Some Geometric Properties of the Bakry-Emery-Ricci Tensor

TL;DR

This paper explores the geometric and topological properties of the Bakry-Emery tensor, an extension of the Ricci tensor for manifolds with a measure, highlighting its behavior under various geometric conditions.

Contribution

It establishes topological implications for the Bakry-Emery tensor similar to Ricci curvature and analyzes its behavior under Riemannian submersions and measured Gromov-Hausdorff limits.

Findings

Topological consequences of positive Bakry-Emery tensor are similar to Ricci curvature.

Bakry-Emery tensor is nondecreasing under certain Riemannian submersions.

Relations between Bakry-Emery tensor and measured Gromov-Hausdorff limits are identified.

Abstract

The Bakry-Emery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the Bakry-Emery tensor. We show that the Bakry-Emery tensor is nondecreasing under a Riemannian submersion whose fiber transport preserves measures up to constants. We give some relations between the Bakry-Emery tensor and measured Gromov-Hausdorff limits.