Remarks on Grassmannian Symmetric Spaces

Lenka Zalabová and Vojtěch Žádník

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


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.