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

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