# Submanifold (differential sense)

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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. $f(p) = f(q) \implies p = q$
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 $f_1 = f_2 \circ g$.