Quarter-symmetric connection on an almost Hermitian manifold and on a Kähler manifold
Milan Zlatanović, Miroslav Maksimović
TL;DR
This work studies a quarter-symmetric generalized metric connection on generalized Riemannian manifolds, specializing to almost Hermitian and Kähler geometries. It proves that an almost Hermitian manifold with a quarter-symmetric $G$-metric connection that preserves the generalized metric is necessarily Kähler, and it develops a family of curvature tensors $\overset{\theta}{R}$ and associated $\overset{\theta}{H}$-tensors that are independent of the connection generator $\pi$. The authors establish precise relations between Weyl projective and holomorphically projective curvature tensors, showing that $\overset{4}{H}$ coincides with the Weyl tensor and expressing $^{g}W$ and $^{g}P$ as linear combinations of the $\overset{\theta}{H}$ tensors under various hypotheses. The results provide a unified framework for curvature identities in the presence of quarter-symmetric connections and lay groundwork for extensions to almost para-Hermitian and para-Kähler settings, with potential applications in geometric analysis and projective-like mappings.
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 Kähler 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.