a quick question:
Let $(M,g), (N,h) $ be pseudo-Riemannian manifolds, $\gamma:I \rightarrow M$ a curve.
$\gamma^{'}(t_0):= d \gamma \dfrac{\partial}{\partial t} |_{t_0}$
Let $F:M \rightarrow M$ be an isometry.
Is it true then, that
$(F \circ \gamma)^{'}(t_0)= d(F \circ \gamma) \dfrac{\partial}{\partial t}|_{t_0} =dF_{\gamma(t_0)} \gamma^{'}= F_{*} \gamma^{'}$
where $F_{*}$ denotes the push-forward.