Almost Riemann Solitons and gradient almost Riemann solitons on $LP$-Sasakian manifolds

Abstract

The upcoming article aims to investigate almost Riemann soliton and gradient almost Riemann soliton in a $LP$-Sasakian manifold $M^3$. At first, it is proved that if $(g, Z,\lambda)$ be an almost Riemann soliton on a $LP$-Sasakian manifold $M^3$, then it reduces to a Riemann soliton, provided the soliton vector $Z$ has constant divergence. Also, we show that if $Z$ is pointwise collinear with the characteristic vector field $\xi$, then $Z$ is a constant multiple of $\xi$, and the ARS reduces to a Riemann soliton. Furthermore, it is proved that if a $LP$-Sasakian manifold $M^3$ admits gradient almost Riemann soliton, then the manifold is a space form.  Also, we consider a non-trivial example and validate a result of our paper.