31 Oct 2002

Vladimir Matveev (Freiburg University, Germany) "Quantum integrability as geodesic equivalence".

I will show that if two different Riemannian metrics on one $n$-dimensional manifold have the same geodesics (considered as unparameterized curves), then the Beltrami-Laplace operator of every of these metrics admits $n$ independent differential operators of second order commuting with it. The operators are self-adjoint and mutually commute. Using these, one can separate the variables in the equation on the eigenfunction of the Beltrami-Laplace operator.

The inverse is also true: if  the Beltrami-Laplace operator of a metric admits $n$ linearly independent commuting differential operators of a certain form, there exists another metric having the same geodesics.

I will give a complete description of such a situation for the two-dimensional case, and construct a few interesting series of examples (including the metric of the ellipsoid and the metric of the Poincare sphere) in the multidimensional case.