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Remarks on Grassmannian symmetric spaces. (English) Zbl 1212.53054
Summary: The classical concept of affine locally symmetric spaces allows for a generalization for various geometric structures on a smooth manifold. We recall 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 as an example a non-flat Grassmannian symmetric space. Next, we observe that there is a distinguished torsion-free affine connection preserving the Grassmannian structure such that, with respect to this connection, the Grassmannian symmetric space is an affine symmetric space in the classical sense.

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53A40 Other special differential geometries 53C05 Connections (general theory) 53C35 Differential geometry of symmetric spaces
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