On trans-Sasakian $3$-manifolds as $\eta$-Einstein solitons

TL;DR

This paper investigates 3-dimensional trans-Sasakian manifolds that admit $ ext{η}$-Einstein solitons, analyzing curvature conditions, Ricci tensor properties, and providing an example to illustrate the theoretical findings.

Contribution

It studies $ ext{η}$-Einstein solitons on 3D trans-Sasakian manifolds with specific Ricci tensor conditions and curvature constraints, offering new insights and an explicit example.

Findings

Characterization of $ ext{η}$-Einstein solitons under Codazzi and cyclic parallel Ricci tensors

Identification of curvature conditions admitting $ ext{η}$-Einstein solitons

Explicit example of a 3D trans-Sasakian manifold with an $ ext{η}$-Einstein soliton

Abstract

The present paper is to deliberate the class of $3$-dimensional trans-Sasakian manifolds which admits $\eta$-Einstein solitons. We have studied $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds where the Ricci tensors are Codazzi type and cyclic parallel. We have also discussed some curvature conditions admitting $\eta$-Einstein solitons on $3$-dimensional trans-Sasakian manifolds and the vector field is torse-forming. We have also shown an example of $3$-dimensional trans-Sasakian manifold with respect to $\eta$-Einstein soliton to verify our results.