# Manifold

The sphere can be "flattened" with several images. Accordingly, the earth's surface can be represented in an atlas.

In mathematics, a manifold is understood to be a topological space that locally resembles Euclidean space . Globally, however, the manifold does not have to resemble a Euclidean space (not be homeomorphic to it). ${\ displaystyle \ mathbb {R} ^ {n}}$

Manifolds are the central subject of differential geometry ; they have important applications in theoretical physics .

## Introductory example

A popular example of a manifold is a sphere (= spherical surface), illustratively the earth's surface:

Each region of the world can use a map to a level ( to be shown). If you approach the edge of the map, you should switch to another map that shows the adjacent area. Thus a manifold can be completely described by a complete set of cards; you need rules as to how the cards overlap when you change cards. On the other hand, there is no single map on which the entire surface of the sphere can be completely represented without "tearing up" the latter; World maps also always have “edges”, or they depict parts of the earth twice. The dimension of a manifold corresponds to the dimension of a local map; all cards have the same dimension. ${\ displaystyle \ mathbb {R} ^ {2}}$

Another example is the torus (“ lifebuoy ”, “ donut ”).

## historical overview

The concept of manifolds arose in the 19th century, in particular through research in geometry and function theory . While differential geometricians examined local concepts such as the curvature of curves and surfaces , function theorists looked at global problems. They found that properties of functions are related to topological invariants of the set for certain . These sets are manifolds (see theorem of regular value ). ${\ displaystyle F}$${\ displaystyle F ^ {- 1} (c)}$${\ displaystyle c}$${\ displaystyle F ^ {- 1} (c)}$

The concept of manifold goes back to Bernhard Riemann . In his habilitation lecture on the hypotheses on which geometry is based , which he gave to Carl Friedrich Gauss in 1854 , he introduced the concept of manifolds . He speaks of discrete and continuous manifolds that are n-fold expanded , so at this time it is limited to structures that are embedded in the . One can measure angles and distances on these manifolds. In later works he developed the Riemann surfaces , which were probably the first abstract manifolds. Manifolds are sometimes called abstract for delimitation, to express that they are not subsets of Euclidean space. ${\ displaystyle \ mathbb {R} ^ {n}}$

Henri Poincaré began his work with the investigation of three-dimensional manifolds , while up to then mainly two-dimensional manifolds (surfaces) had been dealt with. In 1904 he created the Poincaré conjecture named after him . It says that every simply-connected , compact three-dimensional manifold is homeomorphic to the 3-sphere . For this assumption, Grigori Jakowlewitsch Perelman published a proof in 2002, which was not published in a refereed journal, but only on the Internet, but which is considered correct by the specialist public.

The definition commonly used today first appeared in 1913 by Hermann Weyl in Riemannsche surfaces . However, it was not until the 1936 publications of Hassler Whitney that manifolds became an established mathematical object. His best-known result is Whitney's embedding theorem .

## Types of manifolds

### Topological manifolds

Be a topological space . One calls a (topological) manifold of the dimension or briefly a manifold if the following properties are fulfilled: ${\ displaystyle M}$${\ displaystyle M}$${\ displaystyle n}$${\ displaystyle n}$

1. ${\ displaystyle M}$is a Hausdorff room .
2. ${\ displaystyle M}$satisfies the second axiom of countability .
3. ${\ displaystyle M}$is locally Euclidean, that is, every point has a neighborhood which is homeomorphic to an open subset of .${\ displaystyle \ mathbb {R} ^ {n}}$

Manifolds inherit many local properties from Euclidean space: they are locally path-connected, locally compact and locally metrizable . Manifolds that are homeomorphic to one another are considered equal (or equivalent). This gave rise to the question of the classification , i.e. the question of how many non-equivalent manifolds there are.

### Differentiable manifolds

#### Homeomorphism, Map and Atlas

In order to consider differentiable functions, the structure of a topological manifold is not sufficient. Let it be such a topological manifold without a boundary. If an open subset of is given, on which a homeomorphism is defined for an open set of , then this homeomorphism is called a map . A set of maps whose archetypes overlap is called atlas of . Different maps induce a homeomorphism (a so-called map change or coordinate change) between open subsets of . If all such map changes can be differentiated-times for an atlas , then one calls an -atlas. Two atlases (of the same manifold) are called compatible with each other if and only if their union forms an atlas again . This compatibility is an equivalence relation . A -manifold is a topological manifold together with an -atlas (actually with an equivalence class of -atlases). Smooth manifolds are manifolds of type . If all card changes are analytical , then the manifold is also called analytical or also -manifold. ${\ displaystyle M}$${\ displaystyle n}$${\ displaystyle M}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle M}$${\ displaystyle M}$${\ displaystyle \ theta, \ eta}$${\ displaystyle \ theta \ circ \ eta ^ {- 1}}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle k}$${\ displaystyle {\ mathcal {A}}}$${\ displaystyle C ^ {k}}$${\ displaystyle C ^ {k}}$${\ displaystyle C ^ {k}}$${\ displaystyle C ^ {k}}$${\ displaystyle C ^ {k}}$${\ displaystyle C ^ {k}}$${\ displaystyle C ^ {\ infty}}$${\ displaystyle C ^ {\ omega}}$

On a -manifold , a function is called- times differentiable ( ) if and only if it is- times differentiable on every map . ${\ displaystyle C ^ {k}}$${\ displaystyle M}$${\ displaystyle f \ colon M \ to \ mathbb {R}}$${\ displaystyle s}$${\ displaystyle s \ leq k}$${\ displaystyle s}$

For every (paracompact) -manifold ( ) there is an atlas that can be differentiated or even analytical as often as required. In fact, this structure is even unambiguous, that is, it is not a restriction of generality to assume that every manifold is analytic (if one speaks of differentiable manifolds). ${\ displaystyle C ^ {r}}$${\ displaystyle r> 1}$

However, this statement is no longer necessarily correct for topological manifolds of dimension or higher: There are both -manifolds that have no differentiable structure, and -manifolds (or also -manifolds, see above) that are different as differentiable manifolds, but as topological manifolds are equal. The best-known example of the second case are the so-called exotic spheres, which are all homeomorphic to (but not diffeomorphic to one another). Since the topological and the differentiable categories coincide in the lower dimension, such results are difficult to illustrate. ${\ displaystyle 4}$${\ displaystyle C ^ {0}}$${\ displaystyle C ^ {1}}$${\ displaystyle C ^ {\ omega}}$${\ displaystyle 7}$${\ displaystyle \ mathbb {S} ^ {7}}$

#### Tangential bundle

At every point of a -dimensional, differentiable (but not a topological) manifold one finds a tangent space . In a map you simply attach one to this point and then consider that the differential of a change in coordinates defines a linear isomorphism at each point, which transforms the tangent space into the other map. In abstract terms, the tangent space an is defined either as the space of the derivations at this point or the space of equivalence classes of differentiable curves (where the equivalence relation indicates when the velocity vectors of two curves an should be equal). ${\ displaystyle p}$${\ displaystyle n}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle p}$${\ displaystyle p}$

The union of all tangent spaces of a manifold forms a vector bundle , which is called a tangent bundle. The tangent space of a manifold at the point is usually denoted by, the tangential bundle by . ${\ displaystyle M}$${\ displaystyle p}$${\ displaystyle T_ {p} M}$${\ displaystyle TM}$

### Complex manifolds

A topological manifold is called a complex manifold of the (complex) dimension if every point has an open neighborhood that is homeomorphic to an open set . Furthermore, one demands that the card change for every two cards${\ displaystyle X}$${\ displaystyle n}$${\ displaystyle x \ in X}$${\ displaystyle U \ subset X}$${\ displaystyle V \ subset \ mathbb {C} ^ {n}}$${\ displaystyle \ theta _ {i} \ colon U_ {i} \ rightarrow V_ {i}, x \ in U_ {i}}$

${\ displaystyle \ theta _ {j} \ circ \ theta _ {i} ^ {- 1} \ colon V_ {ij} \ rightarrow V_ {ji}}$

is holomorphic . Here denote the amount . ${\ displaystyle V_ {ij} \ subset \ mathbb {C} ^ {n}}$${\ displaystyle \ theta _ {i} (U_ {i} \ cap U_ {j})}$

The essential difference to ordinary differentiable manifolds lies less in the difference between and , but in the much stronger requirement of the complex differentiability of map change maps. ${\ displaystyle \ mathbb {C} ^ {n}}$${\ displaystyle \ mathbb {R} ^ {2n}}$

(Connected) complex manifolds of dimension 1 are called Riemann surfaces . Other special complex manifolds are the Stein manifolds and the Kahler manifolds , which are complex Riemannian manifolds.

### Riemannian manifolds

In order to speak of a differentiable manifold of lengths, distances, angles and volumes, one needs an additional structure. A Riemannian metric (also called metric tensor ) defines a scalar product in the tangent space of every point of the manifold . A differentiable manifold with a Riemannian metric is called a Riemannian manifold . The scalar products initially define the lengths of vectors and angles between vectors, and then, based on this, lengths of curves and distances between points on the manifold.

If, instead of a scalar product, only one (not necessarily symmetrical) norm is defined in each tangent space , one speaks of a Finsler metric and a Finsler manifold . Lengths and distances are defined on Finsler manifolds, but angles are not.

### Semi-Riemannian manifolds

Other generalizations of Riemannian manifolds are semi-Riemannian manifolds (also called pseudo-Riemannian manifolds ), which appear, for example, in general relativity .

Here the symmetrical bilinear form defined by the metric in each tangential space does not need to be positive definite, but only not degenerate . According to Sylvester's law of inertia, such a bilinear form can be represented as a diagonal matrix with entries of . If entries are +1 and entries -1, one speaks of a metric with a signature . If the signature of the metric (or according to another convention ), where the dimension is the manifold, one speaks of a Lorentz manifold . In the general theory of relativity, spacetime is described by a four-dimensional Lorentz manifold, i.e. with the signature (3,1) (or (1,3)). ${\ displaystyle \ pm 1}$${\ displaystyle r}$${\ displaystyle s}$ ${\ displaystyle (r, s)}$${\ displaystyle (m-1.1)}$${\ displaystyle (1, m-1)}$${\ displaystyle m}$

### Banach manifold

A Banach manifold is a Hausdorff space which satisfies the second countability axiom and which is locally homeomorphic to any Banach space and which satisfies the usual map change condition of a differentiable manifold. The card changes can be cheekily differentiable and the dimensions of these manifolds can be infinitely dimensional. Thus this type of manifold can be understood as a generalization of a differentiable manifold.

### Lie groups

A Lie group is both a differentiable manifold and a group , whereby the group multiplication (or addition) and the inversion of a group element must be differentiable mappings. The tangent space of a Lie group at the neutral element with respect to the commutator completed , and forms an associated to the Lie group Lie algebra . ${\ displaystyle [.,.]}$

A simple example of a non- compact Lie group is Euclidean vector space together with normal vector space addition. The unitary group is an example of a compact Lie group. (This manifold can be imagined as a circle and the group operation is a rotation of this circle.) In physics (see quantum chromodynamics ), the groups , the “special unitary groups of order ” (e.g. ) occur above all . ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle U (1)}$${\ displaystyle SU (n)}$${\ displaystyle n}$${\ displaystyle n = 3}$

## Manifold with margin

On the left side there are manifolds without a border and on the right side manifolds with a border are shown.

Manifolds, which have been discussed so far in this article, are not bounded. Bounded manifolds are also not manifolds in the above sense, but their definition is very similar. For this, let us again be a topological Hausdorff space that satisfies the second axiom of countability . The space is called a manifold with a boundary, if every point has a neighborhood which is homeomorphic to a subset of the "nonnegative -dimensional half-space" : ${\ displaystyle M}$ ${\ displaystyle M}$${\ displaystyle n}$${\ displaystyle \ mathbb {R} _ {+} ^ {n}}$

${\ displaystyle \ mathbb {R} _ {+} ^ {n} = \ {x \ in \ mathbb {R} ^ {n} \ mid x_ {n} \ geq 0 \}}$.

This (non-compact) manifold is bounded by the -dimensional plane . ${\ displaystyle (n-1)}$${\ displaystyle \ {x \ in \ mathbb {R} ^ {n} \ mid x_ {n} = 0 \}}$

An example of a compact, bounded manifold is the closed full sphere, which has the sphere as a boundary. This is itself an unrestricted manifold. Additional structures can be defined on bounded manifolds in a manner similar to that on non-bounded manifolds. For example, it is possible to define a differentiable structure on certain manifolds with a boundary or to speak of orientability.

## Manifolds with Orientation

Another essential property of bounded or unbounded manifolds concerns the orientability or non-orientability of the manifold. It can also be defined “by card” (whereby the compatibility is met by itself).

As the following examples show, all four combinations occur with or without a border and with or without orientation.

## Examples

### Discreet space

Every countable discrete topological space is a zero-dimensional topological manifold. The cards of these manifolds are the pairs with and . ${\ displaystyle S}$${\ displaystyle ({s}, \ phi _ {s})}$${\ displaystyle \ phi _ {s} \ colon s \ to 0}$${\ displaystyle s \ in S}$

### sphere

The sphere is an unrestricted oriented manifold of dimension . An atlas of this diversity is given by the two stereographic projections${\ displaystyle \ mathbb {S} ^ {n}}$${\ displaystyle n}$

{\ displaystyle {\ begin {aligned} \ phi _ {1} \ colon \ mathbb {S} ^ {n} \ setminus {N} \ to \ mathbb {R} ^ {n}, & \ quad \ phi _ { 1} (x_ {1}, \ dotsc, x_ {n + 1}) = \ left ({\ frac {x_ {1}} {(1-x_ {n + 1})}}, \ dotsc, {\ frac {x_ {n}} {(1-x_ {n + 1})}} \ right) \\\ phi _ {2} \ colon \ mathbb {S} ^ {n} \ setminus {S} \ to \ mathbb {R} ^ {n}, & \ quad \ phi _ {2} (x_ {1}, \ dotsc, x_ {n + 1}) = \ left ({\ frac {x_ {1}} {(1 + x_ {n + 1})}}, \ dotsc, {\ frac {x_ {n}} {(1 + x_ {n + 1})}} \ right), \ end {aligned}}}

where denote the north pole and the south pole of the sphere. The resulting initial topology is the same that would be induced by on as a subspace topology . In addition to mathematics, the sphere is also examined in other sciences, for example in cartography or in theoretical physics with the so-called Bloch sphere . ${\ displaystyle N}$${\ displaystyle x_ {n + 1} = + 1}$${\ displaystyle S}$${\ displaystyle x_ {n + 1} = - 1}$${\ displaystyle \ mathbb {R} ^ {n + 1}}$${\ displaystyle \ mathbb {S} ^ {n}}$

### rectangle

An orientable manifold with a border: a rectangle with length and width (as well as the diagonal )${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle d}$

A simple example of a bounded and orientable manifold concerns a (closed) rectangle like in the adjacent sketch. The edge consists of the sides of the rectangle; the two orientations are " counter -clockwise" (+) or " in the clockwise direction" (-). In the first case the following circuit is considered: From A to B and further to C and D, from there back to A; everything counterclockwise.

### Möbius band

A non-orientable two-dimensional manifold: the Möbius band

If you glue the sides and the rectangle discussed above together in such a way that A rest on B and C on D, then you get an orientable, bounded manifold that is homeomorphic to , that is, to the Cartesian product of the closed unit interval and the edge of the circle . This can be embedded in the three-dimensional Euclidean space as the lateral surface of a cylinder . ${\ displaystyle {\ overline {AD}}}$${\ displaystyle {\ overline {BC}}}$${\ displaystyle [0,1] \ times S ^ {1}}$

If, on the other hand, points A and C as well as D and B are glued together, which is possible after “twisting” the narrow sides, and if the “gluing” is seamless , a non-orientable two-dimensional manifold with a border is created. This is called the Möbius strip .

The edge of this manifold corresponds to an "8", that is, with the characteristic crossover in the middle. First, z. B. go counterclockwise through the lower semicircle of figure 8 (= from A to B), then the crossing follows (this corresponds to pasting over with twisting); after crossing over, the upper circle follows the 8, running through in the other direction of rotation, i.e. not from C to D, but from D to C.

### Klein bottle

A non-orientable two-dimensional manifold without a rim: the Klein bottle

In an analogous way, by suitably gluing two strips together in spaces with at least three dimensions, a non-orientable two-dimensional manifold without any border is obtained, analogous to the surface of a "sphere with a handle", i.e. a structure that resembles a torus , but which could of course be orientated :

This non-orientable manifold without a rim is called the Klein bottle.

## Classification and invariants of manifolds

At the beginning of the article it was shown that manifolds can carry different structures of a general nature. When classifying manifolds, these structures must of course be taken into account. So two manifolds can be equivalent from a topological point of view, which means that there is a homeomorphism that converts one manifold into the other, but these two manifolds can have different, incompatible differentiable structures , then they are not equivalent from the perspective of differential geometry ; from the topology point of view, on the other hand, they can be equivalent. If two manifolds are equivalent from a given point of view, they also have the same invariants that fit this point of view , for example the same dimension or the same fundamental group .

The connected one-dimensional manifolds are either diffeomorphic and thus also homeomorphic to the real number line or to the circle . ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {S} ^ {1}}$

The classification of closed manifolds is also known in dimensions two and three. Manifolds of this dimension, just like the one-dimensional manifolds, have the special property that every topological manifold allows exactly one differentiable structure. The consequence of this is that topological and differential geometric methods can be combined when investigating such manifolds. In the theory of two-dimensional, closed manifolds there is the classification theorem for 2-manifolds . Two closed surfaces of the same gender are diffeomorphic to one another if they are both orientable or both non-orientable. Closed surfaces are completely determined by the invariants orientability and gender. For three-dimensional, closed manifolds, the important “conjecture about the geometry of 3-manifolds ” by Grigori Perelman has been proven. This theory contains the Poincaré conjecture as a special case .

In the case of four-dimensional manifolds, the classification is very complicated and generally impossible, even in the case of simply connected manifolds, because every finitely presented group occurs as a fundamental group of a four-manifold and the classification of finitely presented groups is algorithmically impossible. Is called the Euclidean space , the sphere and the hyperbolic space model spaces (in English: model spaces ) because their geometry is relatively easy to describe. In dimension four, these spaces are also very complex. It is not known whether the sphere has two incompatible differentiable structures; it is assumed that it has an infinite number. The (not closed) Euclidean space even has an uncountable number. For this reason, the fourth dimension is a specialty, because in all other dimensions only exactly one differentiable structure can be defined. From dimension five the classification, at least for simply connected manifolds, turns out to be a bit simpler. However, there are still many unanswered questions here, and the classification is still very complex. For this reason, one often restricts oneself to examining whether manifolds belong to different classes, i.e. whether they have different invariants. Among other things, techniques from algebraic topology , such as homotopy theory or homology theories, are used to examine manifolds for invariants, for example an invariant for the “simple connection” . ${\ displaystyle \ mathbb {R} ^ {n}}$ ${\ displaystyle \ mathbb {S} ^ {n}}$ ${\ displaystyle H ^ {n}}$ ${\ displaystyle \ mathbb {S} ^ {4}}$${\ displaystyle \ mathbb {R} ^ {4}}$${\ displaystyle \ mathbb {R} ^ {n}}$

Manifolds with constant curvature (from left to right): the hyperboloid with negative curvature, the cylinder with zero curvature, and the sphere with positive curvature.

Connected differentiable manifolds have no local invariants. That is, these properties apply globally to the whole manifold and are not dependent on a point. This is different for Riemannian manifolds. With the help of their scalar product, curvatures can be defined. The most important curvature term is the Riemann curvature tensor , from which most other curvature terms are derived. The value of the curvature tensor depends on the point of the manifold. Thus the invariants of manifolds with a scalar product are more diverse than those of differentiable manifolds without a scalar product. The section curvature is an important variable derived from the curvature tensor. A classification is known for Riemannian manifolds with constant sectional curvature. It can be shown that such manifolds are isometric (i.e. equivalent) . Where one of the above mentioned model spaces or stands and a discrete subgroup of the isometry group is the free and properly discontinuously on surgery. In global Riemannian geometry , one examines manifolds with globally restricted curvature for topological properties. A particularly remarkable result from this area is the set of spheres . Here it is concluded from certain topological properties and a limited curvature of section that the manifold is homeomorphic (topologically equivalent) to the sphere. In 2007 it was even possible to prove that under these conditions the manifolds are diffeomorphic. ${\ displaystyle N / \ Gamma}$${\ displaystyle N}$${\ displaystyle \ mathbb {S} _ {R} ^ {n}, \, \ mathbb {R} ^ {n}}$${\ displaystyle \ mathbb {H} _ {R} ^ {n}}$${\ displaystyle \ Gamma}$ ${\ displaystyle J (N)}$${\ displaystyle N}$${\ displaystyle {\ tfrac {1} {4}} ${\ displaystyle K}$

## Applications

Manifolds play an important role in theoretical physics , theoretical biology , engineering and geosciences , e.g. B. in integration over surfaces and multidimensional integration domains, especially manifolds with a boundary and with orientation (see e.g. the article by Stokes' theorem ).

In general relativity and astrophysics, as well as in relativistic quantum field theories , Lorentz manifolds , i.e. those of the signature (3,1), play a special role in the mathematical modeling of space-time and the many related quantities.

In evolutionary biology, one considers, among other things, the Wright manifold, as the set of allele frequencies in a population that are in a genetic coupling equilibrium.