Quarter-symmetric connection on an almost Hermitian manifold and on a K\"ahler manifold

TL;DR

This paper explores the properties of quarter-symmetric connections on almost Hermitian and Kähler manifolds, revealing conditions under which these manifolds exhibit specific curvature tensor behaviors and establishing relations between different curvature tensors.

Contribution

It demonstrates that an almost Hermitian manifold with a quarter-symmetric connection preserving the metric is actually a Kähler manifold and constructs curvature tensors independent of the connection generator.

Findings

Almost Hermitian manifold with such connection is Kähler.

Constructed curvature tensors independent of the connection generator.

Established relations between Weyl projective and holomorphically projective curvature tensors.

Abstract

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with quarter-symmetric connection preserving the generalized Riemannian metric is actually the K\"ahler manifold. Observing the six linearly independent curvature tensors with respect to the quarter-symmetric connection, we construct tensors that do not depend on the quarter-symmetric connection generator. One of them coincides with the Weyl projective curvature tensor of symmetric metric $g$. Also, we obtain the relations between the Weyl projective curvature tensor and the holomorphically projective curvature tensor. Moreover, we examine the properties of curvature tensors when some tensors are hybrid.