On Static Manifolds and Related Critical Spaces with cyclic parallel Ricci tensor

TL;DR

This paper classifies certain three-dimensional compact Riemannian manifolds with cyclic parallel Ricci tensor that admit solutions to a specific differential equation, focusing on static, volume, and scalar curvature critical metrics.

Contribution

It provides a classification of three-dimensional manifolds with cyclic parallel Ricci tensor satisfying a key differential equation, extending to higher dimensions with non-positive scalar curvature.

Findings

Classified 3D compact manifolds with cyclic parallel Ricci tensor under the given equation.

Identified structures as static triples, and critical metrics of volume and scalar curvature.

Extended classification to n-dimensional cases with non-positive scalar curvature.

Abstract

The aim of this paper is to classify three dimensional compact Riemannian manifolds $(M^{3},g)$ that admits a non-constant solution to the equation $$-\Delta f g+Hess f-fRic=\mu Ric+\lambda g,$$ for some special constants $(\mu, \lambda)$, under assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we will study here will be: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We shall also classify $n$-dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfying the cyclic parallel Ricci tensor condition.