The curvature tensor of (\ka,\mu,\nu)-contact metric manifolds

TL;DR

This paper investigates the Riemann curvature tensor of (,,)-contact metric manifolds, showing it is fully determined in three dimensions and exploring its behavior under deformations, leading to new generalized space forms.

Contribution

It provides a complete characterization of the curvature tensor in 3D and introduces generalized (,,)-space forms with conditions for conformal flatness.

Findings

Curvature tensor is fully determined in 3D.

D_a-homothetic deformations affect the curvature tensor.

Conditions for conformal flatness of generalized space forms.

Abstract

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by D_a-homothetic deformations. This prompts the definition and study of generalized (\kappa,\mu,\nu)-space forms and of the necessary and sufficient conditions for them to be conformally flat.