I got stuck on the following questions. Can anyone give me idea how to proceed?
Suppose $M$ is a Riemannian manifold and $\phi: M \to M$ an isometry map. If $\phi(p)=p$ and $\phi(q)=q$ prove that $(d\phi)_p exp_p^{-1}(q)=exp_p^{-1}(q)$. You can assume that $exp_p$ is well defined on its inputs.
Thanks!