The Bakry-Émery 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-Émery tensor. We show that the Bakry-Émery tensor is nondecreasing under a Riemannian submersion whose fiber transport preserves measures up to constants. We give some relations between the Bakry-Émery tensor and measured Gromov-Hausdorff limits.
Cite this article
John Lott, Some geometric properties of the Bakry-Émery-Ricci tensor. Comment. Math. Helv. 78 (2003), no. 4, pp. 865–883DOI 10.1007/S00014-003-0775-8