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!
I think I figured it out. So as $\phi$ is isometry it fixes the geodesic through $p,q$. Now, let $v=exp_p^{-1}(q)$. We have that $exp_p(tv)$ is the geodesic through connecting $p$ and $q$. By differentiation of $\phi \circ exp_p (tv)=exp_p(tv)$ we get
$$(d\phi)_{exp_p(tv)}(d \: exp_p)_{tv}v=(d \: exp_p)_{tv} v.$$
At $t=0$ we know $(d \: exp_p)_{0}=Id$ and $exp_p(0)=p$, so we get $(d\phi)_pv=v$, which is the required equality.