Cotton tensor on Sasakian 3-manifolds admitting eta-Ricci solitons

Debabrata Kar,Pradip Majhi̇

doi:10.31801/cfsuasmas.769405

Abstract

The object of the present paper is to characterize Cotton tensor on a 3-dimensional Sasakian manifold admitting $\eta$-Ricci solitons. After introduction, we study 3-dimensional Sasakian manifolds and introduce a new notion, namely, Cotton pseudo-symmetric manifolds. Next we deal with the study Cotton tensor on a Sasakian 3-manifold admitting $\eta$-Ricci solitons. Among others we prove that such a manifold is a manifold of constant scalar curvature and Einstein manifold with some appropriate conditions. Also, we classify the nature of the soliton metric. Finally, we give an important remark.

Highlights

In di¤erential geometry, the Weyl conformal curvature tensor vanishes on a 3dimensional pseudo-Riemannian manifold and one can consider an another type of conformal invariant, which is the Cotton tensor
In a 3-dimensional pseudo-Riemannian manifold Cotton tensor vanishes if the metric be conformally ‡at and the idea is given by Eisenhart
The 3-dimensional spaces becoming onto the dignity of interest, as the Cotton tensor restricts the relation between the Ricci tensor and the energy-momentum tensor of matter in the Einstein equations and plays an important role in the Hamiltonian formalism of general relativity

Summary

Introduction

In di¤erential geometry, the Weyl conformal curvature tensor vanishes on a 3dimensional pseudo-Riemannian manifold and one can consider an another type of conformal invariant, which is the Cotton tensor. Ricci soliton, -Ricci soliton, cotton tensor, Sasakian manifold. An -Ricci soliton is a tuple (g; V; ; ), where V is a vector ...eld on M , and are constants, and g is a Riemannian (or pseudo-Riemannian) metric satisfying the equation. COTTON TENSOR ON SASAKIAN 3-M ANIFOLDS ADM ITTING ETA RICCI SOLITONS571 semisymmetric Sasakian 3-manifolds admitting -Ricci solitons and establish a result. We introduce a new notion call Cotton pseudo-symmetric manifold and we study Sasakian 3-manifolds admitting -Ricci solitons.

Three dimensional Sasakian manifolds

Cotton tensor on Sasakian 3-manifolds admitting -Ricci solitons

Cotton flat Sasakian 3-manifolds admitting -Ricci solitons

Cotton pseudo-symmetric Sasakian 3-manifolds admitting -Ricci solitons

Conclusion