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From generalized Kähler to generalized Sasakian structures. (English) Zbl 1207.53058

The author makes a synthesis of some previously published work, concerning several generalized structures on manifolds [I. Vaisman, Adv. Geom. 7, 453–474 (2007; Zbl 1126.53054); Geom. Dedicata 133, 129–154 (2008;Zbl 1138.53033); M. Gualtieri, Ph.D. thesis (2003) arXiv:Math.DG/0401221)]. For a differentiable manifold \(M\), the key constructions are: a “big tangent bundle” \(T^{big}M:=TM \oplus T^*(M)\); a canonical semi-Riemannian metric on \(T^{big}M\), of signature zero; the Courant bracket on \(T^{big}M\). Definitions and properties are given for generalized almost complex manifolds, generalized almost product manifolds, generalized Kähler manifolds, generalized almost contact manifolds and generalized Sasakian manifolds.
There are also two new results, characterizing normal generalized almost contact structures and generalized Sasakian structures, respectively.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
53B35 Local differential geometry of Hermitian and Kählerian structures
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