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Variations on the theme of twistor spaces. (English) Zbl 0926.53020

Author’s abstract: “The construction of the twistor space of an even-dimensional Riemannian manifold was transferred to a similar construction of a reflector space of a neutral manifold, i.e., a manifold with a pseudo-Riemannian metric of signature zero [G. R. Jensen and M. Rigoli, Matematiche 45, 407-443 (1990; Zbl 0757.53035)], and to the study of symplectic twistor spaces [I. Vaisman, J. Geom. Phys. 3, 507-524 (1986; Zbl 0629.53032)] and of Lagrangian-Grassmannian bundles [I. Vaisman, Algebras Groups Geom. 13, 323-341 (1996; Zbl 0873.53023)].
In the present paper, we discuss the common basis of all these constructions, the isosplitting bundles. These bundles are equipped with a pair of complex distributions which are involutive under the conditions known from twistor theory. We give an elementary proof of the involutivity conditions. In the particular case of a pseudo-Riemannian manifold, we study a subbundle of the isosplitting bundle, called the twist-reflector space, which composes twistors and reflectors”.

MSC:

53C28 Twistor methods in differential geometry
53Z05 Applications of differential geometry to physics
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
55R10 Fiber bundles in algebraic topology
58A30 Vector distributions (subbundles of the tangent bundles)
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