Let $S \subset \mathbb{R}^{3}$ be a surface.

(a) In the case where $S$ is compact, define the Euler characteristic $\chi$ and genus $g$ of $S$.

(b) Define the notion of geodesic curvature $k_{g}$ for regular curves $\gamma: I \rightarrow S$. When is $k_{g}=0$ ? State the Global Gauss-Bonnet Theorem (including boundary term).

(c) Let $S=\mathbb{S}^{2}$ (the standard 2-sphere), and suppose $\gamma \subset \mathbb{S}^{2}$ is a simple closed regular curve such that $\mathbb{S}^{2} \backslash \gamma$ is the union of two distinct connected components with equal areas. Can $\gamma$ have everywhere strictly positive or everywhere strictly negative geodesic curvature?

(d) Prove or disprove the following statement: if $S$ is connected with Gaussian curvature $K=1$ identically, then $S$ is a subset of a sphere of radius 1 .