Define the geodesic curvature $k_{g}$ of a regular curve in an oriented surface $S \subset \mathbb{R}^{3}$. When is $k_{g}=0$ along a curve?

Explain briefly what is meant by the Euler characteristic $\chi$ of a compact surface $S \subset \mathbb{R}^{3}$. State the global Gauss-Bonnet theorem with boundary terms.

Let $S$ be a surface with positive Gaussian curvature that is diffeomorphic to the sphere $S^{2}$ and let $\gamma_{1}, \gamma_{2}$ be two disjoint simple closed curves in $S$. Can both $\gamma_{1}$ and $\gamma_{2}$ be geodesics? Can both $\gamma_{1}$ and $\gamma_{2}$ have constant geodesic curvature? Justify your answers.

[You may assume that the complement of a simple closed curve in $S^{2}$ consists of two open connected regions.]