Rolling motions of pseudo-orthogonal groups

The classical definition of a rolling map, describing the rolling motion, without slip or twist, of one Euclidean submanifold over another of the same dimension, as given in Sharpe [8], is generalized for the situation when the embedded space is equipped with a pseudo-Riemannian metric and applied to derive the kinematic equations for the constrained rolling motion of a connected pseudo-Riemannian orthogonal group over its affine tangent spaces at a point. The kinematic equations are solved explicitly when the curve along which the first manifold rolls is a geodesic. We also show that rolling motions along a curve with non-holonomic constraints of not-wist and no-slip encode parallel transport, and derive formulas for the tangent and normal parallel transport of a vector along geodesics. Finally, we make a brief reference on how rolling motions can be used to generate smooth interpolating curves on pseudo-orthogonal groups.