SOME RESULTS ON RICCI SOLITON ON CONTACT METRIC MANIFOLDS

Abstract

The objective of this paper is to study the nature of Ricci soliton admitting various type of contact metric manifolds such as Kenmostu manifold, LP Sasakian manifold and LCS manifold. In this paper, it is proved that a Kemnotsu Ricci soliton is expanding, whereas the LP-Sasakian Ricci soliton is shrinking. Further, the conditions have been obtained on $(LCS)_{n}$ Ricci soliton to be expanding, shrinking and steady and the results are verified by suitable examples. It is also proved that the possible values of soliton constant is the set of all even integers $\ZZ^{2n}$ and the set of negative integres $\ZZ^{-}$ respectively for Kenmostu Ricci soliton and LP Sasakian Ricci Soliton and if the (LCS) Ricci soliton is expanding then soliton constant lies on the interval $(0, \infty)$ and for shrinking it lies on $(-\infty, 0)$. The Projectively flat cases for the above manifolds are also discussed to be expanding, shrinking and steady. Finally, we study these Ricci solitons admitting Ricci semi-symmetric condition $R.S=0$ and prove that the soliton constant $\lambda$ is an eigen value of metric tensor $g$ with respect to associated vector field $\xi$.