局所簡約空間と接代数系

A linearly connected space is called locally reductive if both of the torsion tensor field and the curvature tensor field are parallel. A symmetric space with a canonical connection is such a space of vanishing torsion. On the other hand, every Lie group has a left invariant connection (( - ) -connection) with parallel torsion and vanishing curvature. From this point of view, the geometry of locally reductive spaces has been studied by K. Nomizu in his paper [4] and he showed that a locally reductive space is determined, locally, by its torsion and curvature at a given point.
Observing the tangent algebras of these spaces, K. Yamaguti has introduced in [5] an algebraic system, called general Lie triple system, which is a generalization of both of Lie algebra and Lie triple system, and it has been studied, algebraically, by himself [6] and others.
In the present paper, we shall investigate a correspondence between certain locally reductive spaces and general Lie triple systems as their tangent algebras. In the case of connected, simply connected and complete locally reductive spaces, which can be regarded as homogeneous spaces (Theorem 2), a remarkable correspondence will be seen (Theorem 3). We shall also study certain subspaces of a locally reductive space and subsystems of its tangent algebra. Some results about symmetric spaces will be given as corollaries.