Difference between revisions of "Submanifold (differential sense)"
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Latest revision as of 20:10, 18 May 2008
Definition
Let be a differential manifold. A submanifold of can be viewed as the following data: An abstract differential manifold along with a smooth map from to such that:
 The map is an immersion; in other words, the induced map on the tangent space at any point is injective
 The map is injective i.e.
 The map is a homeomorphism to its image
Note that when is compact, the third condition is redundant, because any injective map from a compact space to a Hausdorff space is an embedding.
To complete the definition, we need to observe that a submanifold is completely determined, upto diffeomorphism, by its settheoretic image in the manifold. In other words, if and are submanifolds with the same settheoretic image, then there is a diffeomorphism such that .