# Myers' type theorem with the Bakry-\'Emery Ricci tensor

## TL;DR

This paper establishes a sharper diameter estimate for complete Riemannian manifolds with a bounded vector field and positive Bakry-Émery Ricci tensor lower bound, using a novel approach based on mean curvature comparison.

## Contribution

It introduces a new Myers' type diameter estimate leveraging generalized mean curvature comparison, improving upon previous results in the context of Bakry-Émery Ricci curvature.

## Key findings

- Proves a sharper diameter bound for manifolds with Bakry-Émery Ricci tensor
- Uses generalized mean curvature comparison instead of classical second variation
- Results applicable to manifolds with bounded vector fields

## Abstract

In this paper we prove a new Myers' type diameter estimate on a complete connected Reimannian manifold which admits a bounded vector field such that the Bakry-\'Emery Ricci tensor has a positive lower bound. The result is sharper than previous Myers' type results. The proof uses the generalized mean curvature comparison applied to the excess function instead of the classical second variation of geodesics.