Variety
Mnogoobra?zie - this , which locally looks like the "usual" Euclidean space Rn. Euclidean space is the simplest example of variety, can serve as a more complex example the earth's surface, on which small regions can be depicted on the flat map, nevertheless it is not possible to compose the united map of its entire surface.
Studies of varieties were begun in the second-half of the 19th century, they naturally arose during the study of differential geometry and group theory of Lie. Nevertheless, the first precision determination were made only in the 30's of the 20th century.
Usually are examined so-called smooth varieties, t. e. those, on which there is the chosen class of "smooth" functions - in such varieties it is possible to speak about the tangential vectors and the tangential spaces. , in order to measure the lengths of curves and the angles, is necessary even more additional structure - Riemann certificate.
In the classical mechanics smooth varieties serve as to fazovy.e of the space. In , four-dimensional pseudo-Riemann varieties are used as model for the space-time.
That which follows below - these are formal determination plus the minimum consideration of varieties.
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Topological varieties
&.lt;.math&.gt;.n&.lt;/.math&.gt;- measured topological variety (without the boundary) this Khausdorfovo topological space in which each point has to the open subset Rn, t. e. &.lt;.math&.gt;.n&.lt;/.math&.gt;- measured Euclidean space.
&.lt;.math&.gt;.n&.lt;/.math&.gt;- measured topological variety with the boundary this Khausdorfovo topological space in which each point has the open environment homeomorphous to the subset of the locked half-space in Rn. The points, which have the open environment homeomorphous to the open subset Rn they are called internal, and the set of all such points the interior of the variety (this always non-empty set). Addition to the interior, is called by the edge, this, &.lt;.math&.gt;(.n- 1) &.lt;/.math&.gt;- measured variety. Surely it should be noted that the concept of edge introduced here completely not equivalent to the concept in the general topology.
In the determinations additionally it is usually assumed that variety or (this is equivalent ), or, which is still stronger, has calculating (this is equivalent so that the variety is packed into the Euclidean space of final dimensionality).
Further we everywhere assume that the variety has the countable base.
Compact variety without the boundary is called locked, noncompact variety is called opened.
The requirement of property of being a hausdorff can seem by superfluous, it is heavy to present the space, which locally homeomorphously Euclidean, but in this case not Khausdorfovo. This example can be built by the gluing of two copies of material straight line on all points, except one.
Smooth varieties
The smooth structure, determined below, usually appears in almost all applications and in this case makes the variety much more convenient in the work.
We begin from the topological variety of &.lt;.matyu&.gt;.M&.lt;/.matyu&.gt; without the boundary. Let us name map the homomorphism of &.lt;.matyu&.gt;\.pyui^{}_{}&.lt;/.matyu&.gt; from the open set &.lt;.matyu&.gt;.U\.subset M&.lt;/.math&.gt; to the open subset Rn. Collection of the maps of those covering entire &.lt;.matyu&.gt;.M&.lt;/.matyu&.gt; it is called atlas. If two maps of &.lt;.matyu&.gt;\.pyui^{}_{}&.lt;/.matyu&.gt; and &.lt;.matyu&.gt;\.psi^{}_{}&.lt;/.matyu&.gt; is covered one point into &.lt;.matyu&.gt;.M&.lt;/.matyu&.gt; that is their composition of &.lt;.matyu&.gt;\.pyui\.chirch\.psi^{-y}&.lt;/.matyu&.gt; is assigned mapping "glueing" from the open set Rn in the open set Rn. If all mappings of glueing from the class of &.lt;.matyu&.gt;.Ch^.k&.lt;/.matyu&.gt; (T.e. the &.lt;.math&.gt;.k&.lt;/.math&.gt;- time of the continuously differentiated functions), then atlas is called &.lt;.matyu&.gt;.Ch^.k&.lt;/.matyu&.gt; atlas (it is possible to also examine &.lt;.matyu&.gt;.k=\.infty&.lt;/.matyu&.gt; or &.lt;.matyu&.gt;\.omega&.lt;/.matyu&.gt;, which corresponds to the infinitely differentiated and analytical glueings).
Example: Sphere can be covered with &.lt;.math&.gt;.C^\.infty&.lt;/.math&.gt;- atlas of two maps during the additions of the north and southern poles by the stereographic projections with respect to these poles.
Two &.lt;.matyu&.gt;.Ch^.k&.lt;/.matyu&.gt; atlas is assigned one &.lt;.math&.gt;.C^.k&.lt;/.math&.gt;- smooth structure if their association it is &.lt;.math&.gt;.C^.k&.lt;/.math&.gt;- atlas.
For such varieties it is possible to introduce the concepts the tangential vector, the tangent and kokasatel'nogo of the spaces and the stratifications.
For the assigned &.lt;.math&.gt;.C^1&.lt;/.math&.gt;- smooth structure it is possible to find &.lt;.math&.gt;.C^\.infty&.lt;/.math&.gt;- smooth structure given to new, &.lt;.math&.gt;.C^\.infty&.lt;/.math&.gt;- atlas, which assigns the same &.lt;.math&.gt;.C^1&.lt;/.math&.gt;- smooth structure. And what is more all such obtained thus varieties are &.lt;.math&.gt;.C^\.infty&.lt;/.math&.gt;- diffeomorphic. Therefore, frequently by smooth structure is understood &.lt;.math&.gt;.C^1&.lt;/.math&.gt;- smooth structure.
Not each topological variety allows smooth structure. Examples of such "rough" varieties appear in already dimensionality four. Also there are examples of topological varieties which they allow several different smooth structures. This first example of nonstandard smooth structure was built by John Milnor (John Milnor) on the seven-dimensional sphere.
Classification of the varieties
Each one-dimensional variety without the boundary of material straight line or circle
The homeomorphous class of the closed connected surface is assigned it By Euler characteristic and by the orientability. (if oreintiruyemo first this sphere with the knobs, if no that the connected sum several copies of projective plane)
The classification of the locked three-dimensional varieties follows from the hypothesis Of terstona, which was recently proven by Grigoriy Perel'man.
If dimensionality is more than three, then classification is impossible; moreover, cannot be built algorithm which determines it does appear the variety . Nevertheless there is a classification of all odnosvyazannykh varieties in all dimensionality? 5.
It is possible to also classify smooth varieties. In dimensionality 2 and 3 any pair of homeomorphous varieties is also diffeomorphic. In dimensionality 4 there are examples of the locked varieties which they allow the infinite number not of equivalent smooth structures, but the open varieties, as, for instance, R4 they allow of different smooth structures. In dimensionality 5 or above any topological variety allows not the more than finite number not of equivalent smooth structures.
Additional structures
Frequently the smooth varieties equip the list of the most frequently met additional structures with additional structures here:
- Riemann certificate
- Simplektichekaya form
- Complex structure
Generalizations
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