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
Best Answer
Let $p\in M$, let $f\in C^\infty\left(N\right)$, and let $X\in T_pM$. By assumption, $\left(F_*X\right)\left(f\right)=X\left(f\circ F\right)=0$. Let $\left(U,\varphi\right)$ be a smooth chart containing $p$. Then
$$X=\sum_iX^i\left.\frac{\partial}{\partial x^i}\right|_p=\sum_iX^i\left(\varphi^{-1}\right)_*\left.\frac{\partial}{\partial x^i}\right|_{\varphi\left(p\right)},$$
which implies that
$$\left(\sum_iX^i\left(\varphi^{-1}\right)_*\left.\frac{\partial}{\partial x^i}\right|_{\varphi\left(p\right)}\right)\left(f\circ F\right)=\sum_iX^i\left.\frac{\partial}{\partial x^i}\right|_{\varphi\left(p\right)}\left(f\circ F\circ\varphi^{-1}\right)=0,$$
which in turn, by basic calculus, implies that $F$ is constant on $U$, i.e., $F\left(x\right)=c$ for some $c\in N$ and every $x\in U$.
Since $M$ is connected, it is the case that $M$ is path connected. Let $q\in M$ and let $\gamma:\left[0,1\right]\to M$ be a path connecting $p$ and $q$. Since, as above, $F$ is constant on each smooth chart $\left(U_{\gamma\left(x\right)},\varphi_{\gamma\left(x\right)}\right)$ containing $\gamma\left(x\right)$ for every $x\in\left[0,1\right]$, it is the case that $F\equiv c$ on $M$ since $F\left(p\right)=c$ and $\gamma$ is continuous.