Submanifold

This article covers submanifolds of abstract manifolds. Submanifolds des are covered in the article Submanifolds des . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

In the differential geometry or differential topology one is submanifold a subset of a manifold associated with the card is compatible to the manifold.

definition

A subset of a -dimensional manifold is a -dimensional (embedded) submanifold if and only if there is a map of with for every point such that the equation ${\displaystyle N}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle k}$ ${\displaystyle p\in N}$ ${\displaystyle (\varphi ,U)}$ ${\displaystyle M}$ ${\displaystyle p\in U}$

${\displaystyle \varphi (N\cap U)=(\mathbb {R} ^{k}\times \{0\}^{n-k})\cap \varphi (U)}$

is satisfied. Each (embedded) submanifold is again a manifold with the maps just given and the induced subspace topology .

There is also a more general definition of immersed submanifolds , these are defined as the image of an injective immersion of a manifold. If submanifolds are spoken of without further addition, however, embedded submanifolds are usually meant.

Orientability and orientation

→ Main article : Orientation (mathematics)

A submanifold is said to be orientable if there is an associated oriented atlas . This is exactly the case when all cards are oriented in the same way. ${\displaystyle N}$

The equi- orientation of atlases thus represents an equivalence relation and we denote the equivalence class as the orientation of M. Both can be combined as a pair . ${\displaystyle \sigma }$ ${\displaystyle (M,\sigma )}$

Examples

Standard examples of submanifolds are the open sets of the (same-dimensional) or the equator of a sphere (lower-dimensional). In general, the archetype of a regular value of a function is a submanifold of , see theorem of regular value . ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle f\colon \,M\to X}$ ${\displaystyle M}$

A Möbius strip parameterized with two cards.

Another example is the Möbius strip. It can also be shown that it is not orientable. The standard proof of this is to run a normal unit vector along the surface. The resulting image is not continuous and therefore the Möbius strip cannot be oriented.

Alternatively, one can show that there is no oriented atlas. To do this, it makes sense to first look for an atlas and show that it is not oriented. However, there could be another atlas which could be oriented. Then one card from each would have to be oriented either in the same way or in opposite directions. However, this is not possible because there are cards that have to be the same in sections and opposite in sections. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$

literature

• Klaus Jänich : Vector analysis. 2nd Edition. Springer-Verlag, Berlin et al. 1993, ISBN 3-540-57142-6 .
• R. Abraham, JE Marsden , T. Ratiu: Manifolds, Tensor Analysis, and Applications (= Applied Mathematical Sciences 75). 2nd edition. Springer-Verlag, New York NY et al. 1988, ISBN 0-387-96790-7 .
• O. Forster: Analysis III, advanced course in mathematics . Vieweg + Teubner Verlag Springer Fachmedien Wiesbaden 2012