Some Results Generalized Curvature Tensor of Viesman-Grey Manifold

Abstract

General Background: The study of differential geometry plays a crucial role in understanding the structural properties of various manifolds, particularly in the context of Riemannian and Hermitian geometry. Specific Background: The Viesman-Grey manifold is a significant class in this field, characterized by its unique conharmonic curvature tensor. Previous studies have examined nearly Hermitian manifolds, conformal invariants, and curvature identities, yet a comprehensive analysis of the generalized conharmonic curvature tensor in Viesman-Grey manifolds remains insufficient. Knowledge Gap: Despite extensive research on curvature tensors in different geometric structures, the specific characteristics and classification of the conharmonic curvature tensor in generalized Viesman-Grey manifolds require further exploration. Aims: This study aims to investigate the properties of the generalized conharmonic curvature tensor in Viesman-Grey manifolds, examining its structure and defining new classifications. Results: The research identifies various classes of Viesman-Grey manifolds based on the components of the conharmonic curvature tensor and establishes their interrelations. The study provides equations characterizing different classes and proves specific properties of the generalized tensor. Novelty: This study introduces new classifications of generalized Viesman-Grey manifolds, offering an extended perspective on conharmonic curvature tensor properties and their implications in differential geometry. Implications: The findings contribute to the theoretical development of Riemannian geometry by refining the classification of nearly Hermitian and Viesman-Grey manifolds. These insights may support future studies in geometric structures, tensor analysis, and mathematical physics.