On the decomposition of a linearly connected manifold with torsion.

Abstract

Let M be a differentiable manifold with a linear connection, and let Φx be the homogeneous holonomy group at a point % e M. If the tangent vector space at x is decomposed into a direct sum of subspaces which are invariant under Φx, then by the parallel displacements along curves on M, parallel distributions are defined on M corresponding to those subspaces. If M is a Riemannian manifold and its connection is Riemannian, then by the de Rham decomposition theorem (Q7] or [_4Γ\ p. 185) the above parallel distributions are completely integrable and, at any point, M is locally isometric to the direct product of leaves through the point. Moreover, if M is simply connected and complete, it is globally isometric to the direct product of those leaves (see also [7] or [4] p. 192). The above local and global decomposition theorems of de Rham are generalized to the case of pseudo-Riemannian manifold by H. Wu ([9]). On the other hand, in [2], S. Kashiwabara generalized the global decomposition theorem to the case of linearly connected manifold without torsion, under the assumption of local decomposability. In the present paper, a linearly connected manifold with torsion will be treated and a condition of local decomposition will be given in terms of curvature and torsion (Theorem 1). Next, in §4, the results will be applied to a reductive homogeneous space with the canonical connection of the second kind, using the notion of algebra introduced by A. A. Sagle in [8]. Finally, in § 5, we shall remark about the decomposition of a local loop with any point in M as its origin (£3]), corresponding to the local decomposition of the linearly connected manifold M. The author wishes to express his hearty thanks to Prof. K. Morinaga for his kind suggestions and encouragement during the preparation of this paper at Hiroshima University.