Generalized CRF-structures. (English) Zbl 1138.53033

Summary: A generalized \(F\)-structure is a complex, isotropic subbundle \(E\) of \(T_cM \oplus T^*_cM\) \((T_cM = TM \otimes_{{\mathbb{R}}} {\mathbb{C}}\) and the metric is defined by pairing) such that \(E \cap \overline{E}^{\perp} = 0\). If \(E\) is also closed by the Courant bracket, \(E\) is a generalized CRF-structure. We show that a generalized \(F\)-structure is equivalent with a skew-symmetric endomorphism \(\Phi\) of \(TM \oplus T^*M\) that satisfies the condition \(\Phi^{3} + \Phi = 0\) and we express the CRF-condition by means of the Courant-Nijenhuis torsion of \(\Phi\). The structures that we consider are generalizations of the \(F\)-structures defined by Yano and of the CR (Cauchy-Riemann) structures. We construct generalized CRF-structures from: a classical \(F\)-structure, a pair \(({\mathcal{V}}, \sigma)\) where \({\mathcal{V}}\) is an integrable subbundle of TM and \(\sigma\) is a 2-form on \(M\), a generalized, normal, almost contact structure of codimension \(h\). We show that a generalized complex structure on a manifold \(M\) induces generalized CRF-structures on some submanifolds \(M \subseteq \widetilde{M}\). Finally, we consider compatible, generalized, Riemannian metrics and we define generalized CRFK-structures that extend the generalized Kähler structures and are equivalent with quadruples \((\gamma , F_{+}, F_{-}, \psi)\), where \((\gamma, F_{\pm}\)) are classical, metric CRF-structures, \(\psi\) is a 2-form and some conditions expressible in terms of the exterior differential \(d \psi\) and the \(\gamma\)-Lévi-Civita covariant derivatives \(\nabla F_{\pm}\) hold. If \(d \psi = 0\), the conditions reduce to the existence of two partially Kähler reductions of the metric \(\gamma\). The paper ends with an Appendix where we define and characterize generalized Sasakian structures.


53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Full Text: DOI arXiv


[1] Barros C.M. (1965). Variétés presque hor-complexes. C. R. Acad. Sci. Paris 260: 1543–1546 · Zbl 0191.20201
[2] Bejancu A. (1986). Geometry of CR-submanifolds, Mathematics and its Applications (East European Series), Vol. 23. D. Reidel Publishing Co., Dordrecht · Zbl 0605.53001
[3] Blair D.E. (1970). Geometry of manifolds with structural group \({\mathcal{U}}(n) \times {\mathcal{O}}(s)\) J. Differential Geom. 4: 155–167 · Zbl 0202.20903
[4] Blair D.E. (2002). Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics, Vol. 203. Birkhäuser Boston, Inc., Boston, MA · Zbl 1011.53001
[5] Courant T.J. (1990). Dirac manifolds. Trans. Am. Math. Soc. 319: 631–661 · Zbl 0850.70212
[6] Dragomir S. and Tomassini G. (2006). Differential Geometry and Analysis on CR Manifolds, Progress in Mathematics, vol. 246. Birkhäuser Verlag, Basel · Zbl 1099.32008
[7] Gualtieri, M.: Generalized complex geometry, Ph.D. thesis, Univ. Oxford (2003); arXiv:math.DG/0401221 · Zbl 1235.32020
[8] Goldberg S.I. and Yano K. (1970). On normal globally framed f-manifolds. Tôhoku Math. J. 22(2): 362–370 · Zbl 0203.54103
[9] Hitchin N.J. (2003). Generalized Calabi-Yau manifolds. Quart. J. Math. 54: 281–308 · Zbl 1076.32019
[10] Iglesias-Ponte D. and Wade A. (2005). Contact manifolds and generalized complex structures. J. Geom. Phys. 53: 249–258 · Zbl 1075.53081
[11] Kobayashi S. and Nomizu K. (1969). Foundations of Differential Geometry, vol. II. Interscience Publ., New York · Zbl 0175.48504
[12] Lindström U., Minasian R., Tomasiello A. and Zabzine M. (2005). Generalized complex manifolds and supersymmetry. Commun. Math. Phys. 257: 235–256 · Zbl 1118.53048
[13] Liu Z.-J., Weinstein A. and Xu P. (1997). Manin triples for Lie bialgebroids. J. Differential Geom. 45: 547–574 · Zbl 0885.58030
[14] Vaisman I. (2007). Reduction and submanifolds of generalized complex manifolds. Differential Geom. Appl. 25: 147–166 · Zbl 1126.53049
[15] Vaisman I. (2007). Dirac structures and generalized complex structures on \(TM \times {\mathbb{R}}^h\) . Adv. Geom. 7: 453–474 · Zbl 1126.53054
[16] Vaisman I. (2007). Isotropic subbundles of \(TM \oplus T^*M\) . Int. J. Geom. Methods Mod. Phys. 4: 487–516 · Zbl 1143.53032
[17] Wade A. (2004). Dirac structures and paracomplex manifolds. C. R. Acad. Sci. Paris, Ser. I 338: 889–894 · Zbl 1057.53019
[18] Yano K. (1963). On a structure defined by a tensor field f of type (1, 1) satisfying f 3 + f = 0. Tensor 14: 99–109 · Zbl 0122.40705
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.