Yamabe Solitons on three-dimensional normal almost paracontact metric manifolds

TL;DR

This paper investigates Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic, and para-Kenmotsu manifolds, establishing conditions for constant scalar curvature, Einstein properties, and the nature of flow vector fields.

Contribution

It provides new results characterizing Yamabe solitons on these specific paracontact manifolds, including conditions for constant curvature and Einstein properties, with explicit examples.

Findings

Yamabe solitons on para-Sasakian manifolds have constant scalar curvature and Killing flow vectors.

On paracosymplectic manifolds, Yamabe solitons are either η-Einstein or Ricci flat.

Para-Kenmotsu manifolds with Yamabe solitons are of constant sectional curvature -1, Einstein, with expanding solitons and Killing vectors.

Abstract

The purpose of the paper is to study Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic and para-Kenmotsu manifolds. Mainly, we proved that *If the semi-Riemannian metric of a three-dimensional para-Sasakian manifold is a Yamabe soliton, then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we proved that either manifold has constant curvature -1 and reduces to an Einstein manifold, or V is an infinitesimal automorphism of the paracontact metric structure on the manifold. *If the semi-Riemannian metric of a three-dimensional paracosymplectic manifold is a Yamabe soliton, then it has constant scalar curvature. Furthermore either manifold is $\eta$-Einstein, or Ricci flat. *If the semi-Riemannian metric on a three-dimensional para-Kenmotsu manifold is a Yamabe soliton, then the manifold is of constant sectional curvature -1, reduces to an Einstein manifold. Furthermore, Yamabe soliton is expanding with $\lambda$=-6 and the vector field V is Killing. Finally, we construct examples to illustrate the results obtained in previous sections.