Manifold with margin

On the left side there are topological manifolds without a border and on the right side those with a border are shown.

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

A cylinder of finite length is a manifold with a margin.

Manifold with margin

With

${\displaystyle \mathbb {H} ^{n}:=\left\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}\mid x_{n}\geq 0\right\}}$

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 . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {H} ^{n}}$

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 . ${\displaystyle n}$ ${\displaystyle V\subset \mathbb {H} ^{n}}$

Generalized map

An open subset together with a homeomorphism , where open is in, is called a generalized map. ${\displaystyle U\subset M}$ ${\displaystyle \phi \colon U\to V\subset \mathbb {H} ^{n}}$ ${\displaystyle V}$ ${\displaystyle \mathbb {H} ^{n}}$

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. ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle x_{n}=0}$ ${\displaystyle M}$ ${\displaystyle \partial \mathbb {H} ^{n}}$ ${\displaystyle M}$ ${\displaystyle \partial M}$

The connected components of called boundary components . ${\displaystyle \partial M}$

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 ${\displaystyle (U,\phi )}$ ${\displaystyle (V,\psi )}$

${\displaystyle \phi \circ \psi ^{-1}|_{\psi (U\cap V)}\colon \psi (U\cap V)\to \phi (U\cap V)}$

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. ${\displaystyle \psi (U\cap V)}$ ${\displaystyle \phi \circ \psi ^{-1}}$ ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \psi (U\cap V)}$ ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle \phi \circ \psi ^{-1}}$

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. ${\displaystyle M}$ ${\displaystyle \partial M}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \partial M}$

Stokes' theorem

→ Main article : 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 ${\displaystyle M}$ ${\displaystyle \omega }$ ${\displaystyle n-1}$

${\displaystyle \int _{M}\mathrm {d} \omega =\int _{\partial M}\omega .}$

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. ${\displaystyle M}$ ${\displaystyle M}$

Manifold with corners

definition

A cube is a manifold with corners

Let be the set of all points of where all coordinates are nonnegative: ${\displaystyle {\overline {\mathbb {R} _{+}^{n}}}}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle {\overline {\mathbb {R} _{+}^{n}}}=\{(x_{1},\dotsc ,x_{n})\in \mathbb {R} ^{n}:x_{1}\geq 0,\dotsc ,x_{n}\geq 0\}.}$

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. ${\displaystyle \mathbb {H} ^{n}}$ ${\displaystyle M}$ ${\displaystyle {\overline {\mathbb {R} _{+}^{n}}}}$ ${\displaystyle M}$ ${\displaystyle (U,\phi )}$ ${\displaystyle U\subset M}$ ${\displaystyle M}$ ${\displaystyle \phi \colon U\to {\tilde {U}}\subset {\overline {\mathbb {R} _{+}^{n}}}}$ ${\displaystyle (U,\phi )}$ ${\displaystyle (V,\psi )}$ ${\displaystyle \phi \circ \psi ^{-1}\colon \psi (U\cap V)\to \phi (U\cap V)}$

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. ${\displaystyle M}$

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 . ${\displaystyle {\overline {\mathbb {R} _{+}^{n}}}}$ ${\displaystyle \mathbb {H} ^{n}}$

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 .