Trying to prove the following proposition which is left as exercise:
Proposition 17.11 : (Pullback of a sum and a product). Let $F : N \to M$ be a $C^{\infty}$ map of manifolds. Suppose $\omega, \tau \in \Omega^1(M)$ and $g \in C^{\infty}(M)$. Then
(1) $F^*(\omega + \tau) = F^* \omega + F^* \tau$
(2) $F^*(g\omega) = (F^* g) (F^* \omega)$
I managed to prove (1) I think, but I can't manage to prove (2) despite the clue given by the textbook.
For (1): Let $X_{p} \in T_p(N)$ $$ F^*(\omega + \tau)(X_p) = (\omega + \tau)(F_* X_p) = \omega(F_* X_p) + \tau(F_* X_p) = (F^*\omega)(X_p) + (F^*\tau)(X_p) $$
So both LHS and RHS act in the same way on a vector.
For (2): I've tried a similar approach
$$ F^*(g\omega)(X_p) = g\omega(F_*X_p) $$
But I'm not sure what to do from here
Can you help? I'm sure it's probably simple, but I can't figure.
It would be helpful to write the pullback definition more explicitly. If $\omega$ is a smooth $1$-form on $M$ and $F:N \to M$ is smooth, then $F^*{\omega}$ is a $1$-form defined by: for all $p \in N$ and all $X_p \in T_pN$, \begin{align} (F^*\omega)(p)(X_p) &:= \omega(F(p))[F_{*,p} X_p] \end{align}
I believe you're having issues because you havent indicated where the forms are being evaluated. Recall that since $\omega$ is a one-form on $M$ it assigns to each point $q \in M$, an element $\omega(q) \in T^*_qM$. So, you first have to evaluate a one-form at a point of the manifold, and after that apply the whole thing to a tangent vector; the final result being a real number.
So, for all $p \in N$ and all $X_p \in T_pN$, we have \begin{align} \bigg(F^*(g \omega)(p) \bigg)(X_p) &:= \bigg((g \omega)(F(p)) \bigg)[F_{*,p} X_p] \\ &:= \bigg(g(F(p)) \cdot \omega(F(p)) \bigg) [F_{*,p} X_p] \\ &:= g(F(p)) \cdot \bigg( \omega(F(p))[F_{*,p} X_p]\bigg) \\ &:= (F^*g)(p) \cdot (F^*\omega)(p)[X_p] \\ &:= \bigg( (F^*g)(p) \cdot (F^*\omega)(p) \bigg)[X_p] \\ &:= \bigg( \left( F^*g \cdot F^* \omega \right) (p) \bigg)[X_p] \end{align} Thus, it follows that $F^*(g \omega) = F^*g \cdot F^* \omega$. I'll leave it to you to figure out why I put in so many equal signs. Each of them is true either by definition of pullback $^*$, or by definition of product of a function and form, or by definition of scalar multiplication in the cotangent space $T_{F(p)}^*M$ etc. Hopefully the bracketing makes it clearer what is being evaluated on what (although for convenience, I may have dropped some brackets).