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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

References

1. Abbena, E., Garbiero, S. and Salamon, S., Almost-Hermitian geometry on six dimensional nilmanifolds, Annali Scuola Norm. Sup. Pisa IV 30 (2001), 147–170.Google Scholar
2. Apostolov, V. and Drăghici, T., The curvature and the integrability of almost-Kähler manifolds: a survey, in Symplectic and contact topology: interactions and perspectives, Fields Institute Communications, Volume 35, pp. 25–53 (American Mathematical Society, Providence, RI, 2003).Google Scholar
3. Apostolov, V., Armstrong, J. and Drăghici, T., Local models and integrability of certain almost-Kähler 4-manifolds, Math. Ann. 323 (2002), 633–666.CrossRefGoogle Scholar
4. Davidov, J. and Muškarov, O., Twistor spaces with Hermitian Ricci tensor, Proc. Am. Math. Soc. 109 (1990), 1115–1120.CrossRefGoogle Scholar
5. de Bartolomeis, P. and Tomassini, A., On formality of some symplectic manifolds, Int. Math. Res. Not. 24 (2001), 1287–1314.CrossRefGoogle Scholar
6. Donaldson, S. K., Two-forms on four-manifolds and elliptic equations, in Inspired by S. S. Chern: a memorial volume in honor of a great mathematician, Nankai Tracts in Mathematics, Volume 11 (World Scientific, 2006).CrossRefGoogle Scholar
7. Falcitelli, M., Farinola, A. and Salamon, S., Almost-Hermitian geometry, Diff. Geom. Applic. 4 (1994), 259–282.CrossRefGoogle Scholar
8. Gauduchon, P., Hermitian connections and Dirac operators, Boll. UMI B11(2, suppl.) (1997), 257–288.Google Scholar
9. Goldberg, S. I., Integrability of almost-Kähler manifolds, Proc. Am. Math. Soc. 21 (1969), 96–100.CrossRefGoogle Scholar
10. Gray, A., Curvature identities for Hermitian and almost-Hermitian manifolds, Tohoku Math. J. 28 (1976), 601–612.CrossRefGoogle Scholar
11. Kirchberg, K.-D., Some integrability conditions for almost-Kähler manifolds, J. Geom. Phys. 49 (2004), 101–115.CrossRefGoogle Scholar
12. Newlander, A. and Nirenberg, L., Complex analytic coordinates in almost-complex manifolds, Annals Math. (2) 65 (1957), 391–404.CrossRefGoogle Scholar
13. Salamon, S., Harmonic and holomorphic maps, in Proc. Geometry Seminar ‘Luigi Bianchi’ II, 1984, Lecture Notes in Mathematics, Volume 1164, pp. 161–224, (Springer, 1985).CrossRefGoogle Scholar
14. Tosatti, V., Weinkove, B. and Yau, S.-T., Taming symplectic forms and the Calabi-Yau equation, Proc. Lond. Math. Soc. 97 (2008), 401–424.CrossRefGoogle Scholar
15. Tricerri, F. and Vanhecke, L., Curvature tensors on almost-Hermitian manifolds, Trans. Am. Math. Soc. 267 (1981), 365–398.CrossRefGoogle Scholar
16. Vezzoni, L., A generalization of the normal holomorphic frames in symplectic manifolds, Boll. UMI B9 (2006), 723–732.Google Scholar
17. Vezzoni, L., On the Hermitian curvature of symplectic manifolds, Adv. Geom. 7 (2007), 207–214.CrossRefGoogle Scholar