arXiv subjects
Publications and source records attributed to Devaraja Mallesha Naik.
The aim of this article is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also some suitable examples of Kenmotsu and $(κ,μ)'$-almost Kenmotsu manifolds are constructed to justify our results.
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 $φ$-sectional curvature, and prove the structure theorem for $ξ$-conformally flat and $φ$-conformally flat Kenmotsu pseudo-metric manifolds. Next, we consider Ricci solitons on this manifolds. In particular, we prove that an $η$-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 $-\varepsilon$.
In this paper, we consider $*$-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if the metric of a Kenmotsu manifold $M$ is a $*$-Ricci soliton, then soliton constant $λ$ is zero. For 3-dimensional case, if $M$ admits a $*$-Ricci soliton, then we show that $M$ is of constant sectional curvature -1. Next, we show that if $M$ admits a $*$-Ricci soliton whose potential vector field is collinear with the characteristic vector field $ξ$, then $M$ is Einstein and soliton vector field is equal to $ξ$. Finally, we prove that if $g$ is a gradient almost $*$-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 $*$-Ricci soliton and gradient almost $*$-Ricci soliton.
We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=\frac{1}{2}£_ξφ$ and $\ell := R(\cdot,ξ)ξ$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $ξ$-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost $CR$ structure $(\mathcal{H}(M), J, θ)$ corresponding to almost contact pseudo-metric manifold $M$ to be $CR$ manifold. Finally, we prove that a contact pseudo-metric manifold $(M,φ,ξ,η,g)$ is Sasakian if and only if the corresponding nondegenerate almost $CR$ structure $(\mathcal{H}(M), J)$ is integrable and $J$ is parallel along $ξ$ with respect to the Bott partial connection.