Devaraja Mallesha Naik

In this paper, we consider $\ast$-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if the metric of a Kenmotsu manifold $M$ is a $\ast$-Ricci soliton, then soliton constant $\lambda$ is zero. For 3-dimensional case, if $M$ admits a $\ast$-Ricci soliton, then we show that $M$ is of constant sectional curvature -1. Next, we show that if $M$ admits a $\ast$-Ricci soliton whose potential vector field is collinear with the characteristic vector field $\xi$, then $M$ is Einstein and soliton vector field is equal to $\xi$. Finally, we prove that if $g$ is a gradient almost $\ast$-Ricci soliton, then either $M$ is Einstein or the potential vector field is collinear with the characteristic vector field on an open set of $M$. We verify our result by constructing examples for both $\ast$-Ricci soliton and gradient almost $\ast$-Ricci soliton.

In this paper, a systematic study of Kenmotsu pseudo-metric manifolds are introduced. After studying the properties of this manifolds, we provide necessary and sufficient condition for Kenmotsu pseudo-metric manifold to have constant $\phi$-sectional curvature, and prove the structure theorem for $\xi$-conformally flat and $\phi$-conformally flat Kenmotsu pseudo-metric manifolds. Next, we consider Ricci solitons on this manifolds. In particular, we prove that an $\eta$-Einstein Kenmotsu pseudo-metric manifold of dimension higher than 3 admitting a Ricci soliton is Einstein, and a Kenmotsu pseudo-metric 3-manifold admitting a Ricci soliton is of constant curvature $-\epsilon$.