Existence in the large of parallelism homomorphisms

ROBERT HERMANN(i) 1. Introduction.All manifolds, tensor-fields, curves, maps, etc., will be of differentiability class C unless mentioned otherwise.All manifolds will be paracompact and connected.Let M be a manifold of dimension n. (1 ^ i, j, ^ it; summation convention.)An everywhere linearly independent set of 1-differential forms, (cOj), on M defines an absolute parallelism for M. We will consider one such parallelism as fixed on M. Let M' be another manifold, with a set of 1-forms (co'i), not necessarily defining a parallelism.Our problem: When is there a map tp:M' -*M such that 0*(a)) = ui'p.(If co is a differential form on M, <f>*(co) is the form on M' induced by 0, denoted by <p(co) in [4].)Suppose that:The cJki are the first order invariants of the parallelism on M. Define the second, third, order invariants cJkih, cJkithiM,---as follows:Suppose further that: 1.3.There are functions c'Jki, c'jkih, on M' such that dm = c'jki co'j A co'k, dc'jki = Cm* o}'h, etc.Let x0 (resp x'0) be a fixed point of M (resp M') and let P (resp P') be the space of(Cco) curves of M starting at x0 (resp at x'0).If 0 exists, with ()(x'0) = x0, it defines a map 0* : P' -+ P. The method of Ambrose and Hicks [1 ; 9] is conversely to first construct 0*, then find the conditions that it pass to the quotient with respect to the natural projections P' -> M' and P^M.Usually, this involves some sort of completeness condition for M, a condition on the fundamental group of M', and a requirement that the corresponding first order invariants are the same along curves corresponding under 0".. Our aim is to build a bridge between this