Address:
Faculty of Applied Informatics, Tomáš Baťa University Zlín, Czech Republic and International Erwin Schrödinger Institute for Mathematical Physics Wien, Austria
Faculty of Education, Masaryk University Brno, Czech Republic
E-mail:
zalabova@math.muni.cz
zadnik@math.muni.cz
Abstract: The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for $|1|$-graded parabolic geometries and for almost Grassmannian structures, in particular. As an application of two general constructions with parabolic geometries, we present an example of non-flat Grassmannian symmetric space. Next we observe there is a distinguished torsion-free affine connection preserving the Grassmannian structure so that, with respect to this connection, the Grassmannian symmetric space is an affine symmetric space in the classical sense.
AMSclassification: primary 53C15; secondary 53A40, 53C05, 53C35.
Keywords: parabolic geometries, Weyl structures, almost Grassmannian structures, symmetric spaces.