# Diversity

 of these articles treats the mathematical term of a diversity. For another meaning see variety.
 differentiable diversity affects the special fields is special case of topological area parakompakter Hausdorff area topological diversity as special cases Euclidean

a diversity is enclosure a topological area, which resembles locally a usual Euclidean area R n. Generally speaking the diversity does not have to correspond to a R n (notto it homöomorph its).

Variousnesses are the central the subject of the Differentialgeometrie; they have important applications in theoretical physics.

## of Einführendes example

a gladly selected example of a diversity is a sphere (= Kugeloberfläche), descriptive for instance the earth's surface:

One knows each region of the earth with a map on one level (R 2) illustrate. If one approaches the edge of the map, one should change to another map, which represents the adjacent area. So one can describe a diversity by a complete sentence of maps completely; one needs thereby rules,overlap themselves as with the map change the maps. On the other hand there is no individual map, on which the entire Kugeloberfläche can be completely represented, without it “tears up”; Maps of the world have also always “edges”, or they illustrate parts of the earth twice.

The dimension of a diversity corresponds to the dimension of a local map; all maps have the same dimension.

## overview

if the map changes are sufficiently smooth, has one a differentiable diversity. From the analysis well-known terms like the derivativeone can transfer to natural kind to differentiable variousnesses. A Riemann diversity (after Bernhard Riemann) possesses an additional structure, which permits it to determine angles and distances with the Riemann Metrik.

Warning: A sphere is an example of oneDiversity, which is embedded into a Euclidean area of higher dimension ( a Untermannigfaltigkeit). Such an imbedding exists for each diversity (see. Imbedding set of Whitney and imbedding set of Nash). The modern mathematical description of variousnesses takes however no purchaseon an imbedding area.

In physics differentiable variousnesses use finds as phase spaces in the classical mechanics and as four-dimensional Pseudoriemann variousnesses for the description of space-time in general relativity theory and cosmology.

## edges, orientation

a sphere (=Kugeloberfläche)is a diversity without edge. A full ball against it is a diversity with edge; their edge is the straight Kugeloberfläche. We do here without a technical definition of the term edge (boundary) and point out that thosein the following given definition of the term diversity only variousnesses without edges includes.

Variousnesses can be orientable. Most well-known examples of non-orientable variousnesses are for instance the Möbiusband and the Klein bottle. Also we will not regard such variousnesses in the further .

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## Topological variousnesses

a topological [itex] n< /math> - diversity is a parakompakter Hausdorff area, in which each point possesses an open environment, homöomorph to an open subset of [itex] \ mathbb {R} ^n< /math> is.

Variousnesses inherit many local characteristics of the Euclidean area: it are to be away-been connected locally,locally compactly and locally metrisierbar.

It is not possible to classify all variousnesses. The coherent 1-dimensionalen variousnesses (without edge) is the real number line [itex] \ mathbb {R}< /math> and the circle [itex] \ mathbb {S} ^1< /math>. The classification of the 2-Mannigfaltigkeiten is likewise well-known, but already for the 3-Mannigfaltigkeitenthe problem is unresolved ( the Geometrisierung of 3-Mannigfaltigkeiten would be a classification; their supposed proof by Grigori Perelman is examined momentarily still by the professional world). The 4-dimensionalen of cases cannot be classified (each group finite-witnessed is as a Fundamentalgruppe of such an arearealizable, and a classification of all groups finite-witnessed is impossible).

## differentiable variousnesses

around differentiable functions to regard, is not sufficient the structure of a topological diversity. It is [itex] M< /math> such topological [itex] n< /math> - diversity without edge. An open subset of[itex] M< /math>, on some Homöomorphismus to an open quantity of [itex] \ mathbb {R} ^n< /math> , calls one is defined a map. A collection of maps, [itex] the M< /math> covers, calls one an Atlas by [itex] M< /math>. Overlapping maps induce themselves a Homöomorphismus (a in such a way specified Map change or coordinate change) between open subsets of [itex] \ mathbb {R} ^n< /math>. If for an Atlas [itex] \ mathcal {A}< /math> all such illustrations [itex] k< /math> - are times differentiable, then one calls [itex] \ mathcal {A}< /math> one [itex] C^k< /math> - Atlas. Two [itex] C^k< /math> - one calls Atlases (the same diversity) exactly compatibly with one another if theirCombination again one [itex] C^k< /math> - Atlas forms. This compatibility is an equivalence relation. One [itex] C^k< /math> - diversity is a topological diversity as well as one [itex] C^k< /math> - Atlas (actually with one Equivalence class of [itex] C^k< /math> - Atlases). Smooth variousnesses is variousnesses of the type [itex] C^ \ infty< /math>. All map changes are even analytic,then one calls the diversity likewise analytically or also [itex] C^ \ omega< /math> - diversity.

On one [itex] C^k< /math> - diversity [itex] M< /math> one calls a function [itex] f: M \ tons \ mathbb {R}< /math> exactly then [itex] s< /math> - times differentiable ([itex] s \ le k< /math>), if it on each map [itex] s< /math> - is times differentiable.

To everyone (parakompakten) [itex] C^r< /math> - diversity ([itex] r> 1< /math>) existsan Atlas, that is arbitrarily often differentiable or even analytic. Indeed is this structure even clearly, i.e. no restriction of the public is to be accepted that each diversity is analytic (if one of differentiable variousnesses talks).

This statement is but for topological variousnesses of the dimension [itex] 4< /math> or more highly no longer absolutely correctly: Thus there are both [itex] C^0< /math> - variousnesses, which do not possess differentiable structure, and [itex] C^1< /math> - variousnesses (or also [itex] C^ \ omega< /math> - M., s.o.), those as differentiable variousnesses differently, but as topological variousnesses directlyare. The most well-known example of the second case are exotic math 7 [/itex] in such a way< specified> - spheres, all this homöomorph too [itex] \ mathbb {S} ^7< /math> (however among themselves not diffeomorph) are. Since the topological and the differentiable category in low dimension agree, are such results unfortunately onlyto illustrate with difficulty.

## complexes variousnesses

a topological diversity [itex] X< /math> complex diversity of the dimension is called [itex] n< /math>, if each point [itex] x \ in X< /math> an open environment [itex] U \ subset X< /math> has, homöomorph to an open quantity [itex] the V \ subset \ mathbbC^n< /math> is. Furthermore one demands that for ever two maps [itex] \ theta_i: U_i \ rightarrow V_i, x \ in U_i, i=1,2< /math> the map change

< math> \ theta_i^ {- 1} \ circ \ theta_j: V_ {ij} \ rightarrow V_ {ji}< /math>

holomorph is. Here designate [itex] V_ {ij} \ subset \ mathbb C^n< /math> the quantity [itex] \ theta_i (U_i \ capU_j)< /math>.

(Coherent one) complexes variousnesses of the dimension 1 are called Riemann surfaces.

## tangential bundles

at each point [itex] p< /math> one finds a tangential area to a differentiable (however not a topological) diversity. In a map one attaches to this pointsimply one [itex] \ mathbb {R} ^n< /math> on and it considers itself then that the differential of a coordinate change at each point defines a linear isomorphicism, which carries the transformation out of the tangential area into the other map. One defines the tangential area on abstractly [itex] p< /math> either as thatArea of the Derivationen at this point or the area of equivalence classes of differentiable curves (whereby the equivalence relation indicates, when the speed vectors of two curves on [itex] p< /math> to be alike are).

The combination of all tangential areas of a diversity forms a vector bundle, the tangential bundleone calls. The tangential area of a diversity [itex] M< /math> in the point [itex] p< /math> becomes usually with [itex] T_p M< /math> designated, the tangential bundle with [itex] TM< /math>.

A Unterbündel of the tangential bundle is the unit tangential bundle< math> UTM< /math>, which consists only of tangential vectors of the length 1. Its fibers are not Vector spaces, but spheres of the dimension [itex] n-1< /math>. Therefore it is not a vector bundle separates only one fiber bundle. As bundle it is independent of the choice of the Metrik on the tangential areas.

## Riemann variousnesses

on a “naked” differentiable diversity is it not possible distances to determine angles or volumes. The most usual kind everything these sizes to specify, is the indication of a dot product at each point of the area (or an equivalent orthonormal basis of tangential vectors).

One calls such a diversity then Riemann diversity. Only if the demand exists after a linear measurement, there is the more general Finsler diversity (also Finsler geometry).

## Lie groups

a Lie group [itex] G< /math> are both a diversity and a group. One demands that both structuresare compatible with one another. These objects describe typical symmetries of geometrical structures and physical systems.