On a Class of Gradient Almost Ricci Solitons
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On a Class of Gradient Almost Ricci Solitons Sinem Güler1 Received: 15 July 2019 / Revised: 30 November 2019 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
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 53C21 · 53C50 · 53C25
1 Introduction Hamilton introduced the concept of the Ricci flow to prove the Poincare Conjecture in the late of twentieth century [1]. Poincare Conjecture was one of the very deep unsolved problems which aims to classify all compact three-dimensional manifolds. In the 1900s, Poincare asked whether a simply connected closed three manifold is necessarily the three sphere S3 . 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 choosing harmonic coordinates, it can be seen that the Ricci flow is a heat-type equation, and the characteristic property of such equations is the maximum principle, which guarantees that this rounding metric
Communicated by Young Jin Suh.
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Sinem Güler [email protected] Department of Industrial Engineering, Istanbul Sabahattin Zaim University, Halkalı, Istanbul, Turkey
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S. Güler
happens in some specific case. Hence, as in the expectation of Hamilton, after the evaluation of the metric, the manifold will become of constant curvature. Moreover, Hamilton proved that for any smooth metric g0 on a compact Riemannian manifold M n , there exists a unique solution g(t) of Ricci flow defined on some interval [0, ε), for some ε > 0, with the initial condition g(0) = g0 [1]. Even in the non-compact case, a complete solution of Ricci flow exists when the sectional curvatures of g0 are bounded [2]. Let (M n , g(t)) be a solution of the Ricci flow and suppose that ϕt : M n → M n is a time-dependent family of diffeomorphisms satisfying ϕ0 = I d and σ (t) is a timedependent scale factor satisfying σ (0) = 1. If g(t) = σ (t)ϕt∗ g(0) holds, then the solution (M n , g(t)) is called a Ricci soliton. Thus, one can regard the Ricci solitons as the fixed points of the Ricci flow, which change only by a diffeomorphism and a rescaling. If we take the derivative of the last relation at t =
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