Riemann solitons on para-Sasakian geometry

Keywords: para-Sasakian manifold, almost Riemann soliton, gradient almost Riemann soliton
Published online: 2022-11-17

Abstract


The goal of the present article is to investigate almost Riemann soliton and gradient almost Riemann soliton on 3-dimensional para-Sasakian manifolds. At first, it is proved that if $(g, Z,\lambda)$ is an almost Riemann soliton on a para-Sasakian manifold $M^3$, then it reduces to a Riemann soliton and $M^3$ is of constant sectional curvature $-1$, provided the soliton vector $Z$ has constant divergence. Besides these, we prove that if $Z$ is pointwise collinear with the characteristic vector field $\xi$, then $Z$ is a constant multiple of $\xi$ and the manifold is of constant sectional curvature $-1$. Moreover, the almost Riemann soliton is expanding. Furthermore, it is established that if a para-Sasakian manifold $M^3$ admits gradient almost Riemann soliton, then $M^3$ is locally isometric to the hyperbolic space $H^{3}(-1)$. Finally, we construct an example to justify some results of our paper.

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How to Cite
(1)
De K., De U. Riemann Solitons on Para-Sasakian Geometry. Carpathian Math. Publ. 2022, 14 (2), 395-405.