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Gray identities, canonical connection and integrability

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  12 August 2010

Antonio J. Di Scala  and
Luigi Vezzoni
Antonio J. Di Scala
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy ([email protected])
Luigi Vezzoni
Affiliation:
Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy ([email protected])
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Abstract

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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.

Keywords

almost Kähler manifoldsquasi-Kähler manifoldsintegrability of almost complex structureGray identities

MSC classification

Secondary: 53B20: Local Riemannian geometry 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 53 , Issue 3 , October 2010 , pp. 657 - 674
Copyright
Copyright © Edinburgh Mathematical Society 2010

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