Homogeneous -symmetric spaces of interior type with simple real fundamental groups and their connection with parabolic spaces.

Summary: Let $𝔤$ be a real semisimple $| k |$ -graded Lie algebra such that the Lie algebra cohomology group $H^{1} (𝔤_{-}, 𝔤)$ is contained in negative homogeneous degrees. We show that if we choose $G = Aut (𝔤)$ and denote by $P$ the parabolic subgroup determined by the grading, there is an equivalence between regular, normal parabolic geometries of type $(G, P)$ and filtrations of the tangent bundle, such that each symbol algebra $gr (T_{x} M)$ is isomorphic to the graded Lie algebra $𝔤_{-}$ . Examples of parabolic geometries determined by filtrations...

Remarks on Grassmannian Symmetric Spaces

Lenka Zalabová, Vojtěch Žádník (2008)

Archivum Mathematicum

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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...

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