Revisiting Arnold's topological proof of the Morse index theorem

TL;DR

This paper revisits Arnold's topological proof of the Morse index theorem for Riemannian manifolds, emphasizing symplectic methods and expanding on the original ideas to provide a clearer exposition.

Contribution

It offers a detailed, self-contained exposition of the Morse index theorem using the Maslov index and symplectic techniques, building upon Arnold's foundational work.

Findings

Clarifies the symplectic approach to the Morse index theorem

Expands Arnold's original proof with detailed explanations

Provides a self-contained exposition for better understanding

Abstract

We give an exposition of the Morse Index Theorem in the Riemannian case in terms of the Maslov index, following and expanding upon Arnold's seminal paper. We emphasize the symplectic arguments in the proof and aim to be as self-contained as possible.