A mild generalization of almost-Schur lemma

TL;DR

This paper proves a new integral geometric inequality in smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature, extending the almost-Schur theorem to a broader setting.

Contribution

It introduces a mild generalization of the almost-Schur lemma within the context of smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature.

Findings

Established an integral geometric inequality in the specified setting.

Extended the almost-Schur theorem to a more general class of spaces.

Provides a new tool for geometric analysis in weighted manifolds.

Abstract

In this short note we establish an integral geometric inequality in a smooth metric measure space of the nonnegative Bakry-\'Emery Ricci curvature. This result can be regarded as a mild generalization of the almost Schur theorem due to De Lellis and Topping (Calc. Var., DOI: 10.1007/s00526-011-0413-z).