Diversity

One Diversity is in topological area, that locally a usual Euclidean area Rⁿ resembles. In the whole one the diversity must not one Rⁿ correspond (not to it homoeomorph its).

Variousnesses are the central article that Differentialgeometrie; they have important applications in that theoretical physics.

Einfuehrendes example

A gladly selected example of a diversity is one Sphere (= kugeloberflaeche), descriptive for instance the earth's surface:

One knows each region of the earth with one Map on one level (R²) 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, as with Map change the maps overlap. On the other hand there is no individual map, on which the entire kugeloberflaeche can be completely represented, without it "tears up"; Maps of the world have also always "edges", or they illustrate parts of the earth twice.

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

Overview

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

Warning: One Sphere an example of a diversity is, in one Euclidean area higher dimension embedded is (one Untermannigfaltigkeit). Such an imbedding exists for each diversity (see Imbedding set of Whitney and Imbedding set of Nash). The modern mathematical description of variousnesses refers however no to an imbedding area.

In that differentiable variousnesses use finds as Phase spaces in that and as four-dimensional Pseudoriemann variousnesses for description that Space-time in that General relativity theory and .

Edges, orientation

One Sphere (= kugeloberflaeche) is a diversity without edge. A full ball against it is a diversity with edge; their edge is the straight kugeloberflaeche. We do here without a technical definition of the term edge (boundary) and it points out that in the following given definition of the term diversity only Variousnesses without Edges includes.

Variousnesses can orientable its. Most well-known examples of non-orientable variousnesses are about that and those . Also such variousnesses we become in the further not regard.

Topological variousnesses

One topological <math>n</math> diversity is in parakompakter Hausdorff area, in which each point possesses an open environment, which <math>\mathbb{R}^n</math> homoeomorph to an open subset of; 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 <math>\mathbb{R}</math> the real number line; and that <math>\mathbb{S}^1</math>. The classification of the 2-Mannigfaltigkeiten is likewise well-known, but for the 3-Mannigfaltigkeiten already is the problem unresolved (for the proof that Poincaré assumption is 1.000.000 US-$ expenditure-praised; a proof of Grigori Perelman momentarily by the professional world one examines). The 4-dimensionalen of cases cannot be classified (each group finite-witnessed is as Fundamentalgruppe such an area realizable).

Differentiable variousnesses

In order to regard differentiable functions, the structure of a topological diversity is not sufficient. It is <math>M</math> such a topological <math>n</math> diversity without edge. One open subset of <math>M</math>, on Homoeomorphismus to an open quantity of <math>\mathbb{R}^n</math> , calls one is defined one Map. A collection of maps, which <math>M</math> covers, calls one one Atlas of <math>M</math>. Overlapping maps induce themselves a Homoeomorphismus (a in such a way specified Map change or Coordinate change) <math>\mathbb{R}^n</math> between open subsets of;. If for an Atlas <math>\mathcal{A}</math> all such illustrations are <math>k</math> times differentiable, then one calls <math>\mathcal{A}</math> an <math>C^k</math> Atlas. One calls two <math>C^k</math> Atlases (the same diversity) then exactly with one another compatibly, if their combination educates again an <math>C^k</math> Atlas. This compatibility is one . One <math>C^k</math> diversity is a topological diversity as well as an <math>C^k</math> Atlas (actually with an equivalence class of <math>C^k</math> Atlases). Smooth variousnesses variousnesses of the type are <math>C^\infty</math>. Are all map changes even analytically, then one calls the diversity likewise analytically or also <math>C^\omega</math> diversity.

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

To each (parakompakten) <math>C^r</math> diversity (<math>r>1</math>) exists an Atlas, that is arbitrarily often differentiable or even analytic. Indeed this structure is even clear, D.h. no restriction of the public is to be accepted that each diversity is analytic (if one of differentiable variousnesses talks).

This statement is however for topological variousnesses of the dimension <math>4</math> or more highly no longer absolutely correctly: Thus there is both to <math>C^0</math>-Mannigfaltigkeiten, which do not possess differentiable structure, and <math>C^1</math>-Mannigfaltigkeiten (or also <math>C^\omega</math> m., s.o.), which as differentiable variousnesses differently, but when topological are equal to variousnesses. The most well-known example of the second case are the exotic <math>7</math>-Sphaeren in such a way specified, all this homoeomorph to <math>\mathbb{S}^7</math> (however among themselves not diffeomorph) are. Since the topological and the differentiable category in low dimension agree, are unfortunately only difficult such results to illustrate.

Complex variousnesses

A topological diversity <math>X</math> is called complex diversity the dimension <math>n</math>, if each point <math>x \in X</math> an open environment <math>U \subset X</math> , homoeomorph to an open quantity <math>V \subset \mathbb the C^n</math> has; is. Furthermore one demands that for two for each maps <math>\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 <math>V_{ij} \subset \mathbb C^n</math> the quantity <math>\theta_i (U_i \cap U_j)</math>.

Complex variousnesses of the dimension 1 becomes as Riemann surfaces designated.

Tangential bundle

At each point <math>p</math> one finds one to a differentiable (however not a topological) diversity Tangential area. In a map one attaches one to this point simply <math>\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 abstractly on <math>p</math> either as the area that 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 <math>p</math> on; to be alike are).

The combination of all tangential areas of a diversity imagines Vector bundle, that Tangential bundle one calls. The tangential area of a diversity <math>M</math> in the point <math>p</math> becomes usually with <math>T_p M</math> designated, the tangential bundle with <math>TM</math>.

A Unterbuendel of the tangential bundle is that Unit tangential bundle <math>UTM</math>, only of tangential vectors of the length 1 consists. Its fibers are not vector spaces, but Spheres the dimension <math>n-1</math>. Therefore it is not a vector bundle only separates in Fiber bundle. As bundle it is independent of the choice of the Metrik on the tangential areas.

Riemann variousnesses

On a "naked" differentiable diversity it is not possible distances to determine angles or volumes. The most usual kind everything these sizes to specify, is the instruction one 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 are the more general Finsler diversity (also Finsler geometry).

Lie groups

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