コロキウム

This talk deals with the geodesic flows of Lie groups with respect to a left-invariant (pseudo-)Riemannian metric. A general framework is given to relate the complete integrability of the Hamiltonian systems for the geodesic flows on the cotangent bundle to the Lie group and that of the Lie-Poisson equations on the dual to the Lie algebra. The algebraic complete integrability is also mentioned. A class of left-invariant metrics on semi-simple Lie groups introduced by Mishchenko-Fomenko around 1980 is then discussed in view of complete integrability and the stability properties of their equilibria. As a related system, another Hamiltonian system on compact semi-simple Lie algebra is also considered in the motivation of information geometry.

The talk is based on joint works with Tudor S. Ratiu (Shanghai Jiao Tong University) and Jean-Pierre Françoise (LJLL, Sorbonne Université).