Integrable geodesic flows on the torus

Daniel Tsodikovich (ISTA)

We will discuss the work of Agapov, Bialy, and Mironov from 2016 about integrable geodesic flows on the torus, in which they prove two results: 

1. Any analytic Liouville metric without magnetic field can be deformed into a non-Liouville metric with non-zero magnetic field, the flow of which is integrable on one level set, with integral  quadratic polynomial in momentum variables.
2. Globally there is rigidity – if there is a quadratic polynomial integral for the whole torus for a magnetic geodesic flow, then up to some change of variables the flow has essentially a unique form.