Marcos Salvai

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 has dimension six and 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 $\lambda\neq k^2$, there is a canonical sub-Riemannian structure on $\mathcal{D}^{\lambda }$. We describe its geodesics. For $k=0,-1$, we compute the lengths of all periodic geodesics of $\left( SO\left( M\right) ,\mathcal{D}^{\lambda }\right) $ in terms of  the lengths and the holonomies of the periodic geodesics of $M$. 

 

It turns out that the notion of rototranslating with constant pitch makes sense for some higher dimensional Riemannian manifolds, for instance, for $\mathbb{R}^{7}$ via the octonionic cross product, or for compact simple Lie groups. We define sub-Riemannian structures analogous to the above and find some of their geodesics.