A note on Almost Riemann Soliton and gradient almost Riemann soliton

TL;DR

This paper investigates almost Riemann solitons and gradient almost Riemann solitons on 3-dimensional non-cosymplectic almost contact metric manifolds, establishing conditions under which the manifold exhibits specific geometric properties such as being quasi-Sasakian or having constant sectional curvature.

Contribution

It provides new results characterizing almost Riemann solitons and gradient almost Riemann solitons in specific geometric settings, including conditions leading to quasi-Sasakian structures or constant curvature.

Findings

Manifold is quasi-Sasakian with constant sectional curvature -λ if Riemann soliton with divergence-free potential exists.

Almost Riemann soliton reduces to Riemann soliton when Z is collinear with ξ and divergence is constant.

Gradient ARS implies the manifold is either quasi-Sasakian or has constant sectional curvature -(^2 - ^2).

Abstract

The quest of the offering article is to investigate \emph{almost Riemann soliton} and \emph{gradient almost Riemann soliton} in a non-cosymplectic normal almost contact metric manifold $M^3$. Before all else, it is proved that if the metric of $M^3$ is Riemann soliton with divergence-free potential vector field $Z$, then the manifold is quasi-Sasakian and is of constant sectional curvature -$\lambda$, provided $\alpha,\beta =$ constant. Other than this, it is shown that if the metric of $M^3$ is \emph{ARS} and $Z$ is pointwise collinear with $\xi $ and has constant divergence, then $Z$ is a constant multiple of $\xi $ and the \emph{ARS} reduces to a Riemann soliton, provided $\alpha,\;\beta =$constant. Additionally, it is established that if $M^3$ with $\alpha,\; \beta =$ constant admits a gradient \emph{ARS} $(\gamma,\xi,\lambda)$, then the manifold is either quasi-Sasakian or is of constant sectional curvature $-(\alpha^2-\beta^2)$. At long last, we develop an example of $M^3$ conceding a Riemann soliton.