Some geometric properties of the Bakry-Émery-Ricci tensor

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Abstract
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.