K -contact metrics as Ricci almost solitons
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K -contact metrics as Ricci almost solitons Dhriti Sundar Patra1 Received: 8 May 2020 / Accepted: 5 October 2020 © The Managing Editors 2020
Abstract In this paper, we prove two fundamental results of Ricci almost soliton on K -contact manifold. First we prove that if a complete K -contact metric represents a gradient Ricci almost soliton, then it is compact Einstein Sasakian and isometric to a unit sphere. Next we prove that if a K -contact metric represents a Ricci almost soliton whose potential vector field V is contact and the Ricci operator commutes with the structure tensor ϕ, then it is Einstein with Einstein constant 2n. Keywords Ricci almost soliton · Einstein manifold · K -contact manifold · Contact vector field Mathematics Subject Classification 53C21 · 53C25 · 53D15
1 Introduction A Riemannian metric g, defined on a smooth manifold M n of dimension n is said to be a Ricci soliton if there exist a vector field V and a constant λ on M satisfy 1 £V g + Ric = λg, 2
(1)
where £V denotes the Lie-derivative in the direction of V and Ric is the Ricci tensor of g. It will be called expanding, steady, or shrinking, respectively, if λ is negative, zero, and positive. Otherwise, it will be indefinite. Indeed, a Ricci soliton can be considered as a generalization of Einstein metric (i.e., the Ricci tensor Ric is a constant multiple of the Riemannian metric g) and often arises as a fixed point of Hamilton’s Ricci flow Hamilton (1988): ∂t∂ gi j = −2 Ricij , viewed as a dynamical system on the space of Riemannian metrics modulo diffeomorphisms and scalings. In Pigola et al. (2011), Pigoli et al. introduced the Ricci almost soliton allowing the constant λ as a smooth
B 1
Dhriti Sundar Patra [email protected]; [email protected] Department of Mathematics, Birla Institute of Technology Mesra, Ranchi 835 215, India
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Beitr Algebra Geom
function on M. If the vector field V is gradient of a smooth function u : M → R, then the Ricci almost soliton is called a gradient Ricci almost soliton. In this case, the Eq. (1) turns out ∇ 2 u + Ric = λg,
(2)
where ∇ 2 u denotes the Hessian of u. If the vector field V is trivial or the potential function u is constant, then the Ricci almost soliton is said to be trivial. Otherwise, it will be a non-trivial Ricci almost soliton. In Perelman (xxx), Perelman proved that every compact Ricci soliton is gradient, and Barros et al. (2014) generalized this result for Ricci almost soliton with constant scalar curvature. On the other hand, Sharma (2008) proved that if a K -contact metric represents a gradient Ricci soliton, then it is Einstein. Using some integral formula, Ghosh (2014) proved that “if a K -contact manifold admits a gradient Ricci almost soliton, then the scalar curvature is constant. Moreover, if M is compact, then it is Einstein, Sasakian and isometric to a unit sphere”. Based on the above results, a natural question can be posed: Is a complete K -contact metric as a gradient Ricci almost soliton Einstein? In this paper, we answer this question affirmativ
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