Manifold with margin
A fringed manifold is a mathematical object from differential geometry . It is not a special case of a manifold , but on the contrary a generalization. Many structures that can be defined on a manifold can be transferred to manifolds with a boundary.
A fringed manifold is a fringed manifold where the boundary is the empty set .
Definitions
Manifold with margin
With
the upper half-space is referred to here. This is provided with the subspace topology of , so in particular as a whole it is both an open and a closed set .
A -dimensional topological manifold with a boundary is a Hausdorff space , which satisfies the second countability axiom and in which every point has an open neighborhood that is homeomorphic to an open subset of the upper half-space .
Generalized map
An open subset together with a homeomorphism , where open is in, is called a generalized map.
edge
The edge of in is the set of points with . If there is a bounded manifold, then the points which are mapped from a (then necessarily every ) map image onto a point of are called an edge point of . The set of all edge points is denoted by.
The connected components of called boundary components .
Structures
Differentiable structure
Similar to an unrestricted manifold, one can also define a differentiable structure on a manifold with a boundary. This consists of an overlap with generalized maps. And for all pairs of such cards and a picture
must be a diffeomorphism . If the definition set of still contains boundary points of , one must select an open set which contains but no longer lies in, in order to examine differentiability. Of course, it is also not possible to define a differentiable structure for every manifold with a boundary. Manifolds with a boundary, like normal manifolds, can also have several different differentiable structures.
orientation
For a bounded (differentiable) manifold , the boundary is a submanifold of . If it is assumed that it is orientable , then the edge can also be orientated. This cannot be taken for granted, as there are submanifolds that cannot be orientated.
Stokes' theorem
With the help of bounded manifolds, one can formulate Stokes's integral theorem concisely and elegantly. Let be an oriented, n-dimensional, differentiable manifold with boundary and let be a differential form of degree that has compact support, then we have
If there is no boundary, then the right integral is zero and if the right integral is a one-dimensional manifold the right integral is a finite sum.
Manifold with corners
definition
Let be the set of all points of where all coordinates are nonnegative:
This subset is homeomorphic, but not diffeomorphic too . Let be a (topological) manifold with a boundary. A vertex manifold is a manifold that is locally diffeomorphic to open subsets des . In this case the cards are called cards with corners . So a card with corners is a pair , where an open subset of and is a homeomorphism. Two cards with corners and are called compatible if smooth.
A smooth structure with corners on a topological manifold with a border is the maximum set of all compatible maps with corners that cover . A topological manifold with a boundary together with a smooth structure with corners is called a manifold with corners.
Remarks
Since is homeomorphic , manifolds with a boundary and manifolds with vertices cannot be distinguished topologically. For this reason it makes no sense to define a manifold with corners without a differentiable structure. An example of a manifold with corners are rectangles .
literature
- John M. Lee: Introduction to Smooth Manifolds (= Graduate Texts in Mathematics 218). Springer-Verlag, New York NY et al. 2003, ISBN 0-387-95448-1 .