scholarly journals A Note on Triple Systems and Totally Geodesic Submanifolds in a Homogeneous Space

1968 ◽  
Vol 32 ◽  
pp. 5-20 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Symmetric Space ◽  
Jordan Algebras ◽  
Riemannian Symmetric Space ◽  
Lie Triple System ◽  
Totally Geodesic ◽  
Triple Systems ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Nonassociative Algebras ◽  
Torsion Free

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.

scholarly journals The index conjecture for symmetric spaces

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Jürgen Berndt ◽  
Carlos Olmos
Keyword(s):  
Symmetric Spaces ◽  
Systematic Approach ◽  
Riemannian Symmetric Space ◽  
Totally Geodesic Submanifold ◽  
Totally Geodesic ◽  
Geodesic Submanifolds ◽  
Geodesic Submanifold ◽  
Totally Geodesic Submanifolds ◽  
Higher Rank ◽  
Differential Geom

AbstractIn 1980, Oniščik [A. L. Oniščik, Totally geodesic submanifolds of symmetric spaces, Geometric methods in problems of algebra and analysis. Vol. 2, Yaroslav. Gos. Univ., Yaroslavl’ 1980, 64–85, 161] introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank {\leq 2}, but for higher rank it was unclear how to tackle the problem. In [J. Berndt, S. Console and C. E. Olmos, Submanifolds and holonomy, 2nd ed., Monogr. Res. Notes Math., CRC Press, Boca Raton 2016], [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], [J. Berndt and C. Olmos, The index of compact simple Lie groups, Bull. Lond. Math. Soc. 49 2017, 5, 903–907], [J. Berndt and C. Olmos, On the index of symmetric spaces, J. reine angew. Math. 737 2018, 33–48], [J. Berndt, C. Olmos and J. S. Rodríguez, The index of exceptional symmetric spaces, Rev. Mat. Iberoam., to appear] we developed several approaches to this problem, which allowed us to calculate the index for many symmetric spaces. Our systematic approach led to a conjecture, formulated first in [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], for how to calculate the index. The purpose of this paper is to verify the conjecture.

scholarly journals A possible symplectic framework for Radon-type transforms

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.

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

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