Symmetries of Parabolic Geometries

We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly $\mathrm{\mid }1\mathrm{\mid }$–graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of Weyl structures to discuss more interesting $\mathrm{\mid }1\mathrm{\mid }$–graded geometries which can carry a symmetry in a point with nonzero curvature. More concretely, we discuss the number of different symmetries which can exist at the point with nonzero curvature.