In this note we show that the total curvature of a geodesic in the manifoldwith-boundary consisting of Euclidean 3-space with a boundary of the form z = f(x, y) has a bound of at most 2p iff satisfies a Lipschitz condition with the Lipschitz constant at most p. This global result immediately yields the local result that such a geodesic has one-sided derivatives everywhere and moreover, that dE/ds = 1 everywhere on the geodesic, where E denotes the Euclidean length and s denotes the arc length. The bound of 2p is clearly the best possible, being achieved, for example, on a right circular cone of vertex half-angle arc cot p.
These results depend on the fact that the geodesic, by virtue of its ability to take locally short paths through 3-space where available, is, broadly speaking, more convex than the boundary manifold; even though the surface can be rather ugly, the geodesic ignores the pathology, skittering over bad folds and retaining an almost pristine differentiability. Certainly, neither our total curvature nor our differentiability results hold if the geodesic is constrained to be a geodesic on the boundary manifold. Appropriately corrugated surfaces yield easy counter-examples. Our result is also qualitatively the best possible in the sense that, even for a differentiable surface, the lack of a uniform Lipschitz constant may allow the total curvature to be unbounded, dE/ds to not be 1, and the one-sided derivative to somewhere fail to exist.
This note represents a portion of work done jointly with R. Alexander, S. Alexander and R. Bishop in a current investigation of manifolds-withboundary. The techniques involved have some resemblance to those used in consideration of geodesics on convex surfaces, as described in, e.g., Busemann's tract [2]; in particular we produce an extension of an idea of Liberman's [3] as described in Busemann. We note that our hard earned result, dE/ds = 1 everywhere, is automatic for a geodesic on a convex surface. R. Alexander and S. Alexander [1] have shown, by elegant local techniques, that a geodesic such as we are considering has one-sided derivatives everywhere, as long as the Lipschitz constant of the boundary is less than 1. F. E. Wolter [-4] has also obtained (two-sided) differentiability of geodesics with appropriate stronger conditions on the boundary surface in a Riemannian setting. The method of Alexander and Alexander goes over to the case of a Riemannian manifold as ambient space, in which setting it is not clear what form our more global results take.