Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

In the study of nonassociative algebras various “triple systems” frequently arise from the associator function and other multilinear objects. In particular Lie triple systems arise in the study of Jordan algebras and a generalization of a Lie triple system arises in Malcev algebras. Lie triple systems also are used to study totally geodesic submanifolds of a Riemannian symmetric space. We shall show how a generalization of Lie triple systems also arises from the study of curvature and geodesies of a torsion free connexion on a manifold and bring out the relation of this to various nonassociative algebras.

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Homotopy of compact symmetric spaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500008764 ◽

1992 ◽

Vol 34 (2) ◽

pp. 221-228 ◽

Cited By ~ 5

Author(s):

John M. Burns

Keyword(s):

Symmetric Spaces ◽

Unified Approach ◽

Topological Characterization ◽

New Approach ◽

Totally Geodesic ◽

Compact Symmetric Space ◽

Geodesic Submanifolds ◽

Totally Geodesic Submanifolds ◽

Compact Symmetric Spaces

In recent years a new approach to the study of compact symmetric spaces has been taken by Nagano and Chen [10]. This approach assigned to each pair of antipodal points on a closed geodesic a pair of totally geodesic submanifolds. In this paper we will show how these totally geodesic submanifolds can be used in conjunction with a theorem of Bott to compute homotopy in compact symmetric spaces. Some of the results are already known (see [1], [5], [11] for example) but we include them here for completeness and to illustrate this unified approach. We also exhibit a connection between the second homotopy group of a compact symmetric space and the multiplicity of the highest root. Using this in conjunction with a theorem of J. H. Cheng [6] we obtain a topological characterization of quaternionic symmetric spaces with antiquaternionic involutive isometry. The author would like to thank Prof T. Nagano for all his help and his detailed descriptions of the totally geodesic submanifolds mentioned above.

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A possible symplectic framework for Radon-type transforms

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887816410024 ◽

2016 ◽

Vol 13 (Supp. 1) ◽

pp. 1641002

Author(s):

Michel Cahen ◽

Thibaut Grouy ◽

Simone Gutt

Keyword(s):

Symmetric Space ◽

Homogeneous Space ◽

Symplectic Geometry ◽

Symmetric Spaces ◽

Symplectic Manifolds ◽

Totally Geodesic ◽

Canonical Connection ◽

Geodesic Submanifolds ◽

Totally Geodesic Submanifolds ◽

Compactly Supported Function

Our project is to define Radon-type transforms in symplectic geometry. The chosen framework consists of symplectic symmetric spaces whose canonical connection is of Ricci-type. They can be considered as symplectic analogues of the spaces of constant holomorphic curvature in Kählerian Geometry. They are characterized amongst a class of symplectic manifolds by the existence of many totally geodesic symplectic submanifolds. We present a particular class of Radon type transforms, associating to a smooth compactly supported function on a homogeneous manifold [Formula: see text], a function on a homogeneous space [Formula: see text] of totally geodesic submanifolds of [Formula: see text], and vice versa. We describe some spaces [Formula: see text] and [Formula: see text] in such Radon-type duality with [Formula: see text] a model of symplectic symmetric space with Ricci-type canonical connection and [Formula: see text] an orbit of totally geodesic symplectic submanifolds.

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