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   \(k=0,1,-1\) $k=0,1,-1$ and let \(SO\left( M\right) \) $SO\left(M\right)$ 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 \( \lambda \in {\mathbb {R}}\) $\lambda \in \mathbb{R}$ , there is a three-dimensional distribution \(\mathcal { D}^{\lambda }\) ${\mathcal{D}}^{\lambda }$ on \(SO\left( M\right) \) $SO\left(M\right)$ accounting for infinitesimal rototranslations of constant pitch \(\lambda \) $\lambda$ . When \(\lambda \ne k^{2}\) $\lambda \ne k^2$ , there is a canonical sub-Riemannian structure on \({\mathcal {D}}^{\lambda }\) ${\mathcal{D}}^{\lambda }$ . We present a geometric characterization of its geodesics, using a previous Lie theoretical description. For \(k=0,-1\) $k=0,-1$ , we compute the sub-Riemannian length spectrum of \(\left( SO\left( M\right) ,{\mathcal {D}} ^{\lambda }\right) \) $\left(S,O,\left(M\right),,,{\mathcal{D}}^{\lambda }\right)$ 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.