Image of a submanifold under a smooth surjective map
Let $f : M \rightarrow N$ be a smooth surjective map, where $M,N$ are smooth manifolds. I have a submanifold $M' \subset M$. If we restrict the map $f|_{M'}:M' \rightarrow N$, is $f(M')$ also a submanifold?
I'm not sure how to give $f(M')$ a smooth structure. Is there a way to do that, or can we find a counter example?
Solution 1:
If $M=\mathbf{R}^3$, $N=\mathbf{R}^2$, $f(x,y,z)=(x,y)$ and $M'=\{x=z=0\}\cup\{y=0,z=1\}$ (which is the union of two disjoint lines), then $f(M')=\{x=0\}\cup\{y=0\}$ is the union of two intersecting lines, and thus not a manifold.