Two smooth maps are related by another smooth map

Question

Let $U$ be an open subset of a smooth manifold $M$ and $f,g:U\rightarrow M$ be smooth maps.

Can we always (When can we) find a smooth map $\Phi: U\rightarrow U$ such that $g=f\circ \Phi$??

I have nothing much to say on this. Just out of curiosity.

Answer 1

I don't know the general case, but here are some thoughts:

• If $f$ is a constant map to a point in $M$, there are many maps $g$ which we cannot relate to $g$ in the described way.
• If $f\colon U\to U$ is a diffeomorphism of $U$, then there is an inverse $f^{-1}\colon U\to U$ such that $f\circ f^{-1} = \mathrm{id}_U$. Obviously, we can then relate $f$ to any $g\colon U\to U$ by setting $\Phi = f^{-1} \circ g$. Then we get $g=f\circ f^{-1} \circ g = f\circ \Phi$
• If $f$ and $g$ have disjoint images, there is no chance to find $\Phi$. More precisely, if you have a $\Phi$, then $\Im (g) = \Im(f\circ \Phi) \subset \Im(f)$. So that would be a very easy obstruction.

Note that those points do not really incorporate the smoothness of those maps.