(i) Define geodesic curvature and state the Gauss-Bonnet theorem.

(ii) Let $\alpha: I \rightarrow \mathbb{R}^{3}$ be a closed regular curve parametrized by arc-length, and assume that $\alpha$ has non-zero curvature everywhere. Let $n: I \rightarrow S^{2} \subset \mathbb{R}^{3}$ be the curve given by the normal vector $n(s)$ to $\alpha(s)$. Let $\bar{s}$ be the arc-length of the curve $n$ on $S^{2}$. Show that the geodesic curvature $k_{g}$ of $n$ is given by

$$k_{g}=-\frac{d}{d s} \tan ^{-1}(\tau / k) \frac{d s}{d \bar{s}},$$

where $k$ and $\tau$ are the curvature and torsion of $\alpha$.

(iii) Suppose now that $n(s)$ is a simple curve (i.e. it has no self-intersections). Show that $n(I)$ divides $S^{2}$ into two regions of equal area.