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
  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.