Functional Analysis and Its Applications

, Volume 36, Issue 3, pp 196–204 | Cite as

Compatible Metrics of Constant Riemannian Curvature: Local Geometry, Nonlinear Equations, and Integrability

  • O. I. Mokhov


The description problem is solved for compatible metrics of constant Riemannian curvature. Nonlinear equations describing all nonsingular pencils of compatible metrics of constant Riemannian curvature are derived and their integrability by the inverse scattering method is proved. In particular, a Lax pair with a spectral parameter is found for these nonlinear equations. We prove that all nonsingular pairs of compatible metrics of constant Riemannian curvature are described by special integrable reductions of the nonlinear equations defining orthogonal curvilinear coordinate systems in spaces of constant curvature.

flat pencil of metrics, compatible metrics, metric of constant Riemannian curvature, nonlinear integrable equation, Lax pair, compatible Poisson brackets 


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  1. 1.
    O. I. Mokhov, “Compatible and almost compatible pseudo-Riemannian metrics,” Funkts. Anal. Prilozhen., 35, No.2, 24–36 (2001); E-print, arXiv: math.DG/0005051.Google Scholar
  2. 2.
    O. I. Mokhov, “On integrability of the equations for nonsingular pairs of compatible flat metrics,” E-print, arXiv: math.DG/0005081.Google Scholar
  3. 3.
    O. I. Mokhov, “Flat pencils of metrics and integrable reductions of the Lamé equations,” Usp. Mat. Nauk, 56, No.2, 221–222 (2001); English transl. in Russian Math. Surveys, 56, No. 2 (2001).Google Scholar
  4. 4.
    E. V. Ferapontov, Compatible Poisson brackets of hydrodynamic type, E-print, arXiv: math.DG/0005221.Google Scholar
  5. 5.
    O. I. Mokhov and E. V. Ferapontov, “Nonlocal Hamiltonian operators of hydrodynamic type related to metrics of constant curvature,” Usp. Mat. Nauk, 45, No.3, 191–192 (1990); English transl. in Russian Math. Surveys, 45, No. 3 (1990).Google Scholar
  6. 6.
    B. A. Dubrovin and S. P. Novikov, “The Hamiltonian formalism of one-dimensional systems of hydrodynamic type and the Bogolyubov-Whitham averaging method,” Dokl. Akad. Nauk SSSR, 270, No.4, 781–785 (1983); English transl. in Soviet Math. Dokl., 27, No. 4, 665–669 (1983).Google Scholar
  7. 7.
    B. Dubrovin, Geometry of 2D topological field theories, Lect. Notes in Math., Vol. 1620, 1996, pp. 120–348; hep-th/9407018.Google Scholar
  8. 8.
    V. E. Zakharov, “Description of the n-orthogonal curvilinear coordinate systems and Hamiltonian integrable systems of hydrodynamic type, I: Integration of the Lamé equations,” Duke Math. J., 94, No.1, 103–139 (1998).Google Scholar
  9. 9.
    I. M. Krichever, “Algebraic-geometric n-orthogonal curvilinear coordinate systems and solutions of the associativity equations,” Funkts. Anal. Prilozhen., 31, No.1, 32–50 (1997).Google Scholar
  10. 10.
    G. Darboux, Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, 2nd ed., Gauthier-Villars, Paris, 1910.Google Scholar

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© Plenum Publishing Corporation 2002

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  • O. I. Mokhov

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