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

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 $$-\Delta fg+Hessf-fRic=\mu Ric+\lambda g,$$ 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.

15 pages

References in corpus (2)

• Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary
• On Critical Point Equation of Compact Manifolds with Zero radial Weyl Curvature

Cited by in corpus (1)

• On the mass of static metrics with positive cosmological constant - I