# Shortest and straightest geodesics in sub-Riemannian geometry

Dates: | 10 May 2019 |
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Times: | 15:00 - 16:00 |

What is it: | Seminar |

Organiser: | Department of Mathematics |

Who is it for: | University staff, External researchers, Adults, Alumni, Current University students |

Speaker: | Professor Dmitri Alekseevsky |

Join us for this research seminar, part of the Geometry, topology and mathematical physics seminar series.

There are several different, but equivalent denitions of geodesics in a Riemannian manifold, They are generalized to sub-Riemannian manifolds, but become non-equivalent. H. R. Herz remarked that there are two main approaches for definition of geodesics: geodesics as shortest curves based on Maupertuis principle of least action (variational approach) and geodesics as straightest curves based on D'Alembert's principle of virtual work (which leads to geometric descriptions based on the notion of parallel transport). We briefly discuss different definitions of sub-Riemannian geodesics and interrelations between them.

A.M. Vershik and L.D. Faddeev showed that for a generic sub-Riemannian manifold Q all shortest geodesics (defined as projections of integral curves of the corresponding Hamiltonian flow) are different from straightest geodesics (defined by Schouten partial connection). They gave the first example when shortest geodesics coincide with straightest Hamiltonian geodesics (with the zero initial covector \lambda \in T^*Q) and stated the problem of characterisation of sub-Riemannian manifolds with such property. We show that this class contains Chaplygin transversally homogeneous systems, defined by the sub-Riemannian metric on the total space Q of a principal bundle \pi: Q \to M = Q/G over a Riemannian manifold (M; g^M), associated with a principal connection. Hamiltonian geodesics of such system describe evolution of a charged particle in Yang--Mills field and straightest geodesics --- the motion of a classical mechanical system with non-holonomic constraints. We describe some classes of homogeneous sub-Riemannian manifolds, where straightest geodesics coincides with shortest geodesics, including sub-Riemannian symmetric spaces.

### Speaker

Professor Dmitri Alekseevsky

Organisation: Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences

Biography: See Professor Dmitri Alekseevsky's profile: