Yamabe and Riemann solitons on Lorentzian para-Sasakian manifolds

Commun. Korean Math. Soc. Published online December 27, 2021

Shruthi Chidananda and Venkatesha Venkatesha
Dept of Mathematics, Kuvempu University, Shankaraghatta, Shimoga, karnataka,India; Dept of Mathematics, Kuvempu University, Shankaraghatta, Shimoga, karnataka,India

Abstract : In the present paper we consider to study Yamabe soliton and Riemann soliton on Lorentzian para-Sasakian manifold. First we prove that, In an $\eta$-Einstein Lorentzian para-Sasakian manifold constancy of scalar curvature $\tau$ implies either $\tau=n(n-1)$ or $\tau=n-1$. Also we construct an example to justify this. Next, we prove that, if a three dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton for $V$ is an infinitesimal contact transformation and $tra \varphi$ is constant, then the soliton is expanding with either $V$ is Killing, or $\lambda=6$. Also we prove that, suppose a $3$-dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton, if tra$\varphi$ is constant and scalar curvature $r$ is harmonic (i. e., $\Delta r =0$), then the soliton constant $\lambda$ is always greater than zero with either $\tau=\lambda$, or $\lambda=6$. Finally, we prove that, if an $\eta$-Einstein Lorentzian para-Sasakian manifold $M$ represents a Riemann soliton for the potential vector field $V$ has constant divergence then $M$ has either constant curvature $1$ or, $V$ is strict infinitesimal contact transformation.