Some comparison results for one extension of the Bakry-\'Emery-Ricci tensor

TL;DR

This paper explores extensions of the Bakry-Émery-Ricci tensor and drifted Laplacian, establishing comparison theorems and geometric results like Myers and splitting theorems for broader contexts.

Contribution

It introduces generalized versions of the Bakry-Émery-Ricci tensor and drifted Laplacian, proving key geometric comparison theorems and classical results in this extended setting.

Findings

Proved a mean curvature comparison theorem for the generalized tensors.

Established a Myers-type theorem under the new framework.

Generalized the Cheeger-Gromoll splitting theorem.

Abstract

We consider generalizations of the drifted Laplacian and the Bakry-\'Emery-Ricci tensor, and we prove a version of the mean curvature comparison theorem. Consequently, we prove a Myers-type theorem and a generalization of the Cheeger-Gromoll splitting theorem.