Journal of the Mathematical Society of Japan

Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics

Vladimir S. MATVEEV

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Abstract

We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove the two-dimensional pseudo-Riemannian version of the projective Obata conjecture.

Article information

Source
J. Math. Soc. Japan, Volume 64, Number 1 (2012), 107-152.

Dates
First available in Project Euclid: 26 January 2012

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1327586976

Digital Object Identifier
doi:10.2969/jmsj/06410107

Mathematical Reviews number (MathSciNet)
MR2879738

Zentralblatt MATH identifier
1251.37058

Subjects
Primary: 37J35: Completely integrable systems, topological structure of phase space, integration methods
Secondary: 53B30: Lorentz metrics, indefinite metrics 53A45: Vector and tensor analysis 53B20: Local Riemannian geometry 53B50: Applications to physics 53C22: Geodesics [See also 58E10] 53D25: Geodesic flows 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60] 70E40: Integrable cases of motion 70H06: Completely integrable systems and methods of integration

Keywords
pseudo-Riemannian metrics geodesic flows quadratic integrals geodesically equivalent metrics projective transformations projective Obata conjecture superintegrability Beltrami problem Lie problem Killing vector field Killing tensor Liouville metrics separation of variables

Citation

MATVEEV, Vladimir S. Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics. J. Math. Soc. Japan 64 (2012), no. 1, 107--152. doi:10.2969/jmsj/06410107. https://projecteuclid.org/euclid.jmsj/1327586976


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