Communications of the
Korean Mathematical Society CKMS

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 $ au=n(n-1)$ or, $ au=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 $ au$ is harmonic (i.e., $Delta au =0$), then the soliton constant $lambda$ is always greater than zero with either $ au=2$, or $ au=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.

Keywords: Lorentzian para-Sasakian manifold, $eta$-Einstein manifold, Yamabe soliton, Riemann soliton

MSC numbers: 53C50, 53C15, 53C25