Mixed curvature almost flat manifolds

TL;DR

This paper extends Gromov's almost flat manifolds theorem to a setting involving mixed curvature bounds, specifically upper sectional and lower Bakry-Emery Ricci curvature, providing new insights into the geometry of such manifolds.

Contribution

It introduces a mixed curvature analogue of Gromov's theorem, combining upper sectional and lower Bakry-Emery Ricci curvature bounds.

Findings

Established a new curvature condition analogue of Gromov's theorem.

Proved that manifolds with these mixed curvature bounds are almost flat.

Extended the understanding of curvature bounds in geometric analysis.

Abstract

We prove a mixed curvature analogue of Gromov's almost flat manifolds theorem for upper sectional and lower Bakry-Emery Ricci curvature bounds.