等質系について(I)

その他

In this paper, homogeneous systems which have been introduced in [4] will be considered on differentiable manifolds. It is intended to show that the various results in [2], [3] for a homogeneous Lie loop G are essentially those results for the homogeneous system of G. Let (G, η) be a differentiable homogeneous system on a connected differentiable manifold G. The canonical connection and the tangent Lie triple algebra of (G, η) are defined in §§1, 2 in the same way as in the case of homogeneous Lie loops [2]. At any point e, G can be expressed as a reductive homogeneous space A/K with the canonical connection and with the decomposition 〓 = 〓 + 〓 of the Lie algebra of A , where 〓 is the tangent L. t. a. of (G, η) at e. In §3 we shall treat of the regular homogeneous system, a geodesic homogeneous system G in which the linear representation of K on 〓 coincides with the holonomy group at e. The following fact will be shown in §4 ; if (G, η) is a regular homogeneous system, then there exists a 1-1 correspondence between the set of invariant subsystem of G and the set of invariant subalgebras of its tangent L. t. a. (Theorem 5).