I’m trying to prove equation (1.25) of Remark 1.22 on page 12 of Bennett Chow et al.’s book “Hamilton’s Ricci Flow”:
$$\frac{\partial}{\partial t} (\phi(t)^*\alpha) = \mathcal L_{X(t)}\phi(t)^*\alpha \tag{1.25}$$
where $$ X(t_0) := \left.\frac{\partial}{\partial t}\right\vert_{t=t_0} \left( \phi(t_0)^{-1}\circ \phi(t)\right) = \left(\phi(t_0)^{-1}\right)_*\left.\frac{\partial}{\partial t}\right\vert_{t=t_0} \phi(t)$$
However, I’m struggling to write down the Lie derivative $\mathcal L_{X(t)}$ in terms of $\phi (t)$. I reason that if I fix $t=t_0$, I get a vector field $X(t_0)$ and, from it, a flow $\psi(t)$ such that $\left.\frac{d}{dt}\psi(t)\right\vert_{t=0} = X(t_0) = \left(\phi(t_0)^{-1}\right)_*\left.\frac{\partial}{\partial t}\right\vert_{t=t_0} \phi(t)$. This means that the RHS of equation (1.25) should be
$$\mathcal L_{X(t_0)} \phi(t_0)^* \alpha = \lim_{\epsilon \rightarrow 0} \frac{1}{\epsilon}\left(\psi(\epsilon)^*\phi(t_0)^* \alpha - \phi(t_0)^*\alpha\right)$$
Unfortunately, with this approach, I then get stuck with the term $\psi(\epsilon)^*$.