## Abstract

A *parabolic space* 2P = 0*[M, H, a) is defined as a C00 manifold *M*, a sub-bundle *H* of the tangent bundle *T* of *M* and a Cm symmetric, bilinear function a: H® which induces a positive-definite quadratic form on each fibre of H. A path t -> f(t) in M is called horizontal if its tangent vector is everywhere in *H*. The Lagrange problem considered is that of finding, in the set Q(P, Q) of piecewise Cl horizontal paths in *M* which join fixed points P, Q, a path f 0 which minimizes the integral §a(f (t)® f (t)) d t.Such an f 0 is called a *geodesic arc*. For each x e M there is an exponential map ex: T* -> M of the set of covectors at *x* into *M* such that, for y e is geodesic, and also ex{N*) — {*}. Here, N* c T* is defined by the exact sequence 0 -> N* -> T* H->0; the epimorphism T* -> H being given by y-> ry, where y(&) — a[cr ® Ty), The behaviour of ex near N* is studied and the following theorems are proved under the hypothesis (*A*) that, for every nonzero local section y of N* (a 1-form on *M*), dy, has maximal rank: (1) there is a neighbourhood Ux of the origin 0 X of T* such that ex UXN* is diffeomorphic, (2) for every C3 horizontal path f: R - + M such that f(0) = x, there exists e >0 such that jf|( — e, e) can be factorized in the form exf x, where f x(0) = Ox and /.(0) exists and is not tangential to N*. The method of proof is to show (without hypothesis (*A*)) that SP determines canonically a parabolic structure 3PM', H', a') on M ' = Hx © Nx such that (primes being used for SP' and *x* being identified with the zero of Hx © Nx) e'x is a first approximation to ex near N'* k, N* when ex, e'x are compared in suitable charts. The geodesic properties of SP' are readily computed and they lead to theorem (1) relative to £P'. The theorem ex ~ e'x then allows this result to be carried over into SP. The approximate location of the set ex(Ux) is found in terms of a chart and it is proved that, for a path /a as in (2), f~ 1ex[Ux) 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 SP, every pair of points* P*, *Q* for which iQ [P,Q) is not empty can be joined by a minimizing geodesic arc. Theorems (1) and (2) imply that a C3 horizontal path / can be approximated by a geodesic polygon pj- which is homotopic to / by a standard homotopy of Morse theory. (No positive lower bound for the lengths of the sides of pf is given—this would be a functional of the curvature of /.) As far as practicable, intrinsic notations are employed.

## Footnotes

This text was harvested from a scanned image of the original document using optical character recognition (OCR) software. As such, it may contain errors. Please contact the Royal Society if you find an error you would like to see corrected. Mathematical notations produced through Infty OCR.

- Received June 15, 1966.

- Scanned images copyright © 2017, Royal Society

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