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A construction of non-flat non-homogeneous symmetric parabolic geometries. (English) Zbl 1389.53023
In previous works, there are several constructions of symmetric parabolic geometries.
Using these constructions, one can obtain in particular:
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symmetric parabolic geometries with semisimple group $$G$$;
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flat and non-flat homogeneous symmetric parabolic geometries on homogeneous fiber bundles over symmetric spaces;
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flat non-homogeneous symmetric parabolic geometries which are not related to symmetric spaces;
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non-flat homogeneous symmetric parabolic geometries on a product of a flat model of a different (non-effective) type of parabolic geometry and a homogeneous space of a nilpotent Lie group.
In a previous work, the present authors proved a simple necessary and sufficient condition for the existence of symmetric parabolic geometries. It was also shown in another work that all non-homogeneous symmetric conformal geometries are flat.
In the present paper, using the combination of methods from previous works, the two series of non-flat non-homogeneous symmetric parabolic geometries are constructed and described in detail.
##### MSC:
 53B15 Other connections 53C10 $$G$$-structures 53C30 Differential geometry of homogeneous manifolds 58J70 Invariance and symmetry properties for PDEs on manifolds
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