Poincaré’s closed geodesic on a convex surface

Abstract

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 ${_{k}\Lambda M} = \operatorname {(briefly)} \Lambda$ of piecewise geodesic closed curves on $M$ with a fixed number $k$ of corners, $k$ chosen sufficiently large. In $\Lambda$ we consider a submanifold $\overset {\approx }{\Lambda }_{0}$ formed by those elements of $\Lambda$ which are simple regular and divide $M$ into two parts of equal total curvature $2\pi$. The main burden of the proof is to show that the energy integral $E$, restricted to $\overset {\approx }{\Lambda }_{0}$, 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 $M$.