The Almost Schur Lemma in Quaternionic Contact Geometry

TL;DR

This paper extends the Almost Schur Lemma to quaternionic contact geometry, providing estimates for the qc scalar curvature based on Ricci tensor components and torsion, under specific positivity conditions.

Contribution

It introduces quaternionic contact versions of the Almost Schur Lemma, linking scalar curvature to Ricci tensor components and torsion in compact qc manifolds.

Findings

Derived bounds for qc scalar curvature in terms of Ricci tensor components.

Established conditions under which the scalar curvature is constant.

Extended classical geometric inequalities to quaternionic contact settings.

Abstract

We establish quaternionic contact (qc) versions of the so called Almost Schur Lemma, which give estimations of the qc scalar curvature on a compact qc manifold to be a constant in terms of the norm of the $[-1]$-component and the norm of the trace-free part of the $[3]$-component of the horizontal qc Ricci tensor and the torsion endomorphism, under certain positivity conditions.