A generalization of almost Schur lemma on CR manifolds

TL;DR

This paper extends the almost Schur lemma to higher-dimensional pseudo-Hermitian manifolds, establishing conditions under which the contact form is pseudo-Einstein and scalar curvature is constant.

Contribution

It generalizes the almost Schur lemma to (2n+1)-dimensional pseudo-Hermitian manifolds for n≥2, providing new geometric insights.

Findings

Contact form is pseudo-Einstein when equality holds.

Pseudo-Hermitian scalar curvature is constant at equality.

The results apply to a broad class of CR manifolds.

Abstract

In this paper, we study a general almost Schur Lemma on pseudo-Hermitian (2n+1)-manifolds $(M,J,\theta)$ for $n\geq2$. When the equality of almost Schur inequality holds, we derive the contact form $\theta$ is pseudo-Einstein and the pseudo-Hermitian scalar curvature is constant.