Manifold

Under a manifold is understood in mathematics a topological space, which resembles the Euclidean space locally. Global but does not have the diversity of a Euclidean space with the same ( not homeomorphic to it).

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

• 7.1 Discrete Space
• 7.2 sphere
• 7.3 square
• 7.4 Mobius band
• 7.5 Klein bottle

Introductory Example

A fond chosen example of a manifold is a sphere ( = sphere ), vividly about the earth's surface:

Each region of the world can be mapped with a map on a plane ( ). Approaching the edge of the map, you should switch to another map that represents the adjacent area. So one can describe a manifold with a complete set of cards completely; you need this rules as to overlap the cards when changing cards. In contrast, there is no single card, on which the entire surface of the sphere can be represented completely without " ripping " to; World maps have indeed always " edges ", or they form parts of the earth twice from. The dimension of a manifold corresponds to the dimension of a local map; all cards have the same dimension.

Another example is the torus ( " lifebuoy ", " Donut" ).

Historical Overview

The concept of manifolds emerged in the 19th century, particularly through research in geometry and the theory of functions. While Differentialgeometer local concepts such as the curvature of curves and surfaces studied, the function under consideration theorists global problems. They found that properties of functions with topological invariants of the amount related to certain. These quantities are manifolds ( cf. Theorem regular value ).

The concept of diversity goes back to Bernhard Riemann. In his habilitation lecture On the Hypotheses which underlie the geometry, which he held in 1854, among others, before Carl Friedrich Gauss, he introduced the concept of manifolds. He speaks of discreten and continuous manifolds that are extended n -fold, is limited at this time, ie on structures that are embedded in the sind.Auf these manifolds can measure angles and distances. In later work, he developed the Riemann surfaces which were probably the first abstract manifolds. Manifolds are sometimes called to distinguish abstract, to express that they are not subsets of Euclidean space.

Henri Poincaré in his work began with the study of three-dimensional manifolds, whereas previously predominantly two-dimensional manifolds ( surfaces ) were treated. In 1904 he set up his namesake Poincaré conjecture. It says that every simply connected, compact three-dimensional manifold is homeomorphic to the 3- sphere. For Grigori Yakovlevich Perelman this conjecture published in 2002 unverified by " referees " evidence, which was indeed not published in a refereed journal, but only on the Internet, but is regarded by the professional community as correct.

Today's standard definition first appeared in 1913 with Hermann Weyl in Riemann surfaces. However, only by the publications of Hassler Whitney were from 1936 manifolds to an established mathematical object. His best-known result of the embedding theorem of Whitney.

Types of manifolds

Topological manifolds

Be a topological space. This is called a ( topological ) manifold of dimension or a short -manifold if the following properties are satisfied:

Manifolds inherit many local properties of Euclidean space: they are locally path-connected, locally compact and locally metrizable. Manifolds which are homeomorphic to each other, are considered to be the same ( or equivalent). From this, the question arose after the classification, ie the question of how many non-equivalent manifolds there.

Differentiable manifolds

To view differentiable functions, the structure of a topological manifold is not sufficient. It was such a topological manifold without boundary. Is an open subset of set on which a homeomorphism defined on an open set of, then this homeomorphism is called a map. A lot of cards that cover their archetypes, called Atlas. Different maps induce a homeomorphism (called a map change or change of coordinates ) between open subsets of. If all such maps are changing times differentiable for an atlas, then it is called an atlas. Two atlases ( the same manifold) is called if and only compatible with each other if their union again forms an atlas. 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. When all cards are changing even analytically, then it is called the multiplicity also analytically or manifold.

On a manifold is called a function if and only times differentiable () when it is on each card - times differentiable.

For every ( paracompact ) manifold () exists an atlas, which is infinitely differentiable or even analytic. In fact, this structure is even unique, ie, there is no loss of generality to assume that each manifold is analytically (if one speaks of differentiable manifolds ).

This statement is but for topological manifolds of dimension or higher no longer necessarily true: there are both manifolds that have no differentiable structure, as well as manifolds (. , Or M, as ) different as differentiable manifolds, but as topological manifolds are the same. The best-known example of the second case, the so-called exotic spheres, all of which are homeomorphic to (but not diffeomorphic to each other ). Since the topological and differentiable category match in lower dimension, such results are difficult to illustrate.

Tangent bundle

At each point one -dimensional, differentiable (but not a topological ) manifold one finds a tangent. In a card you simply attaches to this point at one and then considering that the differential of a coordinate change at each point defines a linear isomorphism, which makes the transformation of the tangent space to the other card. Abstract one defines the tangent to either the space of the derivations of this point or the space of equivalence classes of differentiable curves (the equivalence relation indicating when the velocity vectors of two curves to be of the same).

The union of all tangent spaces of a manifold is a vector bundle, the tangent bundle is called. The tangent space of a manifold in point is usually referred to as, the tangent bundle with.

Complex manifolds

A topological manifold is called complex manifold of (complex) dimension, if every point has an open neighborhood which is homeomorphic to an open set. Furthermore, we require that for any two cards, the exchange

Is holomorphic. Here denotes the set.

The main difference from ordinary differentiable manifolds is less the difference between and, but in the much stronger requirement of complex differentiability of the map change pictures.

( Related ) Complex manifolds of dimension 1 are called Riemann surfaces. Other special complex manifolds are Stein manifolds and Kähler manifolds, which are complex, Riemannian manifolds.

Riemannian manifolds

To speak to a differentiable manifold of lengths, distances, angles and volume, one needs an additional structure. A Riemannian metric (also called Metric tensor ) defined in the tangent space of each point of the manifold a scalar product. A differentiable manifold with a Riemannian metric is called Riemannian manifold. By the scalar products of lengths of angle between vectors, and vectors are initially defined, assuming then curves and lengths of distances between points on the manifold.

If instead of a scalar product in each tangent space only a (not necessarily symmetric ) norm is defined, it is called a Finsler metric and a Finsler manifold. On Finsler manifolds lengths and distances are defined but not angle.

Semi- Riemannian manifolds

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

Here, the symmetric bilinear form defined by the metric at each tangent space need not be positive definite, but only non- degenerate. After the inertia Sylvester's theorem, such a bilinear form can be represented as a diagonal matrix with entries of. Then there are entries 1 and -1 entries, one speaks of a metric with signature. If the signature of the metric (or according to another convention ), the dimension of the manifold is, one speaks of a Lorentzian manifold. In general relativity, spacetime is described by a four-dimensional Lorentzian manifold, that is, with the signature ( 3.1 ) (or (1.3 ) ).

Lie groups

A Lie group is a differentiable manifold, both as a group, the group multiplication (or addition) and the inversion must be a group member differentiable maps. The tangent space of a Lie group at the identity element is closed with respect to the commutator and forms a Lie group associated to the Lie algebra.

A simple example of a non-compact Lie group is the Euclidean vector space together with the normal vector space addition. The unitary group is an example of a compact Lie group. ( One can imagine this manifold as a circle and the group operation is a rotation of the circle. ) In physics (see quantum chromodynamics ) are primarily the groups before, the " special unitary groups of order n" before (eg n = 3, determinant 1).

Topological properties

• For manifolds the notions coincide connected and path-connected. Since manifolds are also locally simply connected, all connected manifolds have a universal covering.
• Each manifold has a countable fundamental group.
• Each manifold of dimension is triangulated. Four-dimensional manifolds are not triangulated in general, and it is still (April 2009) unknown whether manifolds of higher dimension are always triangulated or not.
• Each manifold is metrizable. This follows by the Metrisierbarkeitssatzes of Urysohn from the Zweitabzählbarkeit together with the local compactness or Metrisierbarkeit.

Manifold with boundary

Manifolds, which have until now dealt with in this article are without boundary. Beran Detention manifolds are no multiplicities in the above sense, but its definition is very similar. Be to, so again a topological Hausdorff space which satisfies the second axiom of countability. The space is called a manifold with boundary if every point has a neighborhood which leads to a subset of " non-negative n-dimensional half-space " is homeomorphic:

This ( non-compact ) manifold is bounded by the - dimensional plane.

An example of a compact manifolds with boundary is the closed ball that has the sphere as a boundary. This is itself a unberandete manifold. On bounded manifolds one can define additional structures similar as to unberandeten manifolds. For example it is possible to define on certain manifolds with boundary is a differentiable structure or to speak of orientability.

Manifolds with orientation

Another key feature of a bounded or unberandeten manifolds concerns the orientability or non - orientability of the manifold. It can also be defined "cards as " (where the compatibility is satisfied by itself).

As the following examples show, all four combinations come with or without boundary, and with or without prior orientation.

Examples

Discrete space

Every countable discrete topological space is a zero-dimensional topological manifold. The maps of these manifolds are the couples with and.

Sphere

The sphere is a unberandete oriented manifold of dimension. An atlas of the manifold is given by the two stereographic projections

Where the north pole and the south pole of the sphere, respectively. The resulting initial topology is the same that would be induced by sub-space as a topology. The sphere is also being studied in other sciences except in mathematics, such as in cartography or even in theoretical physics at the so-called Bloch sphere.

Rectangle

A simple example of a bounded and orientable manifold relates to a ( closed ) rectangle as shown in the adjacent sketch. The border consists of the sides of the rectangle; The two orientations are " counter-clockwise " ( ) or "clockwise " (-). In the first case, the next round will be approximately considered: From A to B and on to C and D, then back to A; all counterclockwise.

Mobius band

When the sides and the top covered rectangle so stuck together that A come to lie on D to B and C, then we obtain an orientable, manifold with boundary which is homeomorphic to, that is the Cartesian product of the closed unit interval and the edge of the circle is. It can be embedded in the three-dimensional space as euklididschen surface of a cylinder.

If, however, the points A and C and B and D together glued, which is possible by " twisting " of the narrow sides, and when the " sticking " occurs seamlessly, creates a non-directional edge with two-dimensional manifold. This is called a Möbius strip.

The edge of the manifold corresponds to " 8", that is the characteristic crossover in the middle. First, for example, counter-clockwise, the lower semi-circle of 8 through ( = from A to B ), then follows the crossover ( this corresponds to the paste over with twisting ); after crossing follows the upper circle of 8, by running in the other direction of rotation, that is not from C to D, but from D to C.

Klein bottle

In an analogous manner, by appropriately gluing together two bands in rooms with at least three dimensions of a non- orientable two-dimensional manifold without boundary, analogous to the surface of a " ball with handle ", ie, a structure that resembles a torus, which would of course orientable:

These non- orientable manifold without boundary is called 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. In the classification of manifolds, these structures must be of course taken. Thus, two manifolds from topological point of view to be equivalent, which means that there is a homeomorphism, which transfers the a manifold to the other, but these two Mannigfaltigenkeiten may carry different, non-compatible differentiable structures, then they are not from the viewpoint of differential geometry equivalent; from the point of view of the topology, however, they may be equivalent. If two manifolds from a given point of view equivalent, they also have the same to this view matching invariants, for example, the same dimension or the same fundamental group.

The related one-dimensional manifolds are either diffeomorphic and hence homömorph to the real number line or a circle.

The classification of closed manifolds is also known in dimensions two and three. Manifolds of dimension as well as the one-dimensional manifolds have the special property that every topological manifold allowing only a differentiable structure. This has the consequence that it is possible in the investigation of such manifolds combine topological and differential geometric methods. In the theory of two-dimensional closed manifolds, there is the classification theorem for 2- manifolds. Thus, two closed surfaces with the same gender are diffeomorphic to each other if they are both orientable or both non- orientable. Closed surfaces are thus completely determined by the invariants orientability and gender. Now the important " presumption for geometrization of 3-manifolds " by Grigori Perelman was proven for three-dimensional closed manifolds. This theory contains as a special case of the conjecture of Poincaré.

In four-dimensional manifolds the classification even in the case just related manifolds very complicated and generally impossible, because every finitely presented group occurs as the fundamental group of a 4 -manifold and the classification of finite -presented groups is algorithmically impossible. This is called the Euclidean space, the sphere and the hyperbolic space model spaces ( in English: model spaces ), since their geometry is relatively easy to describe. In dimension four, these rooms are also very complex. It is not known whether the sphere has two non-compatible differentiable structures, it is believed that it has infinitely many. The ( non-closed ) Euclidean space has even uncountably many. For this reason, the fourth dimension is a feature, because in all other dimensions can only be exactly one differentiable structure to define. From the five dimensional classification proves, at least for simply connected manifolds, as something easier. However, there are still many open questions, and the classification is still very complex. For this reason, we restrict ourselves often to investigate whether manifolds belong to different classes, so if they have different invariants. So you use, among other techniques from algebraic topology, such as homotopy or homology theory to investigate manifolds to invariants, such as a invariant for the "simple connection ".

Related differentiable manifolds have no local invariants. That is, these properties are global to the whole manifold and are not dependent on one point. For Riemannian manifolds this is different. With the help of their scalar curvatures can be defined. The most important is the Riemannian curvature tensor of curvature term, from which most other curvature terms are derived. The value of the curvature tensor depends on point of the manifold. Thus the invariants of manifolds with scalar product are more diverse than that of differentiable manifolds without scalar product. The sectional curvature is an important derived from the curvature tensor size. For Riemannian manifolds with constant sectional curvature, a classification is known. It can be shown that such manifolds isometrically (ie equivalent), are. Where one of the above-mentioned model rooms or stands and a discrete subgroup of the isometry group is, which operates freely and properly discontinuously on. In the global Riemannian geometry is examined manifolds with globally bounded curvature on topological properties. A particularly remarkable result from this area is the set of spheres. Here is inferred from certain topological properties and due to limited sectional curvature that the manifold homeomorphic ( topologically equivalent ) to the sphere is. In 2007, could even be proved that under these conditions, the manifolds are diffeomorphic.

Applications

Manifolds play an important role in theoretical physics, theoretical biology, engineering sciences as well as in the geosciences, eg, when integrating over surfaces and multidimensional integration areas, particularly manifolds with boundary and with orientation ( see, eg, the article set Stokes ).

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

In evolutionary biology, considering among other things the Wright manifold, as the amount present in a genetic linkage equilibrium allele frequencies of a population.