On static manifolds and related critical spaces with cyclic parallel Ricci tensor

We classify 3-dimensional compact Riemannian manifolds (M-3, g) that admit a non-constant solution to the equation -Delta fg + Hess f - f Ric = mu Ric +lambda g for some special constants (mu, lambda), under the assumption that the manifold has cyclic parallel Ricci tensor. Namely, the structures that we study here are: positive static triples, critical metrics of the volume functional, and critical metrics of the total scalar curvature functional. We also classify n-dimensional critical metrics of the volume functional with non-positive scalar curvature and satisfying the cyclic parallel Ricci tensor condition.