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General construction of symmetric parabolic structures. (English) Zbl 1350.53068
Summary: First we introduce a generalization of symmetric spaces to parabolic geometries. We provide construction of such parabolic geometries starting with classical symmetric spaces and we show that all regular parabolic geometries with smooth systems of involutive symmetries can be obtained in this way. Further, we investigate the case of parabolic contact geometries in great detail and we provide the full classification of those with semisimple groups of symmetries without complex factors. Finally, we explicitly construct all non-trivial contact geometries with non-complex simple groups of symmetries. We also indicate geometric interpretations of some of them.

MSC:
53C35 Differential geometry of symmetric spaces
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
53C30 Differential geometry of homogeneous manifolds
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