We discuss which semisimple locally symmetric spaces admit an AHS-structure invariant under local symmetries. We classify them for all types of AHS-structures and determine possible equivalence classes of such AHS-structures.
A reflexion space is generalization of a symmetric space introduced by O. Loos in [4]. We generalize locally symmetric spaces to local reflexion spaces in the similar way. We investigate, when local reflexion spaces are equivalently given by a locally flat Cartan connection of certain type.
In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In particular, we show that each symmetric conformal geometry is either locally flat or covered by a pseudo-Riemannian symmetric space, where the covering is a conformal map. We construct examples of locally flat symmetric conformal geometries that are not pseudo-Riemannian...
We construct series of examples of non-flat non-homogeneous parabolic geometries that carry a symmetry of the parabolic geometry at each point.
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