I have just me this problem in my class on smooth manifolds from Lee's introduction to smooth manifolds, from the chapter on the tangent bundle stating the following:
Let $M, N$ be smooth manifolds, with $M$ being connected. Now we have a smooth map $ F : M \to N $ such that its push forward is the zero map. We are to show that the map $F$ is constant
I thought about it for a while, I figured maybe I should assume to get contradiction the map is not constant that would entail that in a connected neighborhood of M the coordinate representation of this map is non constant so due to smoothness some derivation on it is not zero, but how would I connect this with the push forward known to be the zero map? This is where I am stcuk. I thank all helpers on this