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Funktsional. Anal. i Prilozhen., 2001, Volume 35, Issue 2, Pages 24–36 (Mi faa243)  

This article is cited in 21 scientific papers (total in 21 papers)

Compatible and Almost Compatible Pseudo-Riemannian Metrics

O. I. Mokhov

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

Abstract: In the present paper, the notions of compatible and almost compatible Riemannian and pseudo-Riemannian metrics are introduced. These notions are motivated by the theory of compatible Poisson structures of hydrodynamic type (local and nonlocal) and generalize the notion of flat pencils of metrics, which plays an important role in the theory of integrable systems of hydrodynamic type and Dubrovin's theory of Frobenius manifolds. Compatible metrics generate compatible Poisson structures of hydrodynamic type (these structures are local for flat metrics and nonlocal if the metrics are not flat). For the “nonsingular” case in which the eigenvalues of a pair of metrics are distinct, we obtain a complete explicit description of compatible and almost compatible metrics.

DOI: https://doi.org/10.4213/faa243

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English version:
Functional Analysis and Its Applications, 2001, 35:2, 100–110

Bibliographic databases:

UDC: 517.9+514.7
Received: 27.04.2000

Citation: O. I. Mokhov, “Compatible and Almost Compatible Pseudo-Riemannian Metrics”, Funktsional. Anal. i Prilozhen., 35:2 (2001), 24–36; Funct. Anal. Appl., 35:2 (2001), 100–110

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. O. I. Mokhov, “Compatible Metrics of Constant Riemannian Curvature: Local Geometry, Nonlinear Equations, and Integrability”, Funct. Anal. Appl., 36:3 (2002), 196–204  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. O. I. Mokhov, “Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Hierarchies Related to Them”, Theoret. and Math. Phys., 132:1 (2002), 942–954  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. O. I. Mokhov, “Compatible Dubrovin–Novikov Hamiltonian Operators, Lie Derivative, and Integrable Systems of Hydrodynamic Type”, Theoret. and Math. Phys., 133:2 (2002), 1557–1564  mathnet  crossref  crossref  mathscinet  isi  elib
    4. O. I. Mokhov, “The Lax pair for non-singular pencils of metrics of constant Riemannian curvature”, Russian Math. Surveys, 57:3 (2002), 603–605  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. O. I. Mokhov, “Integrable bi-Hamiltonian hierarchies generated by compatible metrics of constant Riemannian curvature”, Russian Math. Surveys, 57:5 (2002), 999–1001  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    6. O. I. Mokhov, “Lax pairs for compatible non-local Hamiltonian operators of hydrodynamic type”, Russian Math. Surveys, 57:6 (2002), 1234–1235  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    7. 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
    8. 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
    9. 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
    10. Strachan, IAB, “Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures”, Differential Geometry and Its Applications, 20:1 (2004), 67  crossref  mathscinet  zmath  isi  scopus
    11. Liu, SQ, “Deformations of semisimple bihamiltonian structures of hydrodynamic type”, Journal of Geometry and Physics, 54:4 (2005), 427  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. O. I. Mokhov, “The classification of multidimensional Poisson brackets of hydrodynamic type”, Russian Math. Surveys, 61:2 (2006), 356–358  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. Dubrovin, B, “On Hamiltonian perturbations of hyperbolic systems of conservation laws I: Quasi-triviality of bi-Hamiltonian perturbations”, Communications on Pure and Applied Mathematics, 59:4 (2006), 559  crossref  mathscinet  zmath  isi
    14. Abenda, S, “Reciprocal transformations and flat metrics on Hurwitz spaces”, Journal of Physics A-Mathematical and Theoretical, 40:35 (2007), 10769  crossref  mathscinet  zmath  adsnasa  isi  scopus
    15. O. I. Mokhov, “The Classification of Nonsingular Multidimensional Dubrovin–Novikov Brackets”, Funct. Anal. Appl., 42:1 (2008), 33–44  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. Dubrovin B., “Hamiltonian Perturbations of Hyperbolic PDEs: from Classification Results to the Properties of Solutions”, New Trends in Mathematical Physics, 2009, 231–276  crossref  mathscinet  zmath  isi
    17. O. I. Mokhov, “Riemann invariants of semisimple non-locally bi-Hamiltonian systems of hydrodynamic type and compatible metrics”, Russian Math. Surveys, 65:6 (2010), 1183–1185  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. 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
    19. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175  mathnet  crossref  elib
    20. O. I. Mokhov, “O metrikakh diagonalnoi krivizny”, Fundament. i prikl. matem., 21:6 (2016), 171–182  mathnet
    21. 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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