: 9494: 9491: 9489: 9486: 9484: 9481: 9479: 9476: 9474: 9471: 9469: 9466: 9465: 9463: 9461: 9457: 9451: 9448: 9446: 9445:Tensor bundle 9443: 9441: 9438: 9436: 9433: 9431: 9428: 9426: 9423: 9421: 9418: 9416: 9413: 9411: 9408: 9406: 9403: 9402: 9400: 9394: 9388: 9385: 9383: 9380: 9378: 9375: 9373: 9370: 9368: 9365: 9363: 9360: 9358: 9355: 9353: 9350: 9348: 9345: 9344: 9342: 9338: 9328: 9325: 9323: 9320: 9318: 9315: 9313: 9310: 9308: 9305: 9304: 9302: 9297: 9294: 9292: 9289: 9288: 9285: 9279: 9276: 9274: 9271: 9269: 9266: 9264: 9261: 9259: 9256: 9254: 9251: 9249: 9246: 9244: 9241: 9240: 9238: 9236: 9232: 9229: 9225: 9221: 9220: 9214: 9210: 9203: 9198: 9196: 9191: 9189: 9184: 9183: 9180: 9168: 9165: 9163: 9162:Supermanifold 9160: 9158: 9155: 9153: 9150: 9146: 9143: 9142: 9141: 9138: 9136: 9133: 9131: 9128: 9126: 9123: 9121: 9118: 9116: 9113: 9111: 9108: 9107: 9105: 9101: 9095: 9092: 9090: 9087: 9085: 9082: 9080: 9077: 9075: 9072: 9070: 9067: 9066: 9064: 9060: 9050: 9047: 9045: 9042: 9040: 9037: 9035: 9032: 9030: 9027: 9025: 9022: 9020: 9017: 9015: 9012: 9010: 9007: 9005: 9002: 9001: 8999: 8997: 8993: 8987: 8984: 8982: 8979: 8977: 8974: 8972: 8969: 8967: 8964: 8962: 8959: 8957: 8953: 8949: 8947: 8944: 8942: 8939: 8937: 8933: 8929: 8927: 8924: 8922: 8919: 8917: 8914: 8912: 8909: 8907: 8904: 8902: 8899: 8898: 8896: 8894: 8890: 8884: 8883:Wedge product 8881: 8879: 8876: 8872: 8869: 8868: 8867: 8864: 8862: 8859: 8855: 8852: 8851: 8850: 8847: 8845: 8842: 8840: 8837: 8835: 8832: 8828: 8827:Vector-valued 8825: 8824: 8823: 8820: 8818: 8815: 8811: 8808: 8807: 8806: 8803: 8801: 8798: 8796: 8793: 8792: 8790: 8786: 8780: 8777: 8775: 8772: 8770: 8767: 8763: 8760: 8759: 8758: 8757:Tangent space 8755: 8753: 8750: 8748: 8745: 8743: 8740: 8739: 8737: 8733: 8730: 8728: 8724: 8718: 8715: 8713: 8709: 8705: 8703: 8700: 8698: 8694: 8690: 8686: 8684: 8681: 8679: 8676: 8674: 8671: 8669: 8666: 8664: 8661: 8659: 8656: 8654: 8651: 8647: 8644: 8643: 8642: 8639: 8637: 8634: 8632: 8629: 8627: 8624: 8622: 8619: 8617: 8614: 8612: 8609: 8607: 8604: 8602: 8599: 8597: 8594: 8592: 8588: 8584: 8582: 8578: 8574: 8572: 8569: 8568: 8566: 8560: 8554: 8551: 8549: 8546: 8544: 8541: 8539: 8536: 8534: 8531: 8529: 8526: 8522: 8521:in Lie theory 8519: 8518: 8517: 8514: 8512: 8509: 8505: 8502: 8501: 8500: 8497: 8495: 8492: 8491: 8489: 8487: 8483: 8477: 8474: 8472: 8469: 8467: 8464: 8462: 8459: 8457: 8454: 8452: 8449: 8447: 8444: 8442: 8439: 8437: 8434: 8433: 8431: 8428: 8424:Main results 8422: 8416: 8413: 8411: 8408: 8406: 8405:Tangent space 8403: 8401: 8398: 8396: 8393: 8391: 8388: 8386: 8383: 8381: 8378: 8374: 8371: 8369: 8366: 8365: 8364: 8361: 8357: 8354: 8353: 8352: 8349: 8348: 8346: 8342: 8337: 8333: 8326: 8321: 8319: 8314: 8312: 8307: 8306: 8303: 8297: 8293: 8289: 8286: 8283: 8279: 8275: 8274: 8269: 8265: 8264: 8253: 8247: 8243: 8242: 8236: 8233: 8229: 8226: 8223: 8219: 8218:0-8053-9021-9 8215: 8211: 8210: 8205: 8202: 8197: 8193: 8192: 8188: 8185: 8181: 8180: 8176: 8175: 8173: 8172:3-253-03059-8 8169: 8165: 8161: 8158: 8155: 8151: 8147: 8143: 8140: 8139:0-13-181629-2 8136: 8132: 8128: 8125: 8108: 8093: 8092:0-201-51035-9 8089: 8085: 8084: 8079: 8076: 8073: 8072:0-691-04833-9 8069: 8065: 8061: 8058: 8055: 8054:0-387-90271-6 8051: 8047: 8043: 8040: 8037: 8033: 8032:0-387-95495-3 8029: 8025: 8024: 8019: 8016: 8015:0-387-98759-2 8012: 8008: 8004: 8001: 7997: 7996:0-691-08190-5 7993: 7989: 7985: 7982: 7979: 7978:0-387-90148-5 7975: 7971: 7967: 7964: 7961: 7960:0-8218-3695-1 7957: 7953: 7949: 7946: 7945:0-13-212605-2 7942: 7938: 7934: 7931: 7928: 7927:0-691-08577-3 7924: 7920: 7916: 7913: 7912: 7899: 7894: 7890: 7886: 7882: 7875: 7867: 7863: 7859: 7855: 7850: 7845: 7841: 7837: 7830: 7822: 7818: 7814: 7810: 7806: 7802: 7797: 7792: 7788: 7784: 7780: 7773: 7765: 7761: 7757: 7753: 7749: 7745: 7738: 7730: 7726: 7722: 7718: 7714: 7710: 7706: 7700: 7690: 7685: 7681: 7677: 7673: 7669: 7663: 7655: 7651: 7645: 7638: 7632: 7624: 7622:0-8218-1045-6 7618: 7614: 7609: 7608: 7599: 7591: 7589:9780534133092 7585: 7581: 7580: 7572: 7570:9789812791818 7566: 7562: 7561: 7552: 7548: 7535: 7532: 7529: 7526: 7523: 7520: 7517: 7514: 7513: 7505: 7502: 7499: 7496: 7494: 7491: 7488: 7485: 7482: 7479: 7476: 7473: 7472: 7448: 7445: 7442: 7419: 7399: 7379: 7359: 7356: 7352: 7346: 7342: 7338: 7335: 7332: 7327: 7323: 7318: 7314: 7311: 7291: 7288: 7283: 7279: 7275: 7272: 7269: 7264: 7260: 7234: 7231: 7210: 7205: 7195: 7185: 7181: 7178: 7158: 7138: 7116: 7106: 7103: 7083: 7063: 7055: 7052: 7048: 7044: 7041: 7038: 7033: 7029: 7026: 7023: 7019: 7015: 7011: 7007: 7003: 6999: 6996: 6993: 6989: 6988: 6982: 6978: 6977: 6971: 6967: 6963: 6959: 6955: 6951: 6948: 6944: 6940: 6935: 6932: 6929: 6926: 6925:finite groups 6922: 6918: 6914: 6913:singularities 6910: 6906: 6903: 6900: 6896: 6892: 6888: 6884: 6883:Banach spaces 6880: 6876: 6872: 6868: 6864: 6860: 6857: 6856: 6850: 6848: 6844: 6840: 6836: 6832: 6828: 6824: 6820: 6816: 6812: 6808: 6803: 6801: 6797: 6792: 6779: 6768: 6765: 6762: 6734: 6731: 6728: 6705: 6702: 6699: 6691: 6686: 6677: 6675: 6671: 6667: 6663: 6658: 6656: 6652: 6648: 6644: 6640: 6636: 6632: 6628: 6624: 6620: 6614: 6607: 6603: 6599: 6598:Morin surface 6594: 6585: 6583: 6579: 6575: 6574:Betti numbers 6571: 6567: 6557: 6555: 6551: 6543: 6542:Boy's surface 6539: 6534: 6529: 6519: 6517: 6511: 6500: 6495: 6493: 6486: 6477: 6468: 6466: 6462: 6458: 6453: 6451: 6450: 6431: 6416: 6415:ordered basis 6398: 6381: 6374:Orientability 6366: 6364: 6360: 6356: 6352: 6351: 6346: 6342: 6340: 6336: 6332: 6328: 6323: 6321: 6317: 6313: 6312:orientability 6309: 6305: 6301: 6296: 6294: 6289: 6287: 6283: 6282: 6277: 6272: 6268: 6266: 6260: 6258: 6254: 6250: 6246: 6242: 6237: 6233: 6220: 6216: 6215: 6210: 6207: 6203: 6199: 6198: 6193: 6190: 6186: 6182: 6178: 6174: 6173: 6168: 6165: 6161: 6160:Banach spaces 6157: 6153: 6136: 6121: 6120: 6115: 6100: 6097: 6077: 6069: 6065: 6061: 6043: 6028: 6027: 6022: 6021: 6015: 6013: 5997: 5994: 5991: 5971: 5967: 5960: 5957: 5954: 5948: 5940: 5936: 5932: 5914: 5910: 5901: 5885: 5882: 5879: 5871: 5867: 5862: 5860: 5856: 5837: 5831: 5820: 5815: 5813: 5809: 5805: 5800: 5790: 5787: 5785: 5766: 5755: 5751: 5745: 5735: 5732: 5725: 5723: 5722:vector fields 5719: 5715: 5711: 5707: 5703: 5699: 5680: 5677: 5674: 5651: 5631: 5608: 5605: 5602: 5592: 5591:inner product 5588: 5587:tangent space 5584: 5578: 5568: 5566: 5562: 5557: 5555: 5551: 5547: 5546: 5541: 5536: 5534: 5530: 5527:. This is an 5526: 5525:tangent space 5522: 5518: 5514: 5508: 5498: 5495: 5491: 5487: 5482: 5480: 5476: 5471: 5468: 5464: 5460: 5456: 5451: 5446: 5430: 5415: 5412: 5408: 5407:homeomorphism 5404: 5386: 5369: 5358: 5348: 5346: 5342: 5338: 5334: 5333:Stephen Smale 5330: 5326: 5322: 5318: 5314: 5310: 5306: 5302: 5298: 5294: 5290: 5286: 5282: 5278: 5274: 5273:Poul Heegaard 5270: 5266: 5256: 5254: 5250: 5246: 5242: 5238: 5234: 5230: 5226: 5224: 5220: 5204: 5201: 5198: 5195: 5190: 5186: 5182: 5177: 5173: 5148: 5144: 5140: 5137: 5132: 5129: 5126: 5102: 5098: 5094: 5091: 5086: 5083: 5059: 5055: 5051: 5048: 5043: 5040: 5037: 5013: 5009: 5005: 5002: 4997: 4994: 4985: 4983: 4979: 4975: 4960: 4957: 4936: 4913: 4910: 4889: 4881: 4864: 4860: 4857: 4853: 4848: 4845: 4821: 4815: 4807: 4802: 4800: 4796: 4792: 4788: 4784: 4779: 4777: 4773: 4763: 4761: 4757: 4753: 4749: 4745: 4741: 4737: 4733: 4729: 4726: 4722: 4718: 4714: 4710: 4706: 4702: 4698: 4693: 4691: 4687: 4683: 4679: 4675: 4671: 4667: 4663: 4659: 4655: 4651: 4646: 4644: 4640: 4636: 4632: 4628: 4624: 4614: 4612: 4608: 4604: 4603:Betti numbers 4600: 4596: 4592: 4588: 4584: 4580: 4576: 4572: 4568: 4564: 4545: 4542: 4539: 4536: 4533: 4530: 4527: 4519: 4515: 4511: 4508: 4504: 4500: 4496: 4492: 4487: 4485: 4481: 4480:ambient space 4477: 4473: 4468: 4466: 4462: 4458: 4454: 4450: 4446: 4442: 4438: 4434: 4430: 4426: 4418: 4416: 4412: 4406: 4392: 4388: 4386: 4382: 4379: × 4378: 4374: 4370: 4365: 4363: 4353: 4351: 4347: 4342: 4339: 4334: 4329: 4319: 4317: 4313: 4308: 4306: 4302: 4298: 4294: 4290: 4286: 4282: 4278: 4273: 4271: 4267: 4263: 4259: 4255: 4249: 4245: 4235: 4233: 4229: 4224: 4211: 4206: 4198: 4191: 4187: 4181: 4178: 4172: 4161: 4145: 4134: 4120:, defined as 4119: 4101: 4086: 4082: 4078: 4075: 4073: 4064: 4059: 4057: 4053: 4052:normal bundle 4049: 4044: 4042: 4038: 4034: 4030: 4020: 4018: 4014: 4009: 4004: 4000: 3996: 3992: 3987: 3982: 3977: 3974: 3969: 3964: 3954: 3951: 3949: 3945: 3941: 3937: 3933: 3929: 3925: 3920: 3904: 3898: 3895: 3892: 3886: 3880: 3877: 3874: 3871: 3868: 3862: 3850: 3832: 3816: 3803: 3799: 3795: 3792: 3787: 3783: 3779: 3774: 3770: 3766: 3761: 3757: 3753: 3748: 3738: 3732: 3729: 3726: 3723: 3720: 3713: 3709: 3706: 3684: 3663: 3661: 3643: 3624: 3619: 3610: 3602: 3588: 3585: 3582: 3562: 3539: 3519: 3516: 3513: 3493: 3473: 3464: 3448: 3443: 3413: 3410: 3405: 3401: 3377: 3357: 3354: 3351: 3343: 3327: 3304: 3296: 3278: 3251: 3231: 3228: 3225: 3205: 3197: 3181: 3167: 3153: 3150: 3145: 3141: 3120: 3117: 3112: 3108: 3084: 3081: 3076: 3072: 3063: 3060: 3055: 3050: 3046: 3031: 3027: 3023: 3020: 3017: 3012: 3008: 3004: 2999: 2995: 2968: 2945: 2942: 2937: 2932: 2928: 2913: 2909: 2905: 2902: 2899: 2894: 2890: 2886: 2881: 2877: 2850: 2841: 2839: 2835: 2831: 2827: 2823: 2820:-manifold. A 2804: 2801: 2798: 2775: 2767: 2763: 2751: 2738: 2729: 2723: 2718: 2708: 2706: 2701: 2699: 2694: 2692: 2688: 2684: 2680: 2676: 2658: 2642: 2633: 2631: 2627: 2623: 2619: 2615: 2597: 2568: 2553: 2542: 2540: 2539:maximal atlas 2535: 2533: 2532: 2525: 2515: 2509: 2491: 2476: 2472: 2468: 2464: 2459: 2457: 2439: 2424: 2420: 2416: 2410: 2400: 2398: 2394: 2390: 2384: 2378: 2368: 2366: 2362: 2358: 2354: 2350: 2348: 2344: 2340: 2336: 2332: 2329:pure manifold 2324: 2320: 2316: 2312: 2303: 2289: 2286: 2283: 2275: 2274:diffeomorphic 2257: 2228: 2201: 2197: 2193: 2190: 2185: 2180: 2176: 2172: 2169: 2166: 2161: 2156: 2152: 2148: 2143: 2138: 2134: 2130: 2125: 2115: 2107: 2103: 2099: 2096: 2093: 2088: 2084: 2080: 2075: 2071: 2063: 2059: 2054: 2040: 2024: 2021: 2018: 2010: 2005: 1987: 1982: 1968: 1964: 1960: 1956: 1952: 1950: 1946: 1942: 1938: 1934: 1930: 1926: 1922: 1920: 1917: 1912: 1910: 1909:diffeomorphic 1906: 1902: 1898: 1893: 1891: 1885: 1875: 1873: 1869: 1863: 1860: 1856: 1852: 1848: 1844: 1840: 1836: 1832: 1828: 1823: 1821: 1797: 1792: 1783: 1781: 1777: 1773: 1769: 1764: 1761: 1758: 1751: 1747: 1740: 1733: 1726: 1721: 1712: 1708: 1704: 1700: 1699: 1698: 1697: 1693: 1689: 1679: 1677: 1673: 1668: 1666: 1662: 1658: 1654: 1636: 1633: 1628: 1625: 1617: 1613: 1609: 1606: + 1605: 1600: 1578: 1574: 1570: 1567: 1562: 1559: 1553: 1551: 1546: 1534: 1530: 1526: 1523: 1516: 1512: 1508: 1505: 1499: 1497: 1492: 1480: 1476: 1472: 1468: 1464: 1460: 1456: 1452: 1448: 1443: 1427: 1424: 1421: 1417: 1412: 1409: 1406: 1400: 1397: 1394: 1371: 1347: 1344: 1341: 1337: 1332: 1329: 1326: 1320: 1317: 1314: 1288: 1274: 1270: 1268: 1267: 1261: 1240: 1236: 1232: 1229: 1224: 1222: 1213: 1205: 1201: 1197: 1194: 1189: 1186: 1182: 1158: 1154: 1152: 1143: 1138: 1135: 1132: 1126: 1123: 1106: 1101: 1077: 1073: 1071: 1063: 1057: 1030: 1027: 1024: 1013: 977: 942: 919: 916: 899: 895: 872: 868: 862: 859: 856: 844: 841: 838: 832: 829: 806: 803: 800: 777: 757: 717: 682: 672: 670: 665: 648: 645: 642: 640: 632: 629: 626: 600: 592: 589: 587: 579: 576: 573: 550: 542: 539: 537: 529: 526: 523: 494: 481: 480: 474: 460: 457: 454: 448: 445: 442: 422: 413: 412:open interval 409: 406: 402: 398: 394: 390: 386: 384: 379: 376: + 375: 371: 361: 347: 345: 340: 338: 334: 330: 326: 322: 318: 315:serve as the 314: 310: 306: 302: 298: 294: 290: 285: 283: 279: 275: 271: 267: 262: 260: 256: 252: 248: 244: 240: 236: 232: 228: 223: 209: 201: 197: 193: 189: 175: 154: 146: 142: 138: 134: 126: 122: 118: 113: 106: 101: 91: 88: 80: 70: 66: 60: 59: 53: 48: 39: 38: 33: 19: 9823:Hermann Weyl 9627:Vector space 9612:Pseudotensor 9596: 9577:Fiber bundle 9530:abstractions 9425:Mixed tensor 9410:Tensor field 9217: 9089:Moving frame 9084:Morse theory 9074:Gauge theory 8866:Tensor field 8795:Closed/Exact 8774:Vector field 8742:Distribution 8683:Hypercomplex 8678:Quaternionic 8415:Vector field 8373:Smooth atlas 8331: 8271: 8240: 8231: 8207: 8195: 8189: 8183: 8177: 8163: 8145: 8130: 8081: 8063: 8060:Milnor, John 8045: 8035: 8021: 8006: 7987: 7969: 7951: 7936: 7918: 7888: 7884: 7874: 7849:math/9304210 7839: 7835: 7829: 7786: 7782: 7772: 7747: 7743: 7737: 7712: 7708: 7699: 7679: 7675: 7662: 7653: 7650:Poincaré, H. 7644: 7631: 7606: 7598: 7578: 7559: 7551: 7510:By dimension 7027:CW-complexes 6991: 6984: 6980: 6973: 6968:are largely 6965: 6964:and English 6961: 6957: 6934:Non-singular 6916: 6804: 6793: 6720: 6659: 6616: 6563: 6549: 6547: 6513: 6510:Klein bottle 6504:Klein bottle 6491: 6488: 6485:Möbius strip 6479:Möbius strip 6471:Möbius strip 6461:Klein bottle 6457:Möbius strip 6454: 6447: 6383: 6348: 6343: 6324: 6297: 6290: 6285: 6279: 6275: 6273: 6269: 6261: 6243:, except in 6241:in principle 6240: 6238: 6235: 6212: 6195: 6170: 6117: 6024: 5939:hyperspheres 5900:real numbers 5863: 5816: 5803: 5802: 5788: 5749: 5747: 5730: 5726: 5582: 5580: 5558: 5554:power series 5543: 5539: 5537: 5528: 5520: 5512: 5510: 5489: 5485: 5483: 5472: 5458: 5454: 5449: 5447: 5371: 5345:Morse theory 5264: 5262: 5241:foundational 5229:Hermann Weyl 5227: 4986: 4928: 4879: 4805: 4803: 4798: 4780: 4775: 4769: 4755: 4751: 4747: 4743: 4735: 4731: 4727: 4724: 4720: 4716: 4712: 4704: 4696: 4694: 4647: 4620: 4594: 4590: 4586: 4578: 4574: 4570: 4566: 4517: 4513: 4506: 4488: 4469: 4427: 4424: 4408: 4384: 4380: 4376: 4366: 4359: 4350:Möbius strip 4343: 4335: 4331: 4309: 4304: 4300: 4296: 4292: 4288: 4284: 4274: 4266:well-behaved 4262:CW complexes 4251: 4231: 4225: 4084: 4080: 4076: 4071: 4068: 4062: 4047: 4045: 4032: 4026: 4016: 4012: 4007: 4002: 3998: 3994: 3990: 3985: 3980: 3978: 3970: 3966: 3952: 3947: 3943: 3939: 3935: 3931: 3927: 3921: 3855:defined by 3848: 3817: 3669: 3659: 3628: 3622: 3608: 3605:Construction 3465: 3294: 3195: 3173: 2842: 2761: 2759: 2749: 2736: 2727: 2705:pseudogroups 2702: 2697: 2695: 2643: 2639: 2629: 2625: 2621: 2617: 2613: 2551: 2548: 2538: 2536: 2529: 2527: 2507: 2470: 2466: 2462: 2460: 2422: 2418: 2414: 2412: 2396: 2392: 2386: 2351: 2338: 2326: 2314: 2310: 2304: 2006: 1959:homeomorphic 1954: 1953: 1944: 1936: 1935:conditions; 1928: 1924: 1923: 1913: 1894: 1887: 1864: 1858: 1854: 1850: 1824: 1817: 1814: cubic. 1786:Other curves 1765: 1762: 1756: 1749: 1745: 1738: 1731: 1724: 1722:: the plane 1717: 1710: 1706: 1702: 1685: 1675: 1671: 1669: 1664: 1660: 1656: 1652: 1615: 1611: 1607: 1603: 1601: 1478: 1474: 1470: 1466: 1462: 1458: 1454: 1450: 1446: 1444: 1279: 1265: 1262: 1011: 673: 666: 477: 475: 392: 388: 382: 377: 373: 366: 341: 317:phase spaces 286: 263: 255:Klein bottle 224: 196:homeomorphic 192:neighborhood 168: 136: 130: 105:Klein bottle 83: 74: 55: 9763:Élie Cartan 9711:Spin tensor 9685:Weyl tensor 9643:Mathematics 9607:Multivector 9398:definitions 9296:Engineering 9235:Mathematics 9034:Levi-Civita 9024:Generalized 8996:Connections 8946:Lie algebra 8878:Volume form 8779:Vector flow 8752:Pushforward 8747:Lie bracket 8646:Lie algebra 8611:G-structure 8400:Pushforward 8380:Submanifold 7952:3-Manifolds 7891:: 251–272. 7750:(1): 1–14. 7705:Whitney, H. 7689:10.4213/rm5 7475:Submanifold 6952:Because of 6923:of various 6839:heat kernel 6800:functionals 6639:submersions 6631:knot theory 6629:, of which 6413:. Given an 6245:dimension 4 6119:CR manifold 6060:holomorphic 5450:topological 5329:John Milnor 5223:submanifold 4670:Hamiltonian 4662:phase space 4573:edges, and 4516:edges, and 4453:Lobachevsky 1963:open subset 1847:cubic curve 1692:unit sphere 385:-coordinate 370:unit circle 268:and modern 200:open subset 133:mathematics 125:South Poles 69:introducing 9838:Categories 9592:Linear map 9460:Operations 9157:Stratifold 9115:Diffeology 8911:Associated 8712:Symplectic 8697:Riemannian 8626:Hyperbolic 8553:Submersion 8461:Hopf–Rinow 8395:Submersion 8390:Smooth map 8268:"Manifold" 7909:References 7796:2303.04200 7789:: 105114. 7528:5-manifold 7522:4-manifold 7516:3-manifold 7032:CW complex 7002:filtration 6970:synonymous 6811:handlebody 6692:of degree 6635:immersions 6627:embeddings 6582:cohomology 6516:cross-caps 6497:See also: 6449:orientable 6281:invariants 5808:Sophus Lie 5804:Lie groups 5793:Lie groups 5718:divergence 5459:consistent 5455:particular 4666:Lagrangian 4569:vertices, 4037:tangential 3342:complement 3317:, denoted 3218:, denoted 2826:1-manifold 2766:2-manifold 2715:See also: 2698:compatible 2465:called an 2381:See also: 2323:dimensions 1868:components 1841:, and the 405:invertible 401:continuous 397:projection 249:, and the 233:, but not 52:references 9844:Manifolds 9731:EM tensor 9567:Dimension 9518:Transpose 9039:Principal 9014:Ehresmann 8971:Subbundle 8961:Principal 8936:Fibration 8916:Cotangent 8788:Covectors 8641:Lie group 8621:Hermitian 8564:manifolds 8533:Immersion 8528:Foliation 8466:Noether's 8451:Frobenius 8446:De Rham's 8441:Darboux's 8332:Manifolds 8278:EMS Press 7968:, (1997) 7821:0393-0440 7764:120977898 7555:E.g. see 7432:. A pair 7357:∈ 7336:… 7315:∘ 7289:∈ 7273:… 7235:∈ 7190:∞ 7182:∈ 7107:⊆ 6960:, German 6904:Orbifolds 6837:, and to 6772:→ 6766:: 6738:→ 6732:: 6602:immersion 6550:antipodes 6286:invariant 6276:different 5995:− 5958:− 5883:× 5832:⁡ 5799:Lie group 5770:‖ 5767:⋅ 5764:‖ 5714:curvature 5684:⟩ 5672:⟨ 5612:⟩ 5609:⋅ 5603:⋅ 5600:⟨ 5494:connected 5486:dimension 5475:Hausdorff 5411:bijective 5325:René Thom 5313:spacetime 5249:Lie group 5196:− 5141:− 5133:− 5095:− 5052:− 5044:− 5006:− 4980:and of a 4846:θ 4816:θ 4783:level set 4740:induction 4635:Jacobians 4617:Synthesis 4531:− 4465:curvature 4373:cylinders 4338:bijection 4258:orbifolds 4203:‖ 4196:‖ 4185:↦ 4167:∖ 4152:→ 4140:∖ 3957:Patchwork 3924:unit disc 3863:χ 3754:∣ 3739:∈ 3586:− 3560:∂ 3517:⁡ 3355:⁡ 3340:, is the 3325:∂ 3229:⁡ 3082:≥ 3043:Σ 3021:… 2925:Σ 2903:… 2802:− 2391:, called 2319:connected 2315:dimension 2170:⋯ 2116:∈ 2097:… 2039:open ball 1945:Hausdorff 1941:long line 1933:point-set 1929:Hausdorff 1839:hyperbola 1831:countable 1820:connected 1509:− 1425:− 1372:χ 1289:χ 1233:− 1198:− 1159:χ 1124:− 1107:χ 1078:χ 978:χ 943:χ 917:− 900:χ 896:∘ 873:χ 851:→ 718:χ 683:χ 601:χ 551:χ 495:χ 423:χ 414:(−1, 1): 333:spacetime 305:distances 188:-manifold 77:July 2021 9597:Manifold 9582:Geodesic 9340:Notation 9135:Orbifold 9130:K-theory 9120:Diffiety 8844:Pullback 8658:Oriented 8636:Kenmotsu 8616:Hadamard 8562:Types of 8511:Geodesic 8336:Glossary 8131:Topology 8000:category 7670:(1998). 7481:Geodesic 7469:See also 7224:, where 6966:manifold 6909:orbifold 6604:used in 6578:homology 6369:Surfaces 6359:homotopy 6355:homology 6329:and the 6316:homology 6251:and the 6204:made by 5866:matrices 5550:analytic 5517:calculus 5277:Max Dehn 5265:surfaces 5221:, every 4961:′ 4914:′ 4861:′ 4849:′ 4725:diskrete 4707:, which 4697:manifold 4510:vertices 4503:polytope 4491:property 4441:Saccheri 4279:, which 4244:Orbifold 4083:-sphere 4029:embedded 4015:= 0 and 3660:manifold 3295:boundary 3196:interior 2469:, whose 2345:), each 1943:, while 1835:parabola 1046:, then: 389:Figure 1 344:topology 297:calculus 291:; their 282:CT scans 266:geometry 239:surfaces 194:that is 137:manifold 9694:Physics 9528:Related 9291:Physics 9209:Tensors 9079:History 9062:Related 8976:Tangent 8954:) 8934:) 8901:Adjoint 8893:Bundles 8871:density 8769:Torsion 8735:Vectors 8727:Tensors 8710:) 8695:) 8691:, 8689:Pseudo− 8668:Poisson 8601:Finsler 8596:Fibered 8591:Contact 8589:) 8581:Complex 8579:) 8548:Section 8280:, 2001 8230:(1999) 8206:(1965) 8129:(2000) 8080:(1991) 8062:(1997) 8044:(1977) 7866:2118532 7801:Bibcode 7729:1968482 7050:theory. 6958:variété 6947:sheaves 6943:Schemes 6921:actions 6331:torsion 5859:modulus 5819:compact 5710:volumes 4929:...les 4776:variété 4476:surface 4439:fails. 4399:History 4074:-sphere 3506:, then 2628:, or a 2518:Atlases 1965:of the 1780:atlases 1713:– 1 = 0 1477:, 1465:, 1453:, 408:mapping 319:in the 295:allows 276:and as 231:circles 65:improve 9622:Vector 9617:Spinor 9602:Matrix 9396:Tensor 9044:Vector 9029:Koszul 9009:Cartan 9004:Affine 8986:Vector 8981:Tensor 8966:Spinor 8956:Normal 8952:Stable 8906:Affine 8810:bundle 8762:bundle 8708:Almost 8631:Kähler 8587:Almost 8577:Almost 8571:Closed 8471:Sard's 8427:(list) 8248: 8216: 8170: 8152: 8137: 8090: 8070: 8052: 8030: 8013: 7994: 7976: 7958: 7943: 7925: 7864: 7819: 7762: 7727: 7619: 7586: 7567: 7020:, and 6885:, and 6668:, and 6645:, and 6492:single 6318:) and 6181:2-form 5941:, are 5933:, the 5784:metric 5702:angles 5540:smooth 4703:term, 4701:German 4549: 4520:faces, 4449:Bolyai 4433:Euclid 4411:curves 4041:normal 3908: 3613:Charts 3293:. The 2981:-ball 2863:-ball 2830:square 2685:. For 2463:charts 2403:Charts 2243:since 1961:to an 1827:closed 1752:< 1 1741:< 0 1734:> 0 1720:charts 1688:sphere 1682:Sphere 479:charts 355:Circle 331:model 309:angles 278:graphs 247:sphere 245:, the 198:to an 54:, but 9542:Basis 9227:Scope 9152:Sheaf 8926:Fiber 8702:Rizza 8673:Prime 8504:Local 8494:Curve 8356:Atlas 7862:JSTOR 7844:arXiv 7791:arXiv 7760:S2CID 7725:JSTOR 7544:Notes 6600:, an 6566:genus 6320:genus 6070:. An 5812:group 5706:areas 4978:chart 4795:graph 4785:of a 4583:torus 4565:with 4445:Gauss 4277:group 2531:atlas 2467:atlas 2423:chart 2397:atlas 2361:sheaf 1890:space 1843:locus 1768:Earth 1445:Here 669:atlas 251:torus 243:plane 227:lines 139:is a 121:North 117:globe 9019:Form 8921:Dual 8854:flow 8717:Tame 8693:Sub− 8606:Flat 8486:Maps 8290:The 8246:ISBN 8214:ISBN 8168:ISBN 8150:ISBN 8135:ISBN 8088:ISBN 8068:ISBN 8050:ISBN 8028:ISBN 8011:ISBN 7992:ISBN 7974:ISBN 7956:ISBN 7941:ISBN 7923:ISBN 7817:ISSN 7617:ISBN 7584:ISBN 7565:ISBN 7056:Let 6985:une 6979:(an 6974:une 6917:e.g. 6653:and 6580:and 6576:and 6536:The 6417:for 6357:and 5754:norm 5716:and 5708:(or 5644:and 5542:and 5484:The 5477:and 5465:and 5457:and 5335:and 5275:and 5247:and 5119:and 4837:and 4734:and 4723:and 4717:many 4668:and 4625:and 4459:and 4451:and 4383:and 4360:The 4314:and 4312:tori 4299:(or 4281:acts 4260:and 4246:and 4069:The 3552:and 3174:Let 3061:< 2943:< 2834:ball 2828:. A 2822:disk 2624:, a 2620:, a 2417:, a 2305:The 2287:> 2191:< 2011:(if 1931:are 1927:and 1837:, a 1736:and 1686:The 1663:and 1655:and 1614:and 1473:to ( 1363:and 770:and 709:and 403:and 307:and 257:and 229:and 135:, a 123:and 103:The 8941:Jet 7893:doi 7854:doi 7840:143 7809:doi 7787:198 7752:doi 7717:doi 7684:doi 6990:(a 6937:an 6907:An 6798:or 6755:or 5857:of 5720:of 5712:), 5698:dot 5409:(a 5343:is 5315:. 5295:'s 5291:). 4801:). 4750:or 4664:in 4435:'s 4307:). 4039:or 4005:= ⁄ 3983:= ⁄ 3514:Int 3466:If 3429:of 3370:in 3352:Int 3344:of 3297:of 3226:Int 3198:of 2840:). 2512:top 2339:not 2302:). 1951:). 1772:map 1727:= 0 1694:of 335:in 323:of 284:). 202:of 131:In 9840:: 8932:Co 8276:, 8270:, 8220:. 8198:). 8174:. 8162:, 7889:18 7887:. 7883:. 7860:. 7852:. 7815:. 7807:. 7799:. 7785:. 7781:. 7758:. 7748:35 7746:. 7723:. 7713:37 7680:53 7615:. 7613:12 7574:; 7045:A 7030:A 6994:). 6877:, 6849:. 6676:. 6664:, 6657:. 6641:, 6637:, 6596:A 6584:. 6267:. 6211:A 6194:A 6169:A 6116:A 6023:A 6014:. 5756:, 5748:A 5724:. 5704:, 5559:A 5481:. 5448:A 5347:. 5331:, 5327:, 4984:. 4637:. 4613:. 4546:2. 4497:. 4447:, 4303:\ 4295:/ 4058:. 3946:, 3938:, 3930:, 3699:: 3601:. 2760:A 2707:. 2700:. 2693:. 2632:. 2413:A 2367:. 2004:. 1857:− 1853:= 1798:: 1757:xy 1748:+ 1709:+ 1705:+ 1269:. 372:, 346:. 339:. 261:. 9201:e 9194:t 9187:v 8950:( 8930:( 8706:( 8687:( 8585:( 8575:( 8338:) 8334:( 8324:e 8317:t 8310:v 8254:. 8156:. 8141:. 8123:. 8109:n 8104:R 8056:. 8038:. 7962:. 7929:. 7901:. 7895:: 7868:. 7856:: 7846:: 7823:. 7811:: 7803:: 7793:: 7766:. 7754:: 7731:. 7719:: 7692:. 7686:: 7639:. 7625:. 7593:. 7452:) 7449:C 7446:, 7443:M 7440:( 7420:C 7400:C 7380:M 7360:C 7353:) 7347:n 7343:f 7339:, 7333:, 7328:1 7324:f 7319:( 7312:H 7292:C 7284:n 7280:f 7276:, 7270:, 7265:1 7261:f 7239:N 7232:n 7211:) 7206:n 7201:R 7196:( 7186:C 7179:H 7159:C 7139:M 7117:M 7112:R 7104:C 7084:M 7064:M 7039:. 7024:. 6901:. 6780:, 6776:C 6769:M 6763:f 6742:R 6735:M 6729:f 6706:5 6703:= 6700:n 6544:. 6432:n 6427:R 6399:n 6394:R 6221:. 6208:. 6191:. 6166:. 6151:. 6137:n 6132:C 6101:n 6098:2 6078:n 6044:n 6039:C 5998:1 5992:n 5972:2 5968:/ 5964:) 5961:1 5955:n 5952:( 5949:n 5915:2 5911:n 5886:n 5880:n 5841:) 5838:1 5835:( 5829:U 5731:n 5681:v 5678:, 5675:u 5652:v 5632:u 5606:, 5529:n 5521:n 5490:n 5431:n 5426:R 5387:n 5382:R 5205:0 5202:= 5199:1 5191:2 5187:y 5183:+ 5178:2 5174:x 5149:2 5145:y 5138:1 5130:= 5127:x 5103:2 5099:y 5092:1 5087:= 5084:x 5060:2 5056:x 5049:1 5041:= 5038:y 5014:2 5010:x 5003:1 4998:= 4995:y 4958:y 4937:y 4911:y 4890:y 4865:) 4858:y 4854:( 4825:) 4822:y 4819:( 4746:( 4730:( 4595:F 4591:E 4587:V 4575:F 4571:E 4567:V 4543:= 4540:F 4537:+ 4534:E 4528:V 4518:F 4514:E 4507:V 4385:S 4381:S 4377:S 4305:M 4301:G 4297:G 4293:M 4289:G 4285:M 4232:n 4212:. 4207:2 4199:x 4192:/ 4188:x 4182:x 4179:: 4176:} 4173:0 4170:{ 4162:n 4157:R 4149:} 4146:0 4143:{ 4135:n 4130:R 4102:n 4097:R 4085:S 4081:n 4077:S 4072:n 4063:n 4017:s 4013:t 4008:t 4003:s 3999:t 3995:t 3991:s 3986:s 3981:t 3948:z 3944:y 3940:z 3936:x 3932:y 3928:x 3905:, 3902:) 3899:y 3896:, 3893:x 3890:( 3887:= 3884:) 3881:z 3878:, 3875:y 3872:, 3869:x 3866:( 3853:χ 3849:z 3833:2 3828:R 3804:. 3800:} 3796:1 3793:= 3788:2 3784:z 3780:+ 3775:2 3771:y 3767:+ 3762:2 3758:x 3749:3 3744:R 3736:) 3733:z 3730:, 3727:y 3724:, 3721:x 3718:( 3714:{ 3710:= 3707:S 3685:3 3680:R 3644:2 3639:R 3623:z 3589:1 3583:n 3563:M 3540:n 3520:M 3494:n 3474:M 3449:n 3444:+ 3439:R 3417:) 3414:0 3411:= 3406:n 3402:x 3398:( 3378:M 3358:M 3328:M 3305:M 3279:n 3274:R 3252:M 3232:M 3206:M 3182:M 3154:0 3151:= 3146:1 3142:x 3121:0 3118:= 3113:1 3109:x 3088:} 3085:0 3077:1 3073:x 3064:1 3056:2 3051:i 3047:x 3040:| 3037:) 3032:n 3028:x 3024:, 3018:, 3013:2 3009:x 3005:, 3000:1 2996:x 2992:( 2989:{ 2969:n 2949:} 2946:1 2938:2 2933:i 2929:x 2922:| 2919:) 2914:n 2910:x 2906:, 2900:, 2895:2 2891:x 2887:, 2882:1 2878:x 2874:( 2871:{ 2851:n 2808:) 2805:1 2799:n 2796:( 2776:n 2753:3 2750:φ 2745:1 2740:2 2737:φ 2731:1 2728:φ 2659:n 2654:R 2614:T 2598:n 2593:R 2569:n 2564:R 2508:x 2492:2 2487:R 2440:n 2435:R 2307:n 2290:0 2284:n 2258:n 2253:R 2229:n 2224:R 2202:. 2198:} 2194:1 2186:2 2181:n 2177:x 2173:+ 2167:+ 2162:2 2157:2 2153:x 2149:+ 2144:2 2139:1 2135:x 2131:: 2126:n 2121:R 2113:) 2108:n 2104:x 2100:, 2094:, 2089:2 2085:x 2081:, 2076:1 2072:x 2068:( 2064:{ 2060:= 2055:n 2050:B 2025:0 2022:= 2019:n 2002:n 1988:, 1983:n 1978:R 1859:x 1855:x 1851:y 1812:■ 1808:■ 1804:■ 1800:■ 1750:y 1746:x 1739:z 1732:z 1725:z 1711:z 1707:y 1703:x 1676:t 1672:s 1665:t 1661:s 1657:t 1653:s 1637:s 1634:1 1629:= 1626:t 1616:t 1612:s 1608:y 1604:x 1579:2 1575:s 1571:+ 1568:1 1563:s 1560:2 1554:= 1547:y 1535:2 1531:s 1527:+ 1524:1 1517:2 1513:s 1506:1 1500:= 1493:x 1479:y 1475:x 1471:s 1467:y 1463:x 1459:t 1455:y 1451:x 1447:s 1428:x 1422:1 1418:y 1413:= 1410:t 1407:= 1404:) 1401:y 1398:, 1395:x 1392:( 1386:s 1383:u 1380:l 1377:p 1348:x 1345:+ 1342:1 1338:y 1333:= 1330:s 1327:= 1324:) 1321:y 1318:, 1315:x 1312:( 1306:s 1303:u 1300:n 1297:i 1294:m 1241:2 1237:a 1230:1 1225:= 1214:) 1206:2 1202:a 1195:1 1190:, 1187:a 1183:( 1176:t 1173:h 1170:g 1167:i 1164:r 1155:= 1144:) 1139:] 1136:a 1133:[ 1127:1 1118:p 1115:o 1112:t 1102:( 1095:t 1092:h 1089:g 1086:i 1083:r 1074:= 1067:) 1064:a 1061:( 1058:T 1034:) 1031:1 1028:, 1025:0 1022:( 1012:a 995:t 992:h 989:g 986:i 983:r 954:p 951:o 948:t 920:1 911:p 908:o 905:t 890:t 887:h 884:g 881:i 878:r 869:= 866:) 863:1 860:, 857:0 854:( 848:) 845:1 842:, 839:0 836:( 833:: 830:T 810:) 807:1 804:, 801:0 798:( 778:y 758:x 735:t 732:h 729:g 726:i 723:r 694:p 691:o 688:t 649:. 646:y 643:= 636:) 633:y 630:, 627:x 624:( 618:t 615:h 612:g 609:i 606:r 593:y 590:= 583:) 580:y 577:, 574:x 571:( 565:t 562:f 559:e 556:l 543:x 540:= 533:) 530:y 527:, 524:x 521:( 515:m 512:o 509:t 506:t 503:o 500:b 461:. 458:x 455:= 452:) 449:y 446:, 443:x 440:( 434:p 431:o 428:t 393:x 383:y 378:y 374:x 210:n 176:n 155:n 127:. 90:) 84:( 79:) 75:( 61:. 34:. 20:)

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