Submanifold: Difference between revisions

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Given any [[embedding]] ''f'' : ''N'' → ''M'' of a manifold ''N'' in ''M'' the image ''f''(''N'') 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 ''M'' be an ''n''-dimensional manifold, and let ''k'' be an integer such that 0 &le; ''k'' &le; ''n''. A ''k''-dimensional embedded submanifold of ''M'' is a subspace ''S'' &sub; ''M'' such that for every point ''p'' &isin; ''S'' there exists a [[chart (topology)|chart]] (''U'' &sub; ''M'', &phi; : ''U'' &rarr; '''R'''<sup>''n''</sup>) containing ''p'' such that &phi;(''S'' &cap; ''U'') is the intersection of a ''k''-dimensional [[plane (mathematics)|plane]] with &phi;(''U''). The pairs (''S'' &cap; ''U'', &phi;|<sub>''S'' &cap; ''U''</sub>) form an [[atlas (topology)|atlas]] for the differential structure on ''S''.
 
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