Simple proof of the global inverse function theorem via the Hopf--Rinow theorem

TL;DR

This paper presents a straightforward proof of Hadamard's global inverse function theorem by leveraging the Hopf--Rinow theorem from Riemannian geometry, simplifying the understanding of the theorem.

Contribution

It offers a simple, geometric proof of the global inverse function theorem using the Hopf--Rinow theorem, connecting differential topology with Riemannian geometry.

Findings

The proof simplifies the understanding of Hadamard's theorem.

It demonstrates the connection between inverse functions and Riemannian geometry.

The approach provides a new perspective on classical results.

Abstract

We explain that Hadamard's global inverse function theorem very simply follows from the Hopf--Rinow theorem in Riemannian geometry.