(a) Let $X$ be a compact surface (without boundary) in $\mathbb{R}^{3}$. State the global GaussBonnet formula for $X$, identifying all terms in the formula.

(b) Let $X \subset \mathbb{R}^{3}$ be a surface. Define what it means for a curve $\gamma: I \rightarrow X$ to be a geodesic. State a theorem concerning the existence of geodesics and define the exponential map.

(c) Let $\psi: X \rightarrow Y$ be an isometry and let $\gamma$ be a geodesic. Show that $\psi \circ \gamma$ is a geodesic. If $K_{X}$ denotes the Gaussian curvature of $X$, and $K_{Y}$ denotes the Gaussian curvature of $Y$, show that $K_{Y} \circ \psi=K_{X}$.

Now suppose $\psi: X \rightarrow Y$ is a smooth map such that $\psi \circ \gamma$ is a geodesic for all $\gamma$ a geodesic. Is $\psi$ necessarily an isometry? Give a proof or counterexample.

Similarly, suppose $\psi: X \rightarrow Y$ is a smooth map such that $K_{Y} \circ \psi=K_{X}$. Is $\psi$ necessarily an isometry? Give a proof or counterexample.