 Solution

For $v\in T_{p}S$ for which the geodesic exists up to time one, the exponential map is ${\exp }_p(v)={\gamma }_v(1)$, where ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$. The initial velocity determines the geodesic uniquely. Since $d({\exp }_p{)}_0$ is the identity, this map is a local diffeomorphism near zero.

Choose an orthonormal basis of $T_{p}S$ and set $e(\theta )=(\cos \theta ,\sin \theta )$. The geodesic polar coordinates arein a normal neighbourhood, away from its centre. For sufficiently small positive $r$, the geodesic circle iswithin that neighbourhood. Outside the injectivity range the distance sphere need not have this smooth coordinate description.