Einstein and scalar flat Riemannian metrics

TL;DR

This paper characterizes critical points of the scalar curvature functional on closed manifolds, showing they are either Einstein or scalar flat, thus clarifying the structure of solutions in Riemannian geometry.

Contribution

It proves that critical points of the scalar curvature functional are precisely Einstein or scalar flat metrics, providing a complete classification in this context.

Findings

Critical points have constant scalar curvature.

Solutions are either Einstein or scalar flat.

Characterizes solutions of the critical point equation.

Abstract

On a given closed connected manifold of dimension two, or greater, we consider the squared $L^2$-norm of the scalar curvature functional over the space of constant volume Riemannian metrics. We prove that its critical points have constant scalar curvature, and use this to show that a metric is a solution of the critical point equation if, and only if, it is either Einstein, or scalar flat.