On warped product manifolds satisfying Ricci-Hessian class type equations

Abstract

We deal with a study of warped product manifold which is also a generalized quasi Einstein manifold. Then, we investigate the relationships between such warped products and certain manifolds that provide some Ricci-Hessian type equations, such as Ric = for some smooth function , where Ric denotes the-Bakery-Emery Ricci tensor. Finally, we obtain some rigidity conditions for such manifolds.

A splitting theorem for Riemannian manifolds of generalised Ricci-Hessian type

“…It is easy to see that, for α = −1 and λ = 0, a function u solves that equation if and only if f := e u solves (1). However the results of [4] cannot be compared with ours since the quasi Einstein condition seems to be interesting only in the case where α > 0 and u > 0.…”

“…Thus we are left with an open problem in case we know about only one such function f . Along the same line, S. Güler and S.A. Demirbag define a Riemannian manifold (M n , g) to be quasi Einstein if and only if there exist smooth functions u, α, λ on M such that Ric + ∇ 2 u − αdu ⊗ ∇u = λ • Id, see [4,Eq. (1.1)].…”

On a Class of Gradient Almost Ricci Solitons

“…Here, we also assume that ∇f is of constant length. Since Cotton tensor is a constant multiple of divergence of Weyl tensor, by virtue of ( 7) and ( 8), we can express the self-duality condition as follows: 3[Q(e i )g(e 1 , ∇f ) − Q(e 1 )g(e i , ∇f )] + Ric(e 1 , ∇f )e i − Ric(e i , ∇f )e 1 (10) +r[g(e i , ∇f )e 1 − g(e 1 , ∇f…”

“…Since ||∇f || = 0, we can normalize ∇f to be a unit and complete it to an orthonormal frame {E i : i = 1, 2, 3, 4}, where E 1 = ∇f ||∇f || . Then, normalizing (10) with respect to this orthonormal frame, we obtain…”

“…In these regards, Ricci solitons and gradient Ricci solitons are natural generalizations of Einstein manifolds. In literature, there are many other different generalizations of Einstein manifolds and gradient Ricci solitons, such as quasi Einstein manifolds [3,4,5], (m, ρ)-quasi Einstein manifolds [6,7,8], generalized quasi Einstein manifolds [9,10,11] and etc. In this paper, we will deal with another generalization of Ricci solitons, given as follows:…”