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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

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

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

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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• http://mi.mathnet.ru/eng/faa243
• https://doi.org/10.4213/faa243
• http://mi.mathnet.ru/eng/faa/v35/i2/p24

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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, “Compatible Metrics of Constant Riemannian Curvature: Local Geometry, Nonlinear Equations, and Integrability”, Funct. Anal. Appl., 36:3 (2002), 196–204
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
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
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
5. O. I. Mokhov, “Integrable bi-Hamiltonian hierarchies generated by compatible metrics of constant Riemannian curvature”, Russian Math. Surveys, 57:5 (2002), 999–1001
6. O. I. Mokhov, “Lax pairs for compatible non-local Hamiltonian operators of hydrodynamic type”, Russian Math. Surveys, 57:6 (2002), 1234–1235
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
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
9. O. I. Mokhov, “Lax Pairs for Equations Describing Compatible Nonlocal Poisson Brackets of Hydrodynamic Type and Integrable Reductions of the Lamé Equations”, Theoret. and Math. Phys., 138:2 (2004), 238–249
10. Strachan, IAB, “Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures”, Differential Geometry and Its Applications, 20:1 (2004), 67
11. Liu, SQ, “Deformations of semisimple bihamiltonian structures of hydrodynamic type”, Journal of Geometry and Physics, 54:4 (2005), 427
12. O. I. Mokhov, “The classification of multidimensional Poisson brackets of hydrodynamic type”, Russian Math. Surveys, 61:2 (2006), 356–358
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
14. Abenda, S, “Reciprocal transformations and flat metrics on Hurwitz spaces”, Journal of Physics A-Mathematical and Theoretical, 40:35 (2007), 10769
15. O. I. Mokhov, “The Classification of Nonsingular Multidimensional Dubrovin–Novikov Brackets”, Funct. Anal. Appl., 42:1 (2008), 33–44
16. Dubrovin B., “Hamiltonian Perturbations of Hyperbolic PDEs: from Classification Results to the Properties of Solutions”, New Trends in Mathematical Physics, 2009, 231–276
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
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
19. I. Kh. Sabitov, “The Moscow Mathematical Society and metric geometry: from Peterson to contemporary research”, Trans. Moscow Math. Soc., 77 (2016), 149–175
20. O. I. Mokhov, “O metrikakh diagonalnoi krivizny”, Fundament. i prikl. matem., 21:6 (2016), 171–182
21. O. I. Mokhov, “Pencils of compatible metrics and integrable systems”, Russian Math. Surveys, 72:5 (2017), 889–937
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