Problem 9-1 in Lee's "Introduction to Riemannian Manifolds", the author asks the following:
Let $(M,g)$ be an oriented Riemannian 2-manifold with nonpositive Gaussian curvature everywhere. Prove that there are no geodesic polygons with exactly 0,1, or 2 ordinary vertices. Give examples of all three if the curvature hypothesis is not satisfied.
I was able to prove the first part of the problem by a simple application of the Gauss-Bonnet formula. I was also able to come up with counter examples for the second part in the case of 0 or 2 ordinary vertices by considering the round 2-sphere (a great circle for the 0 case and two half great circles for the 2 case). I am stuck however coming up with an example for the 1 vertex case. Does anyone have any suggestions?