A note on 4-dimensional locally conformally flat Walker manifolds

Azimpour, Sohrab; Chaichi, M.; Toomanian, Megerdich

doi:10.3103/s1068362307050044

Cited by 1 publication

(2 citation statements)

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“…Putting Y = ∇f in the second equation of ( 19) and using the equations ( 7) and ( 18), we get W (U, W, X, ∇f ) = 0. Combining the last relation with (20), we have either r = 4λ − 2 ∆f f or for all X, g(∇f, X) = 0. But the last one gives rise to ∇f = 0, which is a contradiction.…”

Section: Isotropic Casementioning

confidence: 98%

“…Thus, D is a null-parallel distribution and so (M, g, f, λ) is locally a Walker manifold. Hence, we can state that: Some geometric properties of four-dimensional Walker metrics satisfying c = 0 and c =constant were investiged in [19] and [20], respectively. Here we are intereseted in the particular case by choosing a = c = 0 in the metric g(x, y, z, t) and we construct the following:…”

Section: Isotropic Casementioning

confidence: 99%

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On a Class of Gradient Almost Ricci Solitons

Güler

2020

Bull. Malays. Math. Sci. Soc.

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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

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Section: Isotropic Casementioning

confidence: 98%

Section: Isotropic Casementioning

confidence: 99%