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I was reading these notes about submanifolds. The first theorem proved is this:

Theorem: Let $F: M_1 \rightarrow M_2$ be a morphism of smooth manifolds of dimensions $n_1, n_2$. Let $p \in M_1$, and suppose the rank of the derivative of $F$ at $p$ (that is, for any charts $(U,\phi)$ of $p$ and $(V,\psi)$ of $F(p)$, the rank of the derivative of $\psi \circ F \circ \phi^{-1}: U \rightarrow V$ at $\phi(p)$) is locally constant and equal to $k$. Then there exist charts $(U,\phi)$ of $p$ and $(V,\psi)$ of $F(p)$ such that for all $x = (x_1, ... , x_{n_1}) \in \phi U$, we have

$$\psi \circ F \circ \phi^{-1}(x) = (x_1, ... , x_k, 0, ... , 0)$$

This theorem is meant to motivate the following definitions:

Let $F: M \rightarrow N$ be a smooth morphism of constant rank everywhere (rank defined above). Assume the rank is equal to the dimension of $M$ (in particular, $\textrm{dim } M \leq \textrm{dim } N$). Then we say that $F$ is an immersion. We say that $F$ is an embedding if $F$ is an immersion and is a homeomorphism onto its image.

I think I understand what an embedding is. Another way of saying this is that there is a subset $X$ of $N$, a smooth manifold structure on $X$ in the subspace topology, such that the inclusion map $X \rightarrow N$ is smooth and the rank of the derivative at any point of $X$ is constant and equal to the dimension of $X$. Then an embedding would be a manifold $N$ together with a diffeomorphism $N \cong X$.

I'm less comfortable with the idea of an immersion. What is the intuition behind an immersion? If $F: M \rightarrow N$ is an immersion, is there a submanifold structure we can place on the image $F(M)$?

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In general, the image of an immersion is not a submanifold. For example, let $M$ be the disjoint union of two lines, and let $N$ be the real plane. Then the subset $\{xy=0\}\subset N$ is the image of an immersion $M\to N$.

However, immersions still have "good" properties that distinguish them from general smooth maps. Here are two examples of such properties:

I) Every immersion is a local embedding. That is, if $f:M\to N$ is an immersion and $p\in M$, then $p$ has a neighborhood $U$ such that $f|_U$ is an embedding. This is a consequence of the inverse function theorem.

II) An immersion $f:M\to N$ can be used to define extra structure on $M$. If, for example, $N$ is equipped with a Riemannian metric $g$, then the pullback $f^*g$ is a Riemannian metric on $M$.

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  • $\begingroup$ Any comments on immersed submanifolds? $\endgroup$ – user636532 Jul 21 '19 at 11:40

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