Editing Submanifold (section)

===Embedded submanifolds===

An '''embedded submanifold''' (also called a '''regular submanifold'''), is an immersed submanifold for which the inclusion map is a [[topological embedding]]. That is, the submanifold topology on <math>S</math> is the same as the subspace topology.

Given any [[embedding]] <math>f: N\rightarrow M</math> of a manifold <math>N</math> in <math>M</math> the image <math>f(N)</math> naturally has the structure of an embedded submanifold. That is, embedded submanifolds are precisely the images of embeddings.

There is an intrinsic definition of an embedded submanifold which is often useful. Let <math>M</math> be an <math>n</math>-dimensional manifold, and let <math>k</math> be an integer such that <math>0 \leq k \leq n</math>. A <math>k</math>-dimensional embedded submanifold of <math>M</math> is a subset <math>S \subset M</math> such that for every point <math>p \in S</math> there exists a [[chart (topology)|chart]]  <math>U \subset M, \varphi : U \rightarrow \mathbb{R}^n</math> containing <math>p</math> such that <math>\varphi(S \cap U)</math> is the intersection of a <math>k</math>-dimensional [[plane (mathematics)|plane]] with <math>\varphi(U)</math>. The pairs <math>(S\cap U, \varphi\vert_{S\cap U})</math> form an [[atlas (topology)|atlas]] for the differential structure on <math>S</math>.

[[Alexander's theorem]] and the [[Schoenflies theorem|Jordan–Schoenflies theorem]] are good examples of smooth embeddings.