Rolling geodesics on symmetric semi-Riemannian spaces

This paper is an outgrowth of the results in the domain of rolling obtained in our recent paper written with F. Silva Leite and I. Markina, and the earlier papers on the rollings of spheres produced with J. Zimmerman. We show that the rolling equations associated with a symmetric semi-Riemannian manifold rolling on its tangent space at a fixed point on the manifold essentially have the same structure as the rolling equations for the n-dimensional sphere rolling on the horizontal hyperplane; that is, we show that the rolling equations are described by a left-invariant distribution D on a Lie group G with the Lie bracket growth D + [D,D] + [D, [D,D]] = TG, reminiscent of the growth (2, 3, 5) for the two spheres rolling on the horizontal plane. We then define rolling geodesics on semi-Riemannian spaces as extensions of sub-Riemannian geodesics in the Riemannian symmetric spaces, and show that the rolling geodesics are the projections of the extremal curves, which, remarkably, are the solution curves of a completely integrable Hamiltonian system in the cotangent bundle of the configuration space. Finally, we illustrate the theory with a few noteworthy examples.