On a Class of Gradient Almost Ricci Solitons

Abstract

In this study, we provide some classifications for half-conformally flat gradient f -almost Ricci solitons, denoted by (M, g, f ), in both Lorentzian and neutral signature. First, we prove that if ||∇f || is a non-zero constant, then (M, g, f ) is locally isometric to a warped product of the form I × ϕ N , where I ⊂ R and N is of constant sectional curvature. On the other hand, if ||∇f || = 0, then it is locally a Walker manifold. Then, we construct an example of 4-dimensional steady gradient f -almost Ricci solitons in neutral signature. At the end, we give more physical applications of gradient Ricci solitons endowed with the standard static spacetime metric.Keywords Ricci soliton · Gradient Ricci soliton · Gradient h-almost Ricci soliton · Half-conformally flat manifold · Walker manifold · Standard static spacetime metric Mathematics Subject Classification (2010) 53C21 · 53C50 · 53C25 IntroductionHamilton introduced the concept of the Ricci flow to prove the Poincare Conjecture in the late of 20th century [1]. Poincare Conjecture was one of the very deep unsolved problem which aims to classify all compact three dimensional manifolds. In the 1900's, Poincare asked if a simply-connected closed three manifold is necessarily the three sphere S 3 . For this purpose, Hamilton introduced the Ricci flow as a partial differential equation ∂g(t) ∂t = −2Ric(g(t)), which evolves the metric in a Riemannian manifold to make it rounder. By

Sequential Warped Products and Their Applications

“…Note that, for every smooth function ϕ, the following relation can be verified: Hence, any m-generalized quasi Einstein manifold is a ( −m φ )-almost gradient Ricci soliton. In [15], the author gave some classifications for half-conformally flat f -almost gradient Ricci solitons and physical applications of them in standard static spacetimes. Now, we prove the following result:…”

Imperfect Fluid Generalized Robertson Walker Spacetime Admitting Ricci-Yamabe Metric

“…Again, let us remark that η-Ricci-Yamabe solitons of type ðα, 0Þ or ð1, 0Þ and ð0, βÞ or ð0, 1Þ-type are α − η-Ricci soliton (or η-Ricci soliton) and β-η-Yamabe soliton (or η-Yamabe soliton), respectively; for more details about these particular cases, one can follow ( [19][20][21][22][23][24]).…”

Pseudosymmetric normal paracontact metric space forms admitting (α, β)− type almost η−Ricci-Yamabe solitons

“…Again, let us remark that η−Ricci-Yamabe soliton of type (α, 0) or (1, 0) , (0, β) or (0, 1) −type are α−η−Ricci soliton (or η−Ricci soliton) and β−η−Yamabe soliton (or η−Yamabe soliton) respectively for more details about these particular cases [7][8][9][10][11].…”