Exploring the Wonders of the Viessman-Gray Manifold

Abdulhadi Ahmed Abd

Abdulhadi Ahmed Abd Salah al-Din Education Directorate, Baghdad, Iraq

Keywords: Almost Hermitian Manifold, Conharmonic Curvature tensor, Generalized Viessman -Gray manifold, Locally conformal Kahler, Differential geometry

Abstract

In this study, we investigate the geometric properties of the Generalized Viessman-Gray manifold, focusing on its conharmonic curvature tensor and its relationship with flat conharmonic curvature tensors. Leveraging the fleetness property of the GT-manifold, we utilize Weyl's tensor to derive components of the conformal GT-manifold and establish its classification as a Nearly Kähler manifold due to its zero scalar curvature. Additionally, we uncover the connection between the GT-manifold and conharmonic Para-Kähler manifolds. Through rigorous analysis and proof, we demonstrate that a GT-manifold with flat conharmonic tensors is Nearly Kähler and possesses zero scalar curvature, establishing its classification as a conharmonic Para-Kähler manifold. Our findings contribute to the understanding of these manifold structures and their geometric properties, paving the way for further exploration in related fields.

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