Yamabe and Riemann solitons on Lorentzian para-Sasakian manifolds

Commun. Korean Math. Soc. 2022 Vol. 37, No. 1, 213-228 https://doi.org/10.4134/CKMS.c200365 Published online December 27, 2021 Printed January 31, 2022

Abstract : In the present paper, we aim to study Yamabe soliton and Riemann soliton on Lorentzian para-Sasakian manifold. First, we proved, if the scalar curvature of an $\eta$-Einstein Lorentzian para-Sasakian manifold $M$ is constant, then either $\tau=n(n-1)$ or, $\tau=n-1$. Also we constructed an example to justify this. Next, it is proved that, if a three dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton for $V$ is an infinitesimal contact transformation and $tr\, \varphi$ is constant, then the soliton is expanding. Also we proved that, suppose a $3$-dimensional Lorentzian para-Sasakian manifold admits a Yamabe soliton, if $tr\,\varphi$ is constant and scalar curvature $\tau$ is harmonic (i.e., $\Delta \tau =0$), then the soliton constant $\lambda$ is always greater than zero with either $\tau=2$, or $\tau=6$, or $\lambda=6$. Finally, we proved that, if an $\eta$-Einstein Lorentzian para-Sasakian manifold $M$ represents a Riemann soliton for the potential vector field $V$ has constant divergence then either, $M$ is of constant curvature $1$ or, $V$ is a strict infinitesimal contact transformation.