Submanifold (differential sense)
- 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 .