Generalized Quasi-Einstein Manifolds in Contact Geometry

TL;DR

This paper explores generalized quasi-Einstein structures in normal metric contact pair manifolds, establishing their properties, curvature conditions, and classification, including their local isometry to Hopf manifolds.

Contribution

It introduces new characterizations and curvature conditions for generalized quasi-Einstein normal metric contact pair manifolds, linking them to well-known geometric structures.

Findings

Normal metric contact pair manifolds with generalized quasi-constant curvature are quasi-Einstein.

Such manifolds do not satisfy Codazzi type Ricci tensor conditions.

They are locally isometric to Hopf manifolds.

Abstract

In this study, we investigate generalized quasi-Einstein structure for normal metric contact pair manifolds. Firstly, we deal with elementary properties and examine, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifold. Secondly, the generalized quasi-constant curvature of normal metric contact pair manifolds are studied and it is proven that a normal metric contact pair manifold with generalized quasi-constant curvature is a generalized quasi-Einstein manifold. Normal metric contact pair manifolds satisfying cyclic Ricci tensor and the Codazzi type of Ricci tensor are considered and its shown that a generalized quasi-Einstein normal metric contact pair manifold does not satisfy Codazzi type of Ricci tensor. Finally, we work on normal metric contact pair manifolds satisfying certain curvature conditions related to $ \mathcal{M}- $projective, conformal, and concircular curvature tensors. We show that a normal metric contact pair manifold with generalized quasi-constant curvature is locally isometric to the Hopf manifold $ S^{2n+1}(1) \times S^1 $.