On the geometry of Riemannian manifolds with density

TL;DR

This paper develops a new geometric framework for manifolds with density using affine connections, leading to generalized comparison theorems, rigidity results, and insights into holonomy groups, expanding classical Riemannian geometry concepts.

Contribution

It introduces a novel approach based on affine connections for manifolds with density, generalizing key theorems and exploring holonomy and splitting phenomena.

Findings

Generalized Myers' theorem and Cheng's diameter rigidity.

Holonomy groups are broader than Riemannian holonomy but retain some structure.

Warped or twisted product splittings characterize rigidity phenomena.

Abstract

We introduce a new geometric approach to a manifold equipped with a smooth density function that takes a torsion-free affine connection, as opposed to a weighted measure or Laplacian, as the fundamental object of study. The connection motivates new versions of the volume and Laplacian comparison theorems that are valid for the 1-Bakry-Emery Ricci tensor, a weaker assumption than has previously been considered in the literature. As applications we prove new generalizations of Myers' theorem and Cheng's diameter rigidity result. We also investigate the holonomy groups of the weighted connection. We show that they are more general than the Riemannian holonomy, but also exhibit some of the same structure. For example, we obtain a generalization of the de Rham splitting theorem as well as new rigidity phenomena for parallel vector fields. A general feature of all of our rigidity results is that warped or twisted product splittings are characterized, as opposed to the usual isometric products.