Quantum integrability and complete separation of variables for projectively equivalent metrics on the torus. (English) Zbl 1070.53500

Mladenov, Ivaïlo M. (ed.) et al., Proceedings of the 2nd international conference on geometry, integrability and quantization, Varna, Bulgaria, June 7–15, 2000. Sofia: Coral Press Scientific Publishing (ISBN 954-90618-2-5/pbk). 228-244 (2001).
Summary: Let two Riemannian metrics \(g\) and \(\overline g\) on the torus \(T^n\) have the same geodesics (considered as unparameterized curves). Then we can construct invariantly \(n\) commuting differential operators of second order. The Laplacian \(\Delta_g\) of the metric \(g\) is one of these operators. For any \(x\in T^n\), consider the linear transformation \(G\) of \(T_x T^n\) given by the tensor \(g^{i\alpha}\overline g_{\alpha j}\). If all eigenvalues of \(G\) are different at one point of the torus then they are different at every point; the operators are linearly independent and we can globally separate the variables in the equation \(\Delta_gf=\mu f\) on this torus.
For the entire collection see [Zbl 0957.00038].


53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
53C22 Geodesics in global differential geometry
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics