Development of an extended exterior differential calculus

Abstract

The purpose of this paper is to set up an algebraic machinery for the theory of affine connections on differentiable manifolds and to demonstrate by means of several applications the scope and convenience of this mechanism. We shall associate with a manifold a series of spaces, best described as spaces of multivectors with exterior differential form coefficients, and shall exhibit the algebraic relations between these spaces. It is possible to consider, in a more general fashion, spaces of tensors with differential form coefficients; this is done, in fact, in Cartan [3, Chap. VIII, Sec. II]('), where their use is justified by means of geometrical considerations. We shall define an affine connection as a certain kind of operator on the space of ordinary vector fields to the space of vector fields with differential one-form coefficients. It will be seen that this is simply another formulation of the classical definition. One of our basic results (Theorem 7.1) is that an affine connection induces an operator on each of the series of spaces just mentioned. In Chapter I, we shall summarize the facts that we need about differentiable manifolds and introduce some notation. Chapter II is devoted to the algebraic structure of the series of spaces T;q that we introduce. In Chapter III, we give the calculus associated with an affine connection and applications to a number of identities. In the final chapter, we discuss some applications of our calculus to Riemannian geometry, in particular to the curvatura integra of S. Chern. We hope in the future to give applications of this theory to other parts of differential geometry, possibly to the theory of harmonic integrals and to the theory of Lie groups.