Submanifold: Difference between revisions

→‎Embedded submanifolds: add links to "Alexander's theorem" and "Schoenflies theorem"
(→‎Embedded submanifolds: add links to "Alexander's theorem" and "Schoenflies theorem")
There is an intrinsic definition of an embedded submanifold which is often useful. Let ''M'' be an ''n''-dimensional manifold, and let ''k'' be an integer such that 0 ≤ ''k'' ≤ ''n''. A ''k''-dimensional embedded submanifold of ''M'' is a subset ''S'' ⊂ ''M'' such that for every point ''p'' ∈ ''S'' there exists a [[chart (topology)|chart]] (''U'' ⊂ ''M'', φ : ''U'' → '''R'''<sup>''n''</sup>) containing ''p'' such that φ(''S'' ∩ ''U'') is the intersection of a ''k''-dimensional [[plane (mathematics)|plane]] with φ(''U''). The pairs (''S'' ∩ ''U'', φ|<sub>''S'' ∩ ''U''</sub>) form an [[atlas (topology)|atlas]] for the differential structure on ''S''.
[[Alexander's theorem]] and the [[Schoenflies theorem|Jordan-Schoenflies theorem]] are good examples of smooth embeddings.
===Other variations===