Poincaré’s Closed Geodesic on a Convex Surface

We present a new proof for the existence of a simple closed geodesic on a convex surface M . This result is due originally to Poincaré. The proof uses the 2k-dimensional Riemannian manifold kΛM = (briefly) Λ of piecewise geodesic closed curves on M with a fixed number k of corners, k chosen sufficiently large. In Λ we consider a submanifold ≈ Λ0 formed by those elements of Λ which are simple regular and divide M into two parts of equal total curvature 2π. The main burden of the proof is to show that the energy integral E, restricted to ≈ Λ0, assumes its infimum. At the end we give some indications of how our methods yield a new proof also for the existence of three simple closed geodesics on M .