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 set-theoretic image in the manifold. In other words, if and
are submanifolds with the same set-theoretic image, then there is a diffeomorphism
such that
.