Gray identities, canonical connection and integrability

Abstract

We characterize quasi Kähler 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 Kähler 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 Kähler structures on the Iwasawa manifold having the Hermitian curvature vanishing and the Riemann curvature tensor satisfying the second Gray identity.

16 pages, major revision

References in corpus (4)

• Taming symplectic forms and the Calabi-Yau equation
• Almost Hermitian Geometry on Six Dimensional Nilmanifolds
• Some integrability conditions for almost Kähler manifolds
• A generalization of the normal holomorphic frames in Symplectic Manifolds

Cited by in corpus (5)

• Quasi-Kähler manifolds with trivial Chern Holonomy
• On 3-dimensional almost Einstein manifolds with circulant structures
• Higher-dimensional Osserman metrics with non-nilpotent Jacobi operators
• Curvature properties of Riemannian manifolds with skew-circulant structures
• Curvature identities on almost Hermitian manifolds and applications