The exponential map for the Lagrange problem on differentiable manifolds

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.