Quadratic integrals of geodesic flows and solution of S. Lie problem
Vladimir Matveev (Jena)
In 1882 Sophus Lie asked to describe all 2-dimensional metrics admitting vector fields
whose flows sends (unparameterized) geodesics to geodesics.
I will explain the solution of this problem (part of the results are joint with R.L. Bryant and G. Manno).
The results are published in arXiv:0705.3592, arXiv:0802.2344, and arXiv:0802.2346.
The solution uses a description of metrics whose geodesic flows admit
integrals quadratic in velocities; I will give an overview of this subject.
Besides this theory, the solution consists of a trick and huge calculations. I will explain the trick and
hope you will explain me what other classical problems one can hope to solve by using
similar calculations.