Almost-Schur lemma - Mathematics > Differential Geometry

Abstract: Schur-s lemma states that every Einstein manifold of dimension $n\geq 3$ hasconstant scalar curvature. Here $M,g$ is defined to be Einstein if itstraceless Ricci tensor $$\Rico:=\Ric-\frac{R}{n}g$$ is identically zero. Inthis short note we ask to what extent the scalar curvature is constant if thetraceless Ricci tensor is assumed to be \emph{small} rather than identicallyzero.

Author: Camillo De Lellis, Peter M. Topping

Source: https://arxiv.org/