The sub-Riemannian length spectrum for screw motions of constant pitch on flat and hyperbolic 3-manifolds

Let M be an oriented three-dimensional Riemannian manifold of constant sectional curvature \(k=0,1,-1\) and let \(SO\left( M\right) \) be its direct orthonormal frame bundle (direct refers to positive orientation), which may be thought of as the set of all positions of a small body in M. Given \( \lambda \in {\mathbb {R}}\), there is a three-dimensional distribution \(\mathcal { D}^{\lambda }\) on \(SO\left( M\right) \) accounting for infinitesimal rototranslations of constant pitch \(\lambda \). When \(\lambda \ne k^{2}\), there is a canonical sub-Riemannian structure on \({\mathcal {D}}^{\lambda }\). We present a geometric characterization of its geodesics, using a previous Lie theoretical description. For \(k=0,-1\), we compute the sub-Riemannian length spectrum of \(\left( SO\left( M\right) ,{\mathcal {D}} ^{\lambda }\right) \) in terms of the complex length spectrum of M (given by the lengths and the holonomies of the periodic geodesics) when M has positive injectivity radius. In particular, for two complex length isospectral closed hyperbolic 3-manifolds (even if they are not isometric), the associated sub-Riemannian metrics on their direct orthonormal bundles are length isospectral.