Submanifold (differential sense)

Definition

Let $M$ be a differential manifold. A submanifold of $M$ can be viewed as the following data: An abstract differential manifold $N$ along with a smooth map $f$ from $N$ to $M$ such that:

1. The map $f$ is an immersion; in other words, the induced map $(Df)_{p}$ on the tangent space at any point $p\in M$ is injective
2. The map is injective i.e. $f(p)=f(q)\implies p=q$
3. The map is a homeomorphism to its image

Note that when $N$ 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 $f_{1}:N_{1}\to M$ and $f_{2}:N_{2}\to M$ are submanifolds with the same set-theoretic image, then there is a diffeomorphism $g:N_{1}\to N_{2}$ such that $f_{1}=f_{2}\circ g$.