A THEOREM ON HOLONOMY

Introduction. The object of this paper is to prove Theorem 2 of §2, which shows, for any connexion, how the curvature form generates the holonomy group. We believe this is an extension of a theorem stated without proof by E. Cartan [2, p. 4]. This theorem was proved after we had been informed of an unpublished related theorem of Chevalley and Koszul. We are indebted to S. S. Chern for many discussions of matters considered here. In §1 we give an exposition of some needed facts about connexions; this exposition is derived largely from an exposition of Chern [5 ] and partly from expositions of H. Cartan [3] and Ehresmann [7]. We believe this exposition does however contain one new element, namely Lemma 1 of §1 and its use in passing from H. Cartan's definition of a connexion (the definition given in §1) to E. Cartan's structural equation. 1. Basic concepts. We begin with some notions and terminology to be used throughout this paper. The term "differentiable" will always mean what is usually called "of class C00." We follow Chevalley [6] in general for the definition of tangent vector, differential, etc. but with the obvious changes needed for the differentiable (rather than analytic) case. However if <p is a differentiable mapping we use <j> again instead of d<f> and 8<(> for the induced mappings of tangent vectors and differentials. If AT is a manifold, by which we shall always mean a differentiable manifold but which is not assumed connected, and mGM, we denote the tangent space to M at m by Mm. If W is a vector field at M we denote its value at m by W(m), and if Xi, • • • , xk is a coordinate system of M we always write X\ ■ ■ ■ , Xk for the corresponding partial derivatives, i.e. Xi = d/dxi. We use the word "diffeo" for a 1:1 differentiable mapping of one manifold onto another whose inverse is also differentiable and call the manifolds diffemorphic. If W, W are vector fields we write [W, W] for WW'—W'W (opposite to [ó]). Rk will always denote ¿-dimensional Euclidean space of all ¿-tuples of real numbers and 8i, ■ • • , 8k will always denote the canonical unit elements there, i.e. 8, = (8lj, 82j, • ■ ■ , 8nj). Let G be a Lie group (i.e. a differentiable group) acting differentiably and effectively on a manifold F. It fGF and gGG we write gf for the image of/ under the action of gGG and if i is a tangent vector at/ we write gt for the image of i (which will be a tangent vector of gf) under g. If 0 is a manifold we denote by (0, F, G) the family of all transformations of OXF-^OXF of the form: (o,f)—>(o, t(o,f)) when i is any differentiable mapping of OXF—*F