Generalized Quasi-Einstein Manifolds in Contact Geometry

İnan Ünal

doi:10.3390/math8091592

Abstract

In this study, we investigate generalized quasi-Einstein normal metric contact pair manifolds. Initially, we deal with the elementary properties and existence of generalized quasi-Einstein normal metric contact pair manifolds. Later, we explore the generalized quasi-constant curvature of normal metric contact pair manifolds. It is proved 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 parallel Ricci tensor and the Codazzi type of Ricci tensor are considered, and further prove that a generalized quasi-Einstein normal metric contact pair manifold does not satisfy Codazzi type of Ricci tensor. Finally, we characterize normal metric contact pair manifolds satisfying certain curvature conditions related to 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 S2n+1(1)×S1.

Highlights

An Einstein manifold is a Riemannian manifold ( M, g), which is defined by the Ricci tensorRic = λg for a non-zero constant λ
We present a characterization of generalized quasi-Einstein normal metric contact pair manifolds
We investigate a generalized quasi-Einstein normal metric contact pair manifold under some conditions for Ricci tensor

Summary

Introduction

An Einstein manifold is a Riemannian manifold ( M, g), which is defined by the Ricci tensor. A Riemannian manifold ( M, g) is called a generalized quasi-Einstein manifold if its Ricci tensor has following form: Ric( X1 , X2 ) = λg( X1 , X2 ) + βω ( X1 )ω ( X2 ) + μη ( X1 )η ( X2 ). We consider the generalized quasi-Einstein normal metric contact pair manifolds. We present a characterization of generalized quasi-Einstein normal metric contact pair manifolds. We consider the notion of generalized quasi-constant curvature for normal metric contact pair manifolds and we obtain some results on the sectional curvature. We investigate a generalized quasi-Einstein normal metric contact pair manifold under some conditions for Ricci tensor. We prove that a generalized quasi-Einstein normal metric contact pair manifold does not satisfy Codazzi type of. We characterize normal metric contact pair manifolds satisfying certain curvature conditions related to 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 S2n+1 (1) × S1

Preliminaries

Generalized Quasi-Einstein Normal Contact Pair Manifolds

Normal Metric Contact Pair Manifold Satisfying Certain Curvature Conditions