 Solution

For $v\in T_{p}S$ sufficiently small, let ${\gamma }_v$ be the geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. Define ${\exp }_p(v)={\gamma }_v(1)$. Geodesic uniqueness and rescaling give ${\gamma }_v(t)={\exp }_p(tv)$, soThus its derivative at zero is the identity, an isomorphism. Assuming the stated smoothness, the inverse function theorem proves ${\exp }_p$ is a diffeomorphism between neighbourhoods of zero and $p$.