Applied Analysis Seminar Hopf–Rinow Type Theorems and Periodic Geodesics on Half Lie Groups

Since Arnold’s seminal discovery, many PDEs from mathematical physics have been shown to admit an analogous geometric formulation. This viewpoint often reveals conserved quantities and, in several cases, leads to global weak solutions. Moreover, a Hopf–Rinow type theorem in this geometric setting would imply global well-posedness for the corresponding PDEs.

The geometric formulation of hydrodynamics by Arnold motivated the study of infinite-dimensional manifolds, and more precisely, half Lie groups — topological groups in which right multiplication is smooth while left multiplication is continuous. The main examples are groups of HsHs or CkCkdiffeomorphisms of compact manifolds.

In this talk, we will prove several Hopf–Rinow type theorems for right-invariant magnetic systems and certain Lagrangian systems on half Lie groups, extending the recent work of Bauer–Harms–Michor from the case of geodesic flows to this more general setting. Towards the end, we will show that on a half Lie group that is non-aspherical and is equipped with a strong Riemannian metric, there always exists a contractible periodic geodesic. The technical core of the work relies on the calculus of variations.

This is based on joint work with M. Bauer and F. Ruscelli.