K-contact metrics as Ricci solitons

Abstract

Abstract We study η -Einstein K -contact manifold whose metric is a Ricci soliton. Keywords Ricci soliton · K -contact metric · η -Einstein Mathematics Subject Classification (2000) 53C25 · 53C44 ·53C21 1 Introduction A Riemannian metric g on a smooth manifold is Einstein if its Ricci tensor S is aconstant multiple of g . A Ricci soliton is a generalization of the Einstein metric andis defined on a Riemannian manifold ( M,g )by ( £ V g)(X,Y) +2 S(X,Y) +2 λg(X,Y) = 0(1)for some constant λ , a vector field V , and arbitrary vector fields X,Y on M .TheRicci soliton is said to be shrinking, steady, and expanding according as λ is negative,zero, and positive respectively. Compact Ricci solitons are the fixed points of the Ricciflow: ∂∂t g ij =−2 R ij projected from the space of metrics onto its quotient modulodiffeomorphisms and scalings, and often arise as blow-up limits for the Ricci flow on A. Ghosh ( B )Department of Mathematics, Krishnagar Government College,Krishnanagar, West Bengal, 741101, Indiae-mail: aghosh_70@yahoo.comR. SharmaDepartment of Mathematics, University of New Haven,West Haven, CT 06516, USAe-mail: rsharma@newhaven.edu