The Existence of Infinitely Many Geometrically Distinct Non-Constant Prime Closed Geodesics on Riemannian Manifolds

2018/08/12 by Charles, Sergio
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1808.04017

Abstract

We enumerate a necessary condition for the existence of infinitely many geometrically distinct, non-constant, prime closed geodesics on an arbitrary closed Riemannian manifold M. That is, we show that any Riemannian metric on M admits infinitely many prime closed geodesics such that the energy functional E:ΛM→ℝ has infinitely many non-degenerate critical points on the free loop space ΛM of Sobolev class H¹=W^1,2. This result is obtained by invoking a handle decomposition of free loop space and using methods of cellular homology to study its topological invariants.

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