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- Title
GRAY IDENTITIES, CANONICAL CONNECTION AND INTEGRABILITY.
- Authors
di Scala, Antonio J.; Vezzoni, Luigi
- Abstract
We characterize quasi-Kähler manifolds whose curvature tensor associated to the canonical Hermitian connection satisfies the first Bianchi identity. This condition is related to the third Gray identity and in the almost-Kähler case implies the integrability. Our main tool is the existence of generalized holomorphic frames previously introduced by the second author. 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.
- Subjects
KAHLERIAN manifolds; HERMITIAN forms; HERMITIAN symmetric spaces; HERMITIAN structures; RIEMANN integral; RIEMANN-Roch theorems; DIFFERENTIAL geometry
- Publication
Proceedings of the Edinburgh Mathematical Society, 2010, Vol 53, Issue 3, p657
- ISSN
0013-0915
- Publication type
Article
- DOI
10.1017/S0013091509000157