# Manifolds

## Manifolds Topics

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### Compact manifold

A compact manifold is a manifold that is compact as a topological space. Examples are the circle (the only one-dimensional compact manifold) and the -dimensional sphere and torus. Compact manifolds in two dimensions are completely classified by their orientation and the number of holes (genus). It should be noted that the term "compact manifold" often implies "manifold without boundary," which is the sense in which it is used here. When there is need for a separate term, a compact boundaryless manifold is called a closed manifold.For many problems in topology and geometry, it is convenient to study compact manifolds because of their "nice" behavior. Among the properties making compact manifolds "nice" are the fact that they can be covered by finitely many coordinate charts, and that any continuous real-valued function is bounded on a compact manifold.For any positive integer , a distinct nonorientable..

### Symplectic manifold

A pair , where is a manifold and is a symplectic form on . The phase space is a symplectic manifold. Near every point on a symplectic manifold, it is possible to find a set of local "Darboux coordinates" in which the symplectic form has the simple form(Sjamaar 1996), where is a wedge product.

### Sutured manifold

A sutured manifold is a tool in geometric topology which was first introduced by David Gabai in order to study taut foliations on 3-manifolds. Roughly, a sutured manifold is a pair with a compact, oriented 3-manifold with boundary and with a set of simple closed curves in which are oriented and which divide into pieces and (Juhász 2010).Defined precisely in a seminal work by Gabai (1983), a sutured manifold is a compact oriented 3-manifold together with a set of pairwise disjoint annuli and tori such that each component of contains a homologically nontrivial oriented simple closed curve (called a suture) and such that is oriented. Using this construction, the collection of a sutured manifold effectively splits into disjoint pieces and with , respectively , defined to be the components of whose normal vectors point into, respectively point out of, . Gabai's definition also requires that orientations on be coherent with respect to the..

### Cohomology

Cohomology is an invariant of a topological space, formally "dual" to homology, and so it detects "holes" in a space. Cohomology has more algebraic structure than homology, making it into a graded ring (with multiplication given by the so-called "cup product"), whereas homology is just a graded Abelian group invariant of a space.A generalized homology or cohomology theory must satisfy all of the Eilenberg-Steenrodaxioms with the exception of the dimension axiom.

### Manifold tangent vector

Roughly speaking, a tangent vector is an infinitesimal displacement at a specific point on a manifold. The set of tangent vectors at a point forms a vector space called the tangent space at , and the collection of tangent spaces on a manifold forms a vector bundle called the tangent bundle.A tangent vector at a point on a manifold is a tangent vector at in a coordinate chart. A change in coordinates near causes an invertible linear map of the tangent vector's representations in the coordinates. This transformation is given by the Jacobian, which must be nonsingular in a change of coordinates. Hence the tangent vectors at are well-defined. A vector field is an assignment of a tangent vector for each point. The collection of tangent vectors forms the tangent bundle, and a vector field is a section of this bundle.Tangent vectors are used to do calculus on manifolds. Since manifolds are locally Euclidean, the usual notions of differentiation and integration..

### Submersion

A submersion is a smooth map whengiven that the differential, or Jacobian, is surjective at every in . The basic example of a submersion is the canonical submersion of onto when ,In fact, if is a submersion, then it is possible to find coordinates around in and coordinates around in such that is the canonical submersion written in these coordinates. For example, consider the submersion of onto the circle , given by .

### Manifold orientation

An orientation on an -dimensional manifold is given by a nowhere vanishing differential n-form. Alternatively, it is an bundle orientation for the tangent bundle. If an orientation exists on , then is called orientable.Not all manifolds are orientable, as exemplified by the Möbius strip and the Klein bottle, illustrated above.However, an -dimensional submanifold of is orientable iff it has a unit normal vector field. The choice of unit determines the orientation of the submanifold. For example, the sphere is orientable.Some types of manifolds are always orientable. For instance, complex manifolds, including varieties, and also symplectic manifolds are orientable. Also, any unoriented manifold has a double cover which is oriented.A map between oriented manifolds of the same dimension is called orientation preserving if the volume form on pulls back to a positive volume form on . Equivalently, the differential maps an oriented..

### Closing lemma

Let be a non-wandering point of a diffeomorphism of a compact manifold. The closing lemma concerns if can be arbitrarily well approximated with derivatives of order for each by so that is a periodic point of .The closing lemma is the 10th of Smale's problemsand remains unsettled.

### Submanifold tangent space

The tangent plane to a surface at a point is the tangent space at (after translating to the origin). The elements of the tangent space are called tangent vectors, and they are closed under addition and scalar multiplication. In particular, the tangent space is a vector space.Any submanifold of Euclidean space, and more generally any submanifold of an abstract manifold, has a tangent space at each point. The collection of tangent spaces to forms the tangent bundle . A vector field assigns to every point a tangent vector in the tangent space at .There are two ways of defining a submanifold, and each way gives rise to a different way of defining the tangent space. The first way uses a parameterization, and the second way uses a system of equations.Suppose that is a local parameterization of a submanifold in Euclidean space . Say,(1)where is the open unit ball in , and . At the point , the tangent space is the image of the Jacobian of , as a linear transformation..

### Manifold

A manifold is a topological space that is locally Euclidean (i.e., around every point, there is a neighborhood that is topologically the same as the open unit ball in ). To illustrate this idea, consider the ancient belief that the Earth was flat as contrasted with the modern evidence that it is round. The discrepancy arises essentially from the fact that on the small scales that we see, the Earth does indeed look flat. In general, any object that is nearly "flat" on small scales is a manifold, and so manifolds constitute a generalization of objects we could live on in which we would encounter the round/flat Earth problem, as first codified by Poincaré.More concisely, any object that can be "charted" is a manifold.One of the goals of topology is to find ways of distinguishing manifolds. For instance, a circle is topologically the same as any closed loop, no matter how different these two manifolds may appear. Similarly,..

### Submanifold

A (infinitely differentiable) manifold is said to be a submanifold of a manifold if is a subset of and the identity map of into is an embedding.

### Closed graph theorem

The closed graph theorem states that a linear operator between two Banach spaces and is continuous iff it has a closed graph, where the "graph" is considered closed if it is a closed subset of equipped with the product topology.The closed graph theorem also holds for Fréchetspaces.

### Cheeger's finiteness theorem

Consider the set of compact -Riemannian manifolds with diameter, Volume, and where is the sectional curvature. Then there is a bound on the number of diffeomorphisms classes of this set in terms of the constants , , , and .

### Chart tangent space

From the point of view of coordinate charts, the notion of tangent space is quite simple. The tangent space consists of all directions, or velocities, a particle can take. In an open set in there are no constraints, so the tangent space at a point is another copy of . The set could be a coordinate chart for an -dimensional manifold.The tangent space at , denoted , is the set of possible velocity vectors of paths through . Hence there is a canonical vector basis: if are the coordinates, then are a basis for the tangent space, where is the velocity vector of a particle with unit speed moving inward along the coordinate . The collection of all tangent vectors to every point on the manifold, called the tangent bundle, is the phase space of a single particle moving in the manifold .It seems as if the tangent space at is the same as the tangent space at all other points in the chart . However, while they do share the same dimension and are isomorphic, in a change of coordinates,..

### Stiefel manifold

The Stiefel manifold of orthonormal -frames in is the collection of vectors (, ..., ) where is in for all , and the -tuple (, ..., ) is orthonormal. This is a submanifold of , having dimension .Sometimes the "orthonormal" condition is dropped in favor of the mildly weaker condition that the -tuple (, ..., ) is linearly independent. Usually, this does not affect the applications since Stiefel manifolds are usually considered only during homotopy theoretic considerations. With respect to homotopy theory, the two definitions are more or less equivalent since Gram-Schmidt orthonormalization gives rise to a smooth deformation retraction of the second type of Stiefel manifold onto the first.

### Campbell's theorem

Any -dimensional Riemannian manifold can be locally embedded into an -dimensional manifold with Ricci curvature Tensor . A similar version of the theorem for a pseudo-Riemannian manifold states that any -dimensional pseudo-Riemannian manifold can be locally and isometrically embedded in an -dimensional pseudo-Euclidean space.

### Smooth structure

A smooth structure on a topological manifold (also called a differentiable structure) is given by a smooth atlas of coordinate charts, i.e., the transition functions between the coordinate charts are smooth. A manifold with a smooth structure is called a smooth manifold (or differentiable manifold).A smooth structure is used to define differentiability for real-valued functions on a manifold. This extends to a notion of when a map between two differentiable manifolds is smooth, and naturally to the definition of a diffeomorphism. In addition, the smooth structure is used to define manifold tangent vectors, the collection of which is the tangent bundle.Two smooth structures are considered equivalent if there is a homeomorphism of the manifold which pulls back one atlas to an atlas compatible to the other one, i.e., a diffeomorphism. For instance, any two smooth structures on the circle are equivalent, as can be seen by integration.It is..

### Smooth manifold

Another word for a (infinitely differentiable) manifold, also called a differentiable manifold. A smooth manifold is a topological manifold together with its "functional structure" (Bredon 1995) and so differs from a topological manifold because the notion of differentiability exists on it. Every smooth manifold is a topological manifold, but not necessarily vice versa. (The first nonsmooth topological manifold occurs in four dimensions.) Milnor (1956) showed that a seven-dimensional hypersphere can be made into a smooth manifold in 28 ways.

### Smale theorem

If is a differentiable homotopy sphere of dimension , then is homeomorphic to . In fact, is diffeomorphic to a manifold obtained by gluing together the boundaries of two closed -balls under a suitable diffeomorphism (Milnor).

### Brouwer degree

Let be a map between two compact, connected, oriented -dimensional manifolds without boundary. Then induces a homomorphism from the homology groups to , both canonically isomorphic to the integers, and so can be thought of as a homomorphism of the integers. The integer to which the number 1 gets sent is called the degree of the map .There is an easy way to compute if the manifolds involved are smooth. Let , and approximate by a smooth map homotopic to such that is a "regular value" of (which exist and are everywhere dense by Sard's theorem). By the implicit function theorem, each point in has a neighborhood such that restricted to it is a diffeomorphism. If the diffeomorphism is orientation preserving, assign it the number , and if it is orientation reversing, assign it the number . Add up all the numbers for all the points in , and that is the , the Brouwer degree of . One reason why the degree of a map is important is because it is a homotopy invariant...

### Bordism

A relation between compact boundaryless manifolds (also called closed manifolds). Two closed manifolds are bordant iff their disjoint union is the boundary of a compact -manifold. Roughly, two manifolds are bordant if together they form the boundary of a manifold. The word bordism is now used in place of the original term cobordism.

### Intrinsic tangent space

The tangent space at a point in an abstract manifold can be described without the use of embeddings or coordinate charts. The elements of the tangent space are called tangent vectors, and the collection of tangent spaces forms the tangent bundle.One description is to put an equivalence relation on smooth paths through the point . More precisely, consider all smooth maps where and . We say that two maps and are equivalent if they agree to first order. That is, in any coordinate chart around , . If they are similar in one chart then they are similar in any other chart, by the chain rule. The notion of agreeing to first order depends on coordinate charts, but this cannot be completely eliminated since that is how manifolds are defined.Another way is to first define a vector field as a derivation of the ring of smooth functions . Then a tangent vector at a point is an equivalence class of vector fields which agree at . That is, if for every smooth function . Of course,..

### Blaschke conjecture

The only Wiedersehen surfaces are the standard round spheres. The conjecture was proven by combining the Berger-Kazdan comparison theorem with A. Weinstein's results for even and C. T. Yang's for odd. Green (1963) obtained the first proof of the Blaschke's conjecture in the two-dimensional case.

### Bing's theorem

If is a closed oriented connected 3-manifold such that every simple closed curve in lies interior to a ball in , then is homeomorphic with the hypersphere, .

### Riemannian geometry

The study of manifolds having a complete Riemannian metric. Riemannian geometry is a general space based on the line elementwith for a function on the tangent bundle . In addition, is homogeneous of degree 1 in and of the form(Chern 1996). If this restriction is dropped, the resulting geometry is called Finsler geometry.

### Besov space

A type of abstract space which occurs in spline and rational function approximations. The Besov space is a complete quasinormed space which is a Banach space when , (Petrushev and Popov 1987).

### Riemann sphere

The Riemann sphere, also called the extended complex plane, is a one-dimensional complex manifold (C-star) which is the one-point compactification of the complex numbers , together with two charts. (Here denotes complex infinity.) The notation is also used (Krantz 1999, p. 82; Lorentzen, and Waadeland 2008, p. 3).For all points in the complex plane, the chart is the identity map from the sphere (with infinity removed) to the complex plane. For the point at infinity, the chart neighborhood is the sphere (with the origin removed), and the chart is given by sending infinity to 0 and all other points to .

### Bergman space

Let be an open subset of the complex plane , and let denote the collection of all analytic functions whose complex modulus is square integrable with respect to area measure. Then , sometimes also denoted , is called the Bergman space for . Thus, the Bergman space consists of all the analytic functions in . The Bergman space can also be generalized to , where .

### Harmonic map

A map , between two compact Riemannian manifolds, is a harmonic map if it is a critical point for the energy functionalThe norm of the differential is given by the metric on and and is the measure on . Typically, the class of allowable maps lie in a fixed homotopy class of maps.The Euler-Lagrange differential equation for the energy functional is a non-linear elliptic partial differential equation. For example, when is the circle, then the Euler-Lagrange equation is the same as the geodesic equation. Hence, is a closed geodesic iff is harmonic. The map from the circle to the equator of the standard 2-sphere is a harmonic map, and so are the maps that take the circle and map it around the equator times, for any integer . Note that these all lie in the same homotopy class. A higher-dimensional example is a meromorphic function on a compact Riemann surface, which is a harmonic map to the Riemann sphere.A harmonic map may not always exist in a homotopy class,..

### Beltrami field

A vector field satisfying the vector identitywhere is the cross product and is the curl is said to be a Beltrami field.

### Baire category theorem

Baire's category theorem, also known as Baire's theorem and the category theorem, is a result in analysis and set theory which roughly states that in certain spaces, the intersection of any countable collection of "large" sets remains "large." The appearance of "category" in the name refers to the interplay of the theorem with the notions of sets of first and second category.Precisely stated, the theorem says that if a space is either a complete metric space or a locally compact T2-space, then the intersection of every countable collection of dense open subsets of is necessarily dense in .The above-mentioned interplay with first and second category sets can be summarized by a single corollary, namely that spaces that are either complete metric spaces or locally compact Hausdorff spaces are of second category in themselves. To see that this follows from the above-stated theorem, let be either a complete metric..

### Axiom a diffeomorphism

Let be a diffeomorphism on a compact Riemannian manifold . Then satisfies Axiom A if the nonwandering set of is hyperbolic and the periodic points of are dense in . Although it was conjectured that the first of these conditions implies the second, they were shown to be independent in or around 1977. Examples include the Anosov diffeomorphisms and Smale horseshoe map.In some cases, Axiom A can be replaced by the condition that the diffeomorphism is a hyperbolic diffeomorphism on a hyperbolic set (Bowen 1975, Parry and Pollicott 1990).

### Grassmannian

The Grassmannian is the set of -dimensional subspaces in an -dimensional vector space. For example, the set of lines is projective space. The real Grassmannian (as well as the complex Grassmannian) are examples of manifolds. For example, the subspace has a neighborhood . A subspace is in if and and . Then for any , the vectors and are uniquely determined by requiring and . The other six entries provide coordinates for .In general, the Grassmannian can be given coordinates in a similar way at a point . Let be the open set of -dimensional subspaces which project onto . First one picks an orthonormal basis for such that span . Using this basis, it is possible to take any vectors and make a matrix. Doing this for the basis of , another -dimensional subspace in , gives a -matrix, which is well-defined up to linear combinations of the rows. The final step is to row-reduce so that the first block is the identity matrix. Then the last block is uniquely determined by ...

### Atlas

An atlas is a collection of consistent coordinate charts on a manifold, where "consistent" most commonly means that the transition functions of the charts are smooth. As the name suggests, an atlas corresponds to a collection of maps, each of which shows a piece of a manifold and looks like flat Euclidean space. To use an atlas, one needs to know how the maps overlap. To be useful, the maps must not be too different on these overlapping areas.The overlapping maps from one chart to another are called transition functions. They represent the transition from one chart's point of view to that of another. Let the open unit ball in be denoted . Then if and are two coordinate charts, the composition is a function defined on . That is, it is a function from an open subset of to , and given such a function from to , there are conditions for it to be smooth or have smooth derivatives (i.e., it is a C-k function). Furthermore, when is isomorphic to (in the even dimensional..

### Grassmann manifold

A special case of a flag manifold. A Grassmann manifold is a certain collection of vector subspaces of a vector space. In particular, is the Grassmann manifold of -dimensional subspaces of the vector space . It has a natural manifold structure as an orbit-space of the Stiefel manifold of orthonormal -frames in . One of the main things about Grassmann manifolds is that they are classifying spaces for vector bundles.

### Generalized reeb component

Given a compact manifold and a transversely orientable codimension-one foliation on which is tangent to , the pair is called a generalized Reeb component if the holonomy groups of all leaves in the interior are trivial and if all leaves of are proper. Generalized Reeb components are obvious generalizations of Reeb components.The introduction of the generalized version of the Reeb component facilitates the proof of many significant results in the theory of 3-manifolds and of foliations. It is well-known that generalized Reeb components are transversely orientable and that a manifold admitting a generalized Reeb component also admits a nice vector field (Imanishi and Yagi 1976). Moreover, given a generalized Reeb component , is a fibration over .Like many notions in geometric topology, the generalized Reeb component can be presented in various contexts. One source describes a generalized Reeb component on a closed 3-manifold with foliation..

### Freedman theorem

Two closed simply connected 4-manifolds are homeomorphic iff they have the same bilinear form and the same Kirby-Siebenmann invariant . Any can be realized by such a manifold. If is odd for some , then either value of can be realized also. However, if is always even, then is determined by , being congruent to 1/8 of the signature of . Here, is a symmetric bilinear form with determinant (Milnor).In particular, if is a homotopy sphere, then and , so is homeomorphic to .

### Flat manifold

A manifold with a Riemannian metric that has zero curvature is a flat manifold. The basic example is Euclidean space with the usual metric . In fact, any point on a flat manifold has a neighborhood isometric to a neighborhood in Euclidean space. A flat manifold is locally Euclidean in terms of distances and angles, as well as merely topologically locally Euclidean, as all manifolds are.The simplest nontrivial examples occur as surfaces in four dimensional space. For instance, the flat torus is a flat manifold. It is the image of . A theorem due to Bieberbach says that all compact flat manifolds are tori. More generally, the universal cover of a complete flat manifold is Euclidean space.

### Ambient isotopy

An ambient isotopy from an embedding of a manifold in to another is a homotopy of self diffeomorphisms (or isomorphisms, or piecewise-linear transformations, etc.) of , starting at the identity map, such that the "last" diffeomorphism compounded with the first embedding of is the second embedding of . In other words, an ambient isotopy is like an isotopy except that instead of distorting the embedding, the whole ambient space is being stretched and distorted and the embedding is just "coming along for the ride." For smooth manifolds, a map is isotopic iff it is ambiently isotopic.For knots, the equivalence of manifolds under continuous deformation is independent of the embedding space. Knots of opposite chirality have ambient isotopy, but not regular isotopy.

### Flag manifold

For any sequence of integers , there is a flag manifold of type (, ..., ) which is the collection of ordered sets of vector subspaces of (, ..., ) with and a subspace of . There are also complex flag manifolds with complex subspaces of instead of real subspaces of a real -space.These flag manifolds admit the structure of manifoldsin a natural way and are used in the theory of Lie groups.

### Algebraic manifold

An algebraic manifold is another name for a smooth algebraic variety. It can be covered by coordinate charts so that the transition functions are given by rational functions. Technically speaking, the coordinate charts should be to all of affine space .For example, the sphere is an algebraic manifold, with a chart given by stereographic projection to , and another chart at , with the transition function given by . In this setting, it is called the Riemann sphere. The torus is also an algebraic manifold, in this setting called an elliptic curve, with charts given by elliptic functions such as the Weierstrass elliptic function.

### Abstract manifold

An abstract manifold is a manifold in the context of an abstract space with no particular embedding, or representation in mind. It is a topological space with an atlas of coordinate charts.For example, the sphere can be considered a submanifold of or a quotient space . But as an abstract manifold, it is just a manifold, which can be covered by two coordinate charts and , with the single transition function,defined bywhere . It can also be thought of as two disks glued together at their boundary.

Let be a pair consisting of finite, connected CW-complexes where is a subcomplex of . Define the associated chain complex group-wise for each by setting(1)where denotes singular homology with integer coefficients and where denotes the union of all cells of of dimension less than or equal to . Note that is free Abelian with one generator for each -cell of .Next, consider the universal covering complexes of and , respectively. The fundamental group of can be identified with the group of deck transformations of so that each determines a map(2)which then induces a chain map(3)The chain map turns each chain group into a module over the group ring which is -free with one generator for each -cell of and which is finitely generated over due to the finiteness of .Hence, there is a free chain complex(4)over , the homology groups of which are zero due to the fact that deformation retracts onto . A simple argument shows the existence of a so-called preferred basis..

### Reidemeister torsion

In algebraic topology, the Reidemeister torsion is a notion originally introduced as a topological invariant of 3-manifolds which has now been widely adapted to a variety of contexts. At the time of its discovery, the Reidemeister torsion was the first 3-manifold invariant able to distinguish between manifolds which are homotopy equivalent but not homeomorphic. Since then, the notion has been adapted to higher-dimensional manifolds, knots and links, dynamical systems, Witten's equations, and so on. In particular, it has a number of different definitions for various contexts.For a commutative ring , let be a finite acyclic chain complex of based finitely generated free R-modules of the form(1)The Reidemeister torsion of is the value defined by(2)where is the set of units of , is a chain contraction, is the boundary map, and(3)is a map from to . In this context, Reidemeister torsion is sometimes referred to as the torsion of the complex (Nicolaescu..

### Analytic torsion

Let be a compact -dimensional oriented Riemannian manifold without boundary, let be a group representation of by orthogonal matrices, and let be the associated vector bundle. Suppose further that the Laplacian is strictly negative on where is the linear space of differential k-forms on with values in . In this context, the analytic torsion is defined as the positive real root ofwhere the -function is defined byfor the collection of eigenvalues of , the restriction of to the collection of bundle sections of the sheaf .Intrinsic to the above computation is that is a real manifold. However, there is a collection of literature on analytic torsion for complex manifolds, the construction of which is nearly identical to the construction given above. Analytic torsion on complex manifolds is sometimes called del bar torsion...