Commentarii Mathematici Helvetici

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Volume 78, Issue 4, 2003, pp. 865–883
DOI: 10.1007/s00014-003-0775-8

Published online: 2003-12-31

Some geometric properties of the Bakry-Émery-Ricci tensor

John Lott[1]

(1) University of California, Berkeley, USA

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.

Keywords: Ricci tensor, metric-measure space, Riemannian submersion

Lott John: Some geometric properties of the Bakry-Émery-Ricci tensor. Comment. Math. Helv. 78 (2003), 865-883. doi: 10.1007/s00014-003-0775-8