Abstract
We study Ricci solitons in -Sasakian manifolds and show that it is a shrinking or expanding soliton and the manifold is Einstein with Killing vector field. Further, we prove that if is conformal Killilng vector field, then the Ricci soliton in 3-dimensional -Sasakian manifolds is shrinking or expanding but cannot be steady.
Highlights
A Ricci soliton (g, V, λ) is a generalization of an Einstein metric and is defined on a Riemannian manifold (M, g) by LVg + 2S + 2λg = 0, (1)where V is a complete vector field on M and λ is a constant
Long-existing solutions, that is, solutions which exist on an infinite time interval, are the self-similar solutions, which in Ricci flow are called Ricci soliton
If the vector field V is the gradient of a potential function −f, g is called a gradient Ricci soliton and (1) assumes the form
Summary
If the vector field V is the gradient of a potential function −f, g is called a gradient Ricci soliton and (1) assumes the form. A Ricci soliton on a compact manifold has constant curvature in dimension 2 [1] and in dimension 3 [2]. In [3], Perelman proved that a Ricci soliton on a compact nmanifold is a gradient Ricci soliton. In [4], Sharma studied Ricci solitons in K-contact manifolds, where the structure field ξ is Killing, and he proved that a complete K-contact gradient soliton is compact Einstein and Sasakian. In [5], Tripathi studied Ricci solitons in N(k)-contact metric and (k, μ) manifolds. In [6], Ghosh and Sharma studied K-contact metrics as Ricci solitons. Motivated by the previous studies on Ricci solitons, in this paper, we study Ricci solitons in an αSasakian manifolds, where α is some constant
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