I'm interested in the proof of the following proposition, given in 12.36 of John M. Lee's Introduction to Smooth Manifolds:
Suppose $M$ is a smooth manifold with or without boundary and $V\in \mathfrak{X}(M)$. If $\partial M\not=\varnothing$, assume in addition thaat $V$ is tangent to $\partial M$. Let $\theta$ be the flow of $V$. For any smooth covariant tensor field $A$ and any $(t_0,p)$ in the domain of $\theta$,
$$(*)\qquad\frac{d}{dt}\bigg|_{t=t_0}(\theta_t^*A)_p=\big(\theta_{t_0}^*(\mathscr{L}_VA)\big)_p$$
My only concern has to do with the first line of the proof:
After expanding the definitions of the pullbacks in $(*)$, we see that we have to prove
$$\frac{d}{dt}\bigg|_{t=t_0}d(\theta_t)_p^*\big(A_{\theta_t(p)}\big)=d(\theta_{t_0})_p^*\big((\mathscr{L}_VA)_{\theta_{t_0}(p)}\big)$$
Why proving this statement implies $(*)$? What is the reasoning behind this? Thanks in advance for your attention.
Best Answer
As Lee says, this is just the definition of the pullback: you know that, if $A$ is a $k$-covariant tensor and $X_1,\dots,X_k\in T_pM$,
$$(\theta_t^*A)_p(X_1,\dots,X_k):=A_{\theta_t(p)}((d\theta_t)_p(X_1),\dots,(d\theta_t)_p(X_k))=(d\theta_t)_p^*(A_{\theta_t(p)}),$$
where the last star has to be understood as the linear pullback (if you have a multilinear map $L:F\times\dots\times F\to\mathbb{R}$ and a linear map $f:E\to F$, then you define a multilinear map $f^*L:E\times\dots\times E\to\mathbb{R}$ by $f^*L(V_1,\dots,V_k)=L(f(V_1),\dots,f(V_k))$).