A Note on Triple Systems and Totally Geodesic Submanifolds in a Homogeneous Space
A Note on Triple Systems and Totally Geodesic Submanifolds in a Homogeneous Space
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关于齐次空间中的三重系统和全测地线子流形的注记
DOI：
10.1017/s0027763000026568
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发表时间：
1968
影响因子：
0.8
通讯作者：
A. Sagle
中科院分区：
文献类型：
--
作者：
A. Sagle
在非结合代数的研究中，结合函数和其他多线性对象经常产生各种“三重系统”。特别是在Jordan代数的研究中出现了李三重系统，在Malcev代数中出现了李三重系统的推广。李三元系统也被用来研究黎曼对称空间的全测地线子流形。我们将说明李三系的推广也是如何从流形上无扭转连接的曲率和测地线的研究中产生的，并指出它与各种非结合代数的关系。
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.