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Funktsional. Anal. i Prilozhen., 2002, Volume 36, Issue 3, Pages 36–47 (Mi faa202)  

This article is cited in 9 scientific papers (total in 9 papers)

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

O. I. Mokhov

Landau Institute for Theoretical Physics, Centre for Non-linear Studies

Abstract: 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.

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

DOI: https://doi.org/10.4213/faa202

Full text: PDF file (148 kB)
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English version:
Functional Analysis and Its Applications, 2002, 36:3, 196–204

Bibliographic databases:

UDC: 517.986+512.54
Received: 24.12.2001

Citation: O. I. Mokhov, “Compatible Metrics of Constant Riemannian Curvature: Local Geometry, Nonlinear Equations, and Integrability”, Funktsional. Anal. i Prilozhen., 36:3 (2002), 36–47; Funct. Anal. Appl., 36:3 (2002), 196–204

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. O. I. Mokhov, “Quasi-Frobenius Algebras and Their Integrable $N$-Parameter Deformations Generated by Compatible $(N\times N)$ Metrics of Constant Riemannian Curvature”, Theoret. and Math. Phys., 136:1 (2003), 908–916  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. O. I. Mokhov, “The Liouville Canonical Form for Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies”, Funct. Anal. Appl., 37:2 (2003), 103–113  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. O. I. Mokhov, “Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamй Equations”, Theoret. and Math. Phys., 138:2 (2004), 238–249  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    4. M. V. Pavlov, “The description of pairs of compatible first-order differential geometric poisson brackets”, Theoret. and Math. Phys., 142:2 (2005), 244–258  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. A. E. Mironov, I. A. Taimanov, “Orthogonal Curvilinear Coordinate Systems Corresponding to Singular Spectral Curves”, Proc. Steklov Inst. Math., 255 (2006), 169–184  mathnet  crossref  mathscinet
    6. O. I. Mokhov, “Compatible metrics and the diagonalizability of nonlocally bi-Hamiltonian systems of hydrodynamic type”, Theoret. and Math. Phys., 167:1 (2011), 403–420  mathnet  crossref  crossref  mathscinet  adsnasa  isi
    7. D. A. Berdinskii, I. P. Rybnikov, “On orthogonal curvilinear coordinate systems in constant curvature spaces”, Siberian Math. J., 52:3 (2011), 394–401  mathnet  crossref  mathscinet  isi
    8. O. I. Mokhov, “O metrikakh diagonalnoi krivizny”, Fundament. i prikl. matem., 21:6 (2016), 171–182  mathnet
    9. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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