Gray identities, canonical connection and integrability

TL;DR

This paper characterizes quasi Kähler manifolds with curvature tensors satisfying the first Bianchi identity, linking it to Gray identities and integrability, and provides explicit examples with special curvature properties.

Contribution

It introduces a characterization of quasi Kähler manifolds via the first Bianchi identity and offers new proofs and examples related to Gray identities and curvature tensors.

Findings

Curvature tensor satisfies first Bianchi identity in certain quasi Kähler manifolds.

Explicit examples of quasi Kähler structures with vanishing Hermitian curvature.

Connection between Gray identities and integrability in almost Kähler manifolds.

Abstract

We characterize quasi K\"ahler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related with the third Gray identity and in the almost K\"ahler case implies the integrability. Our main tool is the existence of generalized holomorphic frames introduced by the second author previously. By using such frames we also give a simpler and shorter proof of a Theorem of Goldberg. Furthermore we study almost Hermitian structures having the curvature tensor associated to the canonical Hermitian connection equal to zero. We show some explicit examples of quasi K\"ahler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.