# Morse Index bound of simple closed geodesics on 2-spheres and strong Morse Inequalities

## TL;DR

This paper provides a Morse-theoretic framework to analyze simple closed geodesics on 2-spheres, establishing bounds on their Morse indices and deriving strong Morse inequalities for the length functional.

## Contribution

It introduces a novel Morse-theoretic characterization of simple closed geodesics on Riemannian 2-spheres and proves the existence of geodesics with specific Morse indices under generic metrics.

## Key findings

- Existence of simple closed geodesics with Morse indices 1, 2, and 3
- Strong Morse inequalities for the length functional on simple closed curves
- Morse-theoretic characterization of geodesics on 2-spheres

## Abstract

We give a Morse-theoretic characterization of simple closed geodesics on Riemannian $2$-spheres. On any Riemannian $2$-sphere endowed with a generic metric, we show there exists a simple closed geodesic with Morse index $1$, $2$ and $3$. In particular, for an orientable Riemannian surface we prove strong Morse inequalities for the length functional applied to the space of simple closed curves.