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Marcos Salvai

The length spectrum of the sub-Riemannian geometry of screw motions with constant pitch
Thursday, 30 May, 2024 - 14:00
Résumé : 
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.
Institution de l'orateur : 
Thème de recherche : 
Théorie spectrale et géométrie
Salle : 
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