Define what is meant by a geodesic. Let $S \subset \mathbb{R}^{3}$ be an oriented surface. Define the geodesic curvature $k_{g}$ of a smooth curve $\gamma: I \rightarrow S$ parametrized by arc-length.

Explain without detailed proofs what are the exponential map $\exp _{p}$ and the geodesic polar coordinates $(r, \theta)$ at $p \in S$. Determine the derivative $d\left(\exp _{p}\right)_{0}$. Prove that the coefficients of the first fundamental form of $S$ in the geodesic polar coordinates satisfy

$$E=1, \quad F=0, \quad G(0, \theta)=0, \quad(\sqrt{G})_{r}(0, \theta)=1$$

State the global Gauss-Bonnet formula for compact surfaces with boundary. [You should identify all terms in the formula.]

Suppose that $S$ is homeomorphic to a cylinder $S^{1} \times \mathbb{R}$ and has negative Gaussian curvature at each point. Prove that $S$ has at most one simple (i.e. without selfintersections) closed geodesic.

[Basic properties of geodesics may be assumed, if accurately stated.]