For \(n\in \mathbb{N},\) give a definition of geodesic \(n\)-gon which generalizes the notion of geodesic triangle. For convenience, we will call a geodesic \(n\)-gon simple if it is simple as a piecewise smooth curve and the exterior angles \(\vartheta_i\) satisfy \(-\pi<\vartheta_i<\pi\) for all \(i\in\{1,\dots n\}.\) For each of the following, consider only geodesic \(n\)-gons in the image \(F(U)\) of a local parametrization \(F:U\rightarrow M.\)

1. Does there exist a simple geodesic \(1\)-gon on the sphere?

2. State and prove a generalization of Corollary 4.39 for simple geodesic \(n\)-gons.

3. State and prove a further corollary of part (b), when the Gauss curvature \(K\) is constant.

Use Exercise \(1\) to derive an upper bound and a lower bound on the area of a simple geodesic \(n\)-gon in the image of a local parametrization \(F:U\rightarrow M\) when the Gauss curvature \(K\) is constant. Using these bounds, determine each of the following:

1. Given \(K=0,\) and \(n<3,\) does there exist a simple geodesic \(n\)-gon in \(F(U)\)?

2. Given \(K<0\) a constant and \(n<3,\) does there exist a simple \(n\)-gon in \(F(U)\)?

There exist geodesics on the cylinder and torus admit self-intersections and pairs of geodesics that intersect more than once, why are these not counterexamples to the result shown in part (a)?

Prove that a compact surface \(M\subset\mathbb{R}^3\) contains an open set of elliptic points.