Symmetries of parabolic geometries - Zalabová, Lenka, 1979- (2007)
We introduce and discuss symmetries for the so called parabolic geometries. Our motivation comes from affine locally symmetric spaces. The manifold with affine connection is locally symmetric if and only if the torison vanishes and the curvature is covariantly constant. These geometries can be understood as the special case of reductive Cartan geometries. The parabolic geometries represent another special case of the general Cartan geometries. They are of second order and never reductive. We are interested in $|1|$--graded geometries. In this case, the definition of the symmetry is a generalization of the clasical one and follows the intuitive idea. We show an analogy of the results from the affine locally symmetric spaces and we get more curvature restrictions, which come from the general theory of parabolic geometries. Many types of symmetric $|1|$--graded geometries have to be locally flat. There are also some `interesting' types, which can carry a symmetry in the poin.