Let $I \subset \mathbb{R}$ be an interval, and $S \subset \mathbb{R}^{3}$ be a surface. Assume that $\alpha: I \rightarrow S$ is a regular curve parametrised by arc-length. Define the geodesic curvature of $\alpha$. What does it mean for $\alpha$ to be a geodesic curve?

State the global Gauss-Bonnet theorem including boundary terms.

Suppose that $S \subset \mathbb{R}^{3}$ is a surface diffeomorphic to a cylinder. How large can the number of simple closed geodesics on $S$ be in each of the following cases?

(i) $S$ has Gaussian curvature everywhere zero;

(ii) $S$ has Gaussian curvature everywhere positive;

(iii) $S$ has Gaussian curvature everywhere negative.

In cases where there can be two or more simple closed geodesics, must they always be disjoint? Justify your answer.

[A formula for the Gaussian curvature of a surface of revolution may be used without proof if clearly stated. You may also use the fact that a piecewise smooth curve on a cylinder without self-intersections either bounds a domain homeomorphic to a disc or is homotopic to the waist-curve of the cylinder.]