Let $S$ be a closed, connected regular surface, so that any two points of $S$ can be connected by a geodesic. Suppose that $\phi: S \to S$ is a smooth map with the property that it sends geodesics to geodesics, i.e., for every geodesic $\gamma(t)$, the curve $(\phi\circ\gamma)(t)$ is again a geodesic.
Prove that if there exists $p \in S$ such that $\phi(p) = p$ and $d\phi(p) = \operatorname{Id}$, then $\phi$ is the identity map.
I understand how knowing $\phi(p) = p$ is helpful but do not see how $d\phi(p) = \operatorname{Id}$ helps. Further, I am confused on how to put these conditions together to prove the statement. Please any help would be appreciated.