arXiv 0803.02892008-03math-phmath.MPnlin.SI

Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

Alexey V. Bolsinov, Vladimir S. Matveev, Giuseppe Pucacco

发表信息：: J. Geom. Phys. 59(2009), no. 7, 1048--1062
DOI：: 10.1016/j.geomphys.2009.04.010

英文摘要：

We discuss pseudo-Riemannian metrics on 2-dimensional manifolds such that the geodesic flow admits a nontrivial integral quadratic in velocities. We construct local normal forms of such metrics. We show that these metrics have certain useful properties similar to those of Riemannian Liouville metrics, namely: 1) they admit geodesically equivalent metrics; 2) one can use them to construct a big family of natural systems admitting integrals quadratic in momenta; 3) the integrability of such systems can be generalized to the quantum setting; 4) these natural systems are integrable by quadratures.

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