Generalized CRF-structures. (English) Zbl 1138.53033

Vaisman, Izu

Geom. Dedicata 133, 129-154 (2008).

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.

MSC:

53C15

General geometric structures on manifolds (almost complex, almost product structures, etc.)

Keywords:

Courant bracket; \(F\)-structure; CRF-structure; CRFK-structure; Sasakian structure

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