Topological structure of non-contractible loop space and closed geodesics on real projective spaces with odd dimensions

TL;DR

This paper explores the topological structure of non-contractible loop spaces on odd-dimensional real projective spaces using advanced algebraic topology tools, and applies these findings to closed geodesics.

Contribution

It introduces a novel analysis of non-contractible loop spaces on real projective spaces with odd dimensions, linking topology to geodesic properties.

Findings

Topological description of non-contractible loop space components

Resonance identity for prime closed geodesics

Conditions for finiteness of prime closed geodesics

Abstract

In this paper, we use Chas-Sullivan theory on loop homology and Leray-Serre spectral sequence to investigate the topological structure of the non-contractible component of the free loop space on the real projective spaces with odd dimensions. Then we apply the result to get the resonance identity of non-contractible homologically visible prime closed geodesics on such spaces provided the total number of distinct prime closed geodesics is finite.