Math 6510 • Differentiable Manifolds

True or false (give a short explanation or a counterexample): Let \(\Sigma\) be a
smooth, closed, oriented surface. Assume that \(\Sigma\) has a Riemannian metric of
constant curvature \(K\). Suppose that \(\gamma\subset\Sigma\) is a closed embedded
geodesic loop which separates \(\Sigma\). (That is, \(\Sigma-\gamma\) has two components.)
\begin{enumerate}
\item Then \(K\le0\).
\item Then \(K\ne0\).
\end{enumerate}