## Abstract

A parabolic space $\scr{P}$ = $\scr{P}$(M, H, a) is defined as a C$^{\infty}$ manifold M, a sub-bundle H of the tangent bundle T of M and a C$^{\infty}$ symmetric, bilinear function a: H $\otimes $ H $\rightarrow $ R which induces a positive-definite quadratic form on each fibre of H. A path t $\rightarrow $ f(t) in M is called horizontal if its tangent vector $\dot{f}$(t) is everywhere in H. The Lagrange problem considered is that of finding, in the set $\Omega $(P, Q) of piecewise C$^{1}$ horizontal paths in M which join fixed points P, Q, a path f$_{0}$ which minimizes the integral $\int $a($\dot{f}$(t)$\otimes \dot{f}$(t)) dt. Such an f$_{0}$ is called a geodesic arc. For each x $\in $ M there is an exponential map e$_{x}$: T$_{x}^{\ast}\rightarrow $ M of the set of covectors at x into M such that, for y $\in $ T$_{x}^{\ast}$, t $\rightarrow $ e$_{x}$(ty) is geodesic, and also e$_{x}$(N$_{x}^{\ast}$) = {x}. Here, N$^{\ast}\subset $ T$^{\ast}$ is defined by the exact sequence 0 $\rightarrow $ N$^{\ast}\rightarrow $ T$^{\ast}\rightarrow $ H $\rightarrow $ 0; the epimorphism T$^{\ast}\rightarrow $ H being given by y $\rightarrow \tau _{y}$, where y($\sigma $) = a($\sigma \otimes \tau _{y}$), $\sigma \in $ H. The behaviour of e$_{x}$ near N$_{x}^{\ast}$ is studied and the following theorems are proved under the hypothesis (A) that, for every nonzero local section $\mu $ of N$^{\ast}$ (a 1-form on M), d$\mu $ has maximal rank: (1) there is a neighbourhood U$_{x}$ of the origin O$_{x}$ of T$_{x}^{\ast}$ such that e$_{x}|$U$_{x}\backslash $N$_{x}^{\ast}$ is diffeomorphic, (2) for every C$^{3}$ horizontal path f: R $\rightarrow $ M such that f(0) = x, there exists $\epsilon $ > 0 such that f$|$(-$\epsilon $, $\epsilon $) can be factorized in the form e$_{x}\tilde{f}_{x}$, where $\tilde{f}_{x}$(0) = O$_{x}$ and $\tilde{f}_{x}$(0) exists and is not tangential to N$^{\ast}$. The method of proof is to show (without hypothesis (A)) that $\scr{P}$ determines canonically a parabolic structure $\scr{P}^{\prime}$(M$^{\prime}$, H$^{\prime}$, a$^{\prime}$) on M$^{\prime}$ = H$_{x}$ $\oplus $ N$_{x}$ such that (primes being used for $\scr{P}^{\prime}$ and x being identified with the zero of H$_{x}$ $\oplus $ N$_{x}$) e$_{x}^{\prime}$ is a first approximation to e$_{x}$ near N$_{x}^{\prime \ast}\approx $ N$_{x}^{\ast}$ when e$_{x}$, e$_{x}^{\prime}$ are compared in suitable charts. The geodesic properties of $\scr{P}^{\prime}$ are readily computed and they lead to theorem (1) relative to $\scr{P}^{\prime}$. The theorem e$_{x}\sim $ e$_{x}^{\prime}$ then allows this result to be carried over into $\scr{P}$. The approximate location of the set e$_{x}$(U$_{x}$) is found in terms of a chart and it is proved that, for a path f as in (2), f$^{-1}$e$_{x}$(U$_{x}$) is open. This, after further analysis, yields (2). In the course of the paper various related results are established. In particular, it is proved (3) without assumptions of normality that a sufficiently short geodesic arc is shorter than any other horizontal arc joining its end-points, (4) that, in a complete space $\scr{P}$, every pair of points P, Q for which $\Omega $(P, Q) is not empty can be joined by a minimizing geodesic arc. Theorems (1) and (2) imply that a C$^{3}$ horizontal path f can be approximated by a geodesic polygon p$_{f}$ which is homotopic to f by a standard homotopy of Morse theory. (No positive lower bound for the lengths of the sides of p$_{f}$ is given-this would be a functional of the curvature of f.) As far as practicable, intrinsic notations are employed.