Almost-Schur lemma

TL;DR

This paper investigates how close the scalar curvature of an Einstein manifold is to being constant when the traceless Ricci tensor is small, extending the classical Schur's lemma to near-Einstein conditions.

Contribution

It provides a quantitative analysis of scalar curvature constancy under small traceless Ricci tensor assumptions, extending Schur's lemma.

Findings

Scalar curvature remains nearly constant when traceless Ricci tensor is small

Quantitative bounds relating traceless Ricci tensor size to scalar curvature variation

Extension of classical Schur's lemma to approximate Einstein manifolds

Abstract

Schur's lemma states that every Einstein manifold of dimension $n\geq 3$ has constant scalar curvature. Here $(M,g)$ is defined to be Einstein if its traceless Ricci tensor $$\Rico:=\Ric-\frac{R}{n}g$$ is identically zero. In this short note we ask to what extent the scalar curvature is constant if the traceless Ricci tensor is assumed to be \emph{small} rather than identically zero.