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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.

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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