A short proof to the rigidity of volume entropy

TL;DR

This paper provides a concise proof of the rigidity of volume entropy, showing that manifolds with maximal volume entropy under Ricci curvature bounds are space forms, extending previous results by Ledrappier and Wang.

Contribution

It offers a shorter, more direct proof of a known rigidity theorem for volume entropy under Ricci curvature constraints.

Findings

Manifolds with maximal volume entropy are space forms.

The proof simplifies previous arguments for the rigidity theorem.

The result applies to closed manifolds with Ricci curvature bounded below.

Abstract

In this note we give a short proof to the rigidity of volume entropy. The result says that for a closed manifold with Ricci curvature bounded from below, if the universal cover has maximal volume entropy, then it is the space form. This theorem was first proved by F. Ledrappier and X. Wang in [1].