Spherical complexities, with applications to closed geodesics

TL;DR

This paper introduces new numerical invariants for topological spaces that help estimate critical points of symmetric functions on spheres, leading to novel results on the existence of closed geodesics in Finsler manifolds.

Contribution

It develops new homotopy invariants akin to Lusternik-Schnirelmann category, applicable to loop and sphere spaces, and applies them to prove existence of closed geodesics.

Findings

New invariants provide lower bounds for critical orbits.

Application to closed geodesics on Finsler manifolds.

Proved existence results under curvature and pinching conditions.

Abstract

We construct and discuss new numerical homotopy invariants of topological spaces that are suitable for the study of functions on loop and sphere spaces. These invariants resemble the Lusternik-Schnirelmann category and provide lower bounds for the numbers of critical orbits of SO(n)-invariant functions on spaces of n-spheres in a manifold. Lower bounds on these invariants are derived using weights of cohomology classes. As an application, we prove new existence results for closed geodesics on Finsler manifolds of positive flag curvature satisfying a pinching condition.