Geodesic
Pencils of compatible metrics and integrable systems
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 5, pp. 889-937 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This survey is devoted to the theory of pencils of compatible Riemannian and pseudo-Riemannian metrics, related non-linear integrable systems, and applications. Bibliography: 82 titles.
Keywords: compatible metrics, compatible connections and curvatures, orthogonal curvilinear coordinate systems, Nijenhuis tensor, integrable systems, systems of hydrodynamic type.
Mots-clés : Riemann invariants 
@article{RM_2017_72_5_a2,
     author = {O. I. Mokhov},
     title = {Pencils of compatible metrics and integrable systems},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {889--937},
     year = {2017},
     volume = {72},
     number = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2017_72_5_a2/}
}
TY  - JOUR
AU  - O. I. Mokhov
TI  - Pencils of compatible metrics and integrable systems
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2017
SP  - 889
EP  - 937
VL  - 72
IS  - 5
UR  - http://geodesic.mathdoc.fr/item/RM_2017_72_5_a2/
LA  - en
ID  - RM_2017_72_5_a2
ER  - 
%0 Journal Article
%A O. I. Mokhov
%T Pencils of compatible metrics and integrable systems
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2017
%P 889-937
%V 72
%N 5
%U http://geodesic.mathdoc.fr/item/RM_2017_72_5_a2/
%G en
%F RM_2017_72_5_a2
O. I. Mokhov. Pencils of compatible metrics and integrable systems. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 72 (2017) no. 5, pp. 889-937. http://geodesic.mathdoc.fr/item/RM_2017_72_5_a2/

[1] O. I. Mokhov, “Compatible and almost compatible metrics”, Russian Math. Surveys, 55:4 (2000), 819–821 | DOI | DOI | MR | Zbl

[2] O. I. Mokhov, “Compatible and almost compatible pseudo-Riemannian metrics”, Funct. Anal. Appl., 35:2 (2001), 100–110 ; (2000), 18 pp., arXiv: math/0005051 | DOI | DOI | MR | Zbl

[3] O. I. Mokhov, “Compatible flat metrics”, J. Appl. Math., 2:7 (2002), 337–370 | DOI | MR | Zbl

[4] B. A. Dubrovin, S. P. Novikov, “Hamiltonian formalism of one-dimensional systems of hydrodynamic type, and the Bogolyubov–Whitham averaging method”, Soviet Math. Dokl., 27:3 (1983), 665–669 | MR | Zbl

[5] B. Dubrovin, “Geometry of 2D topological field theories”, Integrable systems and quantum groups (Montecatini Terme, 1993), Lecture Notes in Math., 1620, Springer, Berlin, 1996, 120–348 ; 1994, 204 pp., arXiv: hep-th/9407018 | DOI | MR | Zbl

[6] F. Magri, “A simple model of the integrable Hamiltonian equation”, J. Math. Phys., 19:5 (1978), 1156–1162 | DOI | MR | Zbl

[7] I. M. Gel'fand, I. Ya. Dorfman, “Hamiltonian operators and algebraic structures related to them”, Funct. Anal. Appl., 13:4 (1979), 248–262 | DOI | MR | Zbl

[8] B. Fuchssteiner, “Application of hereditary symmetries to nonlinear evolution equations”, Nonlinear Anal., 3:6 (1979), 849–862 | DOI | MR | Zbl

[9] A. S. Fokas, B. Fuchssteiner, “On the structure of symplectic operators and hereditary symmetries”, Lett. Nuovo Cimento (2), 28:8 (1980), 299–303 | DOI | MR

[10] P. J. Olver, Applications of Lie groups to differential equations, Grad. Texts in Math., 107, Springer-Verlag, New York, 1986, xxvi+497 pp. | DOI | MR | Zbl

[11] I. Dorfman, Dirac structures and integrability of nonlinear evolution equations, Nonlinear Sci. Theory Appl., John Wiley Sons, Ltd., Chichester, 1993, xii+176 pp. | MR

[12] O. I. Mokhov, “Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems”, Russian Math. Surveys, 53:3 (1998), 515–622 | DOI | DOI | MR | Zbl

[13] O. I. Mokhov, Simplekticheskaya i puassonova geometriya na prostranstvakh petel gladkikh mnogoobrazii i integriruemye uravneniya, In-t kompyuternykh issledovanii, M.–Izhevsk, 2004, 248 pp. ; O. I. Mokhov, Symplectic and Poisson geometry on loop spaces of smooth manifolds and integrable equations, Rev. Math. Math. Phys., 11, Part 2, Harwood Acad. Publ., Amsterdam, 2001, ii+204 СЃ. ; Rev. Math. Math. Phys., 13, Part 2, 2nd rev. ed., Cambridge Sci. Publ., Cambridge, 2009, ii+221 pp. | MR | MR | Zbl | Zbl

[14] O. I. Mokhov, “Local third-order Poisson brackets”, Russian Math. Surveys, 40:5 (1985), 233–234 | DOI | MR | Zbl

[15] O. I. Mokhov, “Hamiltonian differential operators and contact geometry”, Funct. Anal. Appl., 21:3 (1987), 217–223 | DOI | MR | Zbl

[16] C. S. Gardner, “Korteweg–de Vries equation and generalizations. IV. The Korteweg–de Vries equation as a Hamiltonian system”, J. Math. Phys., 12:8 (1971), 1548–1551 | DOI | MR | Zbl

[17] V. E. Zakharov, L. D. Faddeev, “Korteweg–de Vries equation: a completely integrable Hamiltonian system”, Funct. Anal. Appl., 5:4 (1971), 280–287 | DOI | MR | Zbl

[18] B. A. Dubrovin, S. P. Novikov, “Hydrodynamics of weakly deformed soliton lattices. Differential geometry and Hamiltonian theory”, Russian Math. Surveys, 44:6 (1989), 35–124 | DOI | MR | Zbl

[19] Y. Nutku, “On a new class of completely integrable nonlinear wave equations. II. Multi-Hamiltonian structure”, J. Math. Phys., 28:11 (1987), 2579–2585 | DOI | MR | Zbl

[20] P. J. Olver, Y. Nutku, “Hamiltonian structures for systems of hyperbolic conservation laws”, J. Math. Phys, 29:7 (1988), 1610–1619 | DOI | MR | Zbl

[21] M. Arik, F. Neyzi, Y. Nutku, P. J. Olver, J. M. Verosky, “Multi-Hamiltonian structure of the Born–Infeld equation”, J. Math. Phys., 30:6 (1989), 1338–1344 | DOI | MR | Zbl

[22] F. Neyzi, “Diagonalization and Hamiltonian structures of hyperbolic systems”, J. Math. Phys., 30:8 (1989), 1695–1698 | DOI | MR | Zbl

[23] H. Gümral, Y. Nutku, “Multi-Hamiltonian structure of equations of hydrodynamic type”, J. Math. Phys., 31:11 (1990), 2606–2611 | DOI | MR | Zbl

[24] E. V. Ferapontov, “Hamiltonian systems of hydrodynamic type and their realizations on hypersurfaces of a pseudo-Euclidean space”, J. Soviet Math., 55:5 (1991), 1970–1995 | DOI | MR | Zbl

[25] E. V. Ferapontov, M. V. Pavlov, “Quasiclassical limit of coupled KdV equations. Riemann invariants and multi-Hamiltonian structure”, Phys. D, 52:2-3 (1991), 211–219 | DOI | MR | Zbl

[26] E. Witten, “On the structure of the topological phase of two-dimensional gravity”, Nuclear Phys. B, 340:2-3 (1990), 281–332 | DOI | MR

[27] E. Witten, “Two-dimensional gravity and intersection theory on moduli space”, Surveys in differential geometry (Cambridge, MA, 1990), v. 1, Lehigh Univ., Bethlehem, PA, 1991, 243–310 | DOI | MR | Zbl

[28] R. Dijkgraaf, H. Verlinde, E. Verlinde, “Topological strings in $d 1$”, Nuclear Phys. B, 352:1 (1991), 59–86 | DOI | MR

[29] O. I. Mokhov, “On compatible Poisson structures of hydrodynamic type”, Russian Math. Surveys, 52:6 (1997), 1310–1311 | DOI | DOI | MR

[30] O. I. Mokhov, “On compatible potential deformations of Frobenius algebras and associativity equations”, Russian Math. Surveys, 53:2 (1998), 396–397 | DOI | DOI | MR | Zbl

[31] O. I. Mokhov, “Compatible Poisson structures of hydrodynamic type and associativity equations”, Proc. Steklov Inst. Math., 225 (1999), 269–284 | MR | Zbl

[32] O. Mokhov, “Compatible Poisson structures of hydrodynamic type and the equations of associativity in two-dimensional topological field theory”, Rep. Math. Phys., 43:1-2 (1999), 247–256 | DOI | MR | Zbl

[33] B. A. Dubrovin, S. P. Novikov, “On Poisson brackets of hydrodynamic type”, Soviet Math. Dokl., 30:3 (1984), 651–654 | MR | Zbl

[34] O. I. Mokhov, “Dubrovin–Novikov type Poisson brackets (DN-brackets)”, Funct. Anal. Appl., 22:4 (1998), 336–338 | DOI | MR | Zbl

[35] O. I. Mokhov, “The classification of multidimensional Poisson brackets of hydrodynamic type”, Russian Math. Surveys, 61:2 (2006), 356–358 | DOI | DOI | MR | Zbl

[36] O. I. Mokhov, “Multidimensional Poisson brackets of hydrodynamic type and flat pencils of metrics”, J. Geom. Symmetry Phys., 10 (2007), 51–71 | MR | Zbl

[37] O. I. Mokhov, “The classification of nonsingular multidimensional Dubrovin–Novikov brackets”, Funct. Anal. Appl., 42:1 (2008), 33–44 | DOI | DOI | MR | Zbl

[38] E. V. Ferapontov, P. Lorenzoni, A. Savoldi, “Hamiltonian operators of Dubrovin–Novikov type in 2D”, Lett. Math. Phys., 105:3 (2015), 341–377 | DOI | MR | Zbl

[39] O. I. Mokhov, E. V. Ferapontov, “Non-local Hamiltonian operators of hydrodynamic type related to metrics of constant curvature”, Russian Math. Surveys, 45:3 (1990), 218–219 | DOI | MR | Zbl

[40] O. I. Mokhov, E. V. Ferapontov, “Hamiltonian pairs associated with skew-symmetric Killing tensors on spaces of constant curvature”, Funct. Anal. Appl., 28:2 (1994), 123–125 | DOI | MR | Zbl

[41] O. I. Mokhov, “Hamiltonian systems of hydrodynamic type and constant curvature metrics”, Phys. Lett. A, 166:3-4 (1992), 215–216 | DOI | MR

[42] E. V. Ferapontov, “Differential geometry of nonlocal Hamiltonian operators of hydrodynamic type”, Funct. Anal. Appl., 25:3 (1991), 195–204 | DOI | MR | Zbl

[43] E. V. Ferapontov, “Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications”, Topics in topology and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 170, Adv. Math. Sci., 27, Amer. Math. Soc., Providence, RI, 1995, 33–58 | DOI | MR | Zbl

[44] V. L. Alekseev, “On non-local Hamiltonian operators of hydrodynamic type connected with Whitham's equations”, Russian Math. Surveys, 50:6 (1995), 1253–1255 | DOI | MR | Zbl

[45] B. Dubrovin, “Flat pencils of metrics and Frobenius manifolds”, Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), World Sci. Publ., River Edge, NJ, 1998, 47–72 | MR | Zbl

[46] 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 | DOI | DOI | MR | Zbl

[47] O. I. Mokhov, “Nonlocal Hamiltonian operators of hydrodynamic type with flat metrics, integrable hierarchies, and the associativity equations”, Funct. Anal. Appl., 40:1 (2006), 11–23 ; (2004), 23 pp., arXiv: math/0406292 | DOI | DOI | MR | Zbl

[48] O. I. Mokhov, “Frobenius manifolds as a special class of submanifolds in pseudo-Euclidean spaces”, Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 224, Adv. Math. Sci., 61, Amer. Math. Soc., Providence, RI, 2008, 213–246 ; 2007, 33 pp., arXiv: 0710.5860 | DOI | MR | Zbl

[49] A. Nijenhuis, “$X_{n-1}$-forming sets of eigenvectors”, Nederl. Akad. Wetensch. Proc. Ser. A., 54, Indag. Math., 13 (1951), 200–212 | DOI | MR | Zbl

[50] J. Haantjes, “On $X_m$-forming sets of eigenvectors”, Nederl. Akad. Wetensch. Proc. Ser. A., 58, Indag. Math., 17 (1955), 158–162 | DOI | MR | Zbl

[51] G. Darboux, Leçons sur les systèmes orthogonaux et les coordonnées curvilignes, 2ème éd., Gauthier-Villars, Paris, 1910, vii+567 pp. | MR | Zbl

[52] É. Cartan, Les systèmes différentiels extérieurs et leurs applications géométriques, Actualités Sci. Ind., 994, Hermann et Cie., Paris, 1945, 214 pp. | MR | Zbl

[53] L. Bianchi, Opere, v. 3, Sistemi tripli ortogonali, Edizioni Cremonese, Roma, 1955, 851 pp. | MR | Zbl

[54] 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., 94:1 (1998), 103–139 | DOI | MR | Zbl

[55] V. Zakharov, “Application of inverse scattering method to problems of differential geometry”, The legacy of the inverse scattering transform in applied mathematics (South Hadley, MA, 2001), Contemp. Math., 301, Amer. Math. Soc., Providence, RI, 2002, 15–34 | DOI | MR | Zbl

[56] I. M. Krichever, “Algebraic-geometric $n$-orthogonal curvilinear coordinate systems and solutions of the associativity equations”, Funct. Anal. Appl., 31:1 (1997), 25–39 | DOI | DOI | MR | Zbl

[57] A. E. Mironov, I. A. Taimanov, “Orthogonal curvilinear coordinate systems corresponding to singular spectral curves”, Proc. Steklov Inst. Math., 255 (2006), 169–184 | DOI | MR | Zbl

[58] D. A. Berdinskii, I. P. Rybnikov, “On orthogonal curvilinear coordinate systems in constant curvature spaces”, Sib. Math. J., 52:3 (2011), 394–401 | DOI | MR | Zbl

[59] O. A. Bogoyavlenskaya, “Ob odnom klasse konechnozonnykh krivolineinykh ortogonalnykh koordinat”, Sib. elektron. matem. izv., 12 (2015), 947–954 | DOI | MR | Zbl

[60] E. V. Glukhov, Algebro-geometricheskie metody postroeniya ortogonalnykh koordinat v rimanovykh prostranstvakh, Diplomnaya rabota. Mekh.-matem. fak-t, MGU im. M. V. Lomonosova, M., 2015, 12 pp.

[61] O. I. Mokhov, “Flat pencils of metrics and integrable reductions of Lamé's equations”, Russian Math. Surveys, 56:2 (2001), 416–418 | DOI | DOI | MR | Zbl

[62] O. I. Mokhov, “Integrability of the equations for nonsingular pairs of compatible flat metrics”, Theoret. and Math. Phys., 130:2 (2002), 198–212 | DOI | DOI | MR | Zbl

[63] V. E. Zakharov, A. B. Shabat, “A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I”, Funct. Anal. Appl., 8:3 (1974), 226–235 | DOI | MR | Zbl

[64] O. I. Mokhov, “The Lax pair for non-singular pencils of metrics of constant Riemannian curvature”, Russian Math. Surveys, 57:3 (2002), 603–605 | DOI | DOI | MR | Zbl

[65] O. I. Mokhov, “Compatible metrics of constant Riemannian curvature: local geometry, nonlinear equations, and integrability”, Funct. Anal. Appl., 36:3 (2002), 196–204 | DOI | DOI | MR | Zbl

[66] E. V. Ferapontov, “Compatible Poisson brackets of hydrodynamic type”, J. Phys. A, 34:11 (2001), 2377–2388 ; (2000), 14 pp., arXiv: math/0005221 | DOI | MR | Zbl

[67] O. I. Mokhov, “Lax pairs for compatible non-local Hamiltonian operators of hydrodynamic type”, Russian Math. Surveys, 57:6 (2002), 1234–1235 | DOI | DOI | MR | Zbl

[68] 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 | DOI | DOI | MR | Zbl

[69] O. I. Mokhov, “Compatible nonlocal Poisson brackets of hydrodynamic type and integrable reductions of the Lamé equations”, Nonlinear physics: theory and experiment (Gallipoli, 2002), v. II, World Sci. Publ., River Edge, NJ, 2003, 200–206 | MR | Zbl

[70] E. V. Ferapontov, “Surfaces with flat normal bundle: an explicit construction”, Differential Geom. Appl., 14:1 (2001), 15–37 ; (1998), 24 pp., arXiv: math/9805012 | DOI | MR | Zbl

[71] L. P. Eisenhart, Transformations of surfaces, 2nd ed., Chelsea Publishing Co., New York, 1962, xi+379 pp. | MR | Zbl

[72] S. P. Tsarev, “The geometry of Hamiltonian systems of hydrodynamic type. The generalized hodograph method”, Math. USSR-Izv., 37:2 (1991), 397–419 | DOI | MR | Zbl

[73] M. V. Pavlov, R. A. Sharipov, S. I. Svinolupov, “Invariant integrability criterion for equations of hydrodynamic type”, Funct. Anal. Appl., 30:1 (1996), 15–22 | DOI | DOI | MR | Zbl

[74] O. I. Mokhov, “Compatible Dubrovin–Novikov Hamiltonian operators and the Lie derivative”, Russian Math. Surveys, 56:6 (2001), 1175–1176 | DOI | DOI | MR | Zbl

[75] 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 | DOI | DOI | MR | Zbl

[76] 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 | DOI | DOI | MR

[77] O. I. Mokhov, “Integrable bi-Hamiltonian systems of hydrodynamic type”, Russian Math. Surveys, 57:1 (2002), 153–154 | DOI | DOI | MR | Zbl

[78] O. I. Mokhov, “Integrable bi-Hamiltonian hierarchies generated by compatible metrics of constant Riemannian curvature”, Russian Math. Surveys, 57:5 (2002), 999–1001 | DOI | DOI | MR | Zbl

[79] 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 | DOI | DOI | MR | Zbl

[80] 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 | DOI | DOI | MR | Zbl

[81] 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 | DOI | DOI | MR | Zbl

[82] A. P. Fordy, O. I. Mokhov, “On a special class of compatible Poisson structures of hydrodynamic type”, Phys. D, 152/153 (2001), 475–490 | DOI | MR | Zbl