Generalized Eta-Einstein and $(\kappa ,\mu )$-structures

TL;DR

This paper explores generalized eta-Einstein and $()$-structures in 3-dimensional manifolds, revealing their properties, classifications, and providing explicit examples of such structures.

Contribution

It characterizes 3D generalized $()$-structures, showing only K-contact types can be eta-Einstein and constructs explicit examples of generalized Jacobi $()$-structures.

Findings

Only K-contact structures can be eta-Einstein in 3D.

Almost regular eta-Einstein structures are found on closed manifolds.

Explicit examples of compact generalized Jacobi $()$-structures are provided.

Abstract

Generalized $(\kappa ,\mu )$ structures occur in dimension 3 only. In this dimension 3, only K-contact structures can occur as generalized Eta-Einstein. On closed manifolds, Eta-Einstein, K-contact structures which are not D-homothetic to Einstein structures are almost regular. We also construct examples of compact, generalized Jacobi $(\kappa ,\mu )$-structures.