Difference between revisions of "Submanifold (differential sense)"
From Diffgeom
m (1 revision) |
|
(No difference)
| |
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 
.
