Geodesic
Remarks on Grassmannian Symmetric Spaces
Archivum mathematicum, Tome 44 (2008) no. 5, pp. 569-585 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 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.
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.
Classification : 53A40, 53C05, 53C15, 53C35
Keywords: parabolic geometries; Weyl structures; almost Grassmannian structures; symmetric spaces 
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Zalabová, Lenka; Žádník, Vojtěch. Remarks on Grassmannian Symmetric Spaces. Archivum mathematicum, Tome 44 (2008) no. 5, pp. 569-585. http://geodesic.mathdoc.fr/item/ARM_2008_44_5_a15/

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