Math 6510 • Differentiable Manifolds

Let \(\Sigma\) be an oriented genus 2 surface smoothly embedded in \(\mathbb{R}^3\).
\begin{enumerate}
\item Describe the Gauss map \(g:\Sigma\to S^2\) and define the Gaussian curvature at a
point of \(\Sigma\) in terms of \(g\).
\item State the Gauss-Bonnet theorem for this situation.
\item Prove that \(\Sigma\) has (many) saddle points, i.e. points of negative curvature.
\end{enumerate}