A study on ricci solitons in almost contact metric manifolds

Abstract

In the present thesis we have studied Ricci solitons in contact geometric frame work. In 1982 R. S. Hamilton introduced Ricci flow; a technique to deform Riemannian 3-manifolds with positive Ricci curvature to a manifold with constant sectional curvature. Being a non-linear parabolic equation, Ricci flow develops singularities and this singularity models are special solution of Ricci flow which moves purely by homothetic and diffeomorphism, called Ricci solitons. Ricci solitons for a fixed time slice can be considered as the generalizations of Einstein metrics, called quasi-Einstein metrics in physics literature. Contact Riemannian manifolds are smooth odd-dimensional manifold with a special kind of non-integrable hyperplane distributions, called contact distribution. It contains a rich class of Einstein metrics and some of its weaker classes. This motivates us to study Ricci solitons on contact geometric frame work. newline newlineIn this thesis we have studied Ricci solitons and some of its generalization on Kenmotsu manifolds, Generalized Sasakian-space-forms, generalized (k, and#956;)-space-forms and para-Sasakian manifolds under some curvature and symmetry conditions. We have obtained several interesting fact about the soliton vector field and geodesic on such spaces. In some cases we have classified the solitons and studied them under some metric and non-metric connections with torsion. We have also verified some of our results by means on concrete examples. newline newline newline newline