Let $S \subset \mathbb{R}^{3}$ be a surface.

(a) Define the exponential map $\exp _{p}$ at a point $p \in S$. Assuming that exp ${ }_{p}$ is smooth, show that $\exp _{p}$ is a diffeomorphism in a neighbourhood of the origin in $T_{p} S$.

(b) Given a parametrization around $p \in S$, define the Christoffel symbols and show that they only depend on the coefficients of the first fundamental form.

(c) Consider a system of normal co-ordinates centred at $p$, that is, Cartesian coordinates $(x, y)$ in $T_{p} S$ and parametrization given by $(x, y) \mapsto \exp _{p}\left(x e_{1}+y e_{2}\right)$, where $\left\{e_{1}, e_{2}\right\}$ is an orthonormal basis of $T_{p} S$. Show that all of the Christoffel symbols are zero at $p$.