Compatible and Almost Compatible Pseudo-Riemannian Metrics

O. I. Mokhov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 35 (2001) no. 2, pp. 24-36.

Résumé

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.

Bibliographie

[1] Dubrovin B., “Geometry of 2D topological field theories”, Lect. Notes Math., 1620, 1996, 120–348 ; arXiv: /hep-th/9407018 | DOI | MR | Zbl

[2] Dubrovin B., Differential geometry of the space of orbits of a Coxeter group, Preprint SISSA-29/93/FM | MR

[3] Dubrovin B., Flat pencils of metrics and Frobenius manifolds, Preprint SISSA 25/98/FM | MR

[4] Ferapontov E. V., “Nonlocal Hamiltonian operators of hydrodynamic type: differential geometry and applications”, Topics in topology and mathematical physics, ed. S. P. Novikov, Amer. Math. Soc., Providence, RI, 1995, 33–58 | MR | Zbl

[5] Mokhov O. I., “O soglasovannykh puassonovykh strukturakh gidrodinamicheskogo tipa”, UMN, 52:6 (1997), 171–172 | DOI | MR

[6] Mokhov O. I., “O soglasovannykh potentsialnykh deformatsiyakh frobeniusovykh algebr i uravneniyakh assotsiativnosti”, UMN, 53:2 (1998), 153–154 | DOI | MR | Zbl

[7] Mokhov O. I., “Soglasovannye puassonovy struktury gidrodinamicheskogo tipa i uravneniya assotsiativnosti”, Trudy MIRAN, 225, 1999, 284–300 | MR | Zbl

[8] Mokhov O. I., “Compatible Poisson structures of hydrodynamic type and the equations of associativity in two-dimensional topological field theory”, Reports on Mathematical Physics, 43:1–2 (1999), 247–256 | DOI | MR | Zbl

[9] Dubrovin B. A., Novikov S. P., “Gamiltonov formalizm odnomernykh sistem gidrodinamicheskogo tipa i metod usredneniya Bogolyubova–Uizema”, DAN SSSR, 270:4 (1983), 781–785 | MR | Zbl

[10] Mokhov O. I., Ferapontov E. V., “O nelokalnykh gamiltonovykh operatorakh gidrodinamicheskogo tipa, svyazannykh s metrikami postoyannoi krivizny”, UMN, 45:3 (1990), 191–192 | MR | Zbl

[11] Ferapontov E. V., “Differentsialnaya geometriya nelokalnykh gamiltonovykh operatorov gidrodinamicheskogo tipa”, Funkts. analiz i ego pril., 25:3 (1991), 37–49 | MR | Zbl

[12] Ferapontov E. V., “Gamiltonovye sistemy gidrodinamicheskogo tipa i ikh realizatsiya na giperpoverkhnostyakh psevdoevklidova prostranstva”, Itogi nauki i tekhniki. Problemy geometrii, 22, VINITI, M., 1990, 59–96 | MR

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

[14] Mokhov O. I., Ferapontov E. V., “Gamiltonovy pary, porozhdaemye kososimmetrichnymi tenzorami Killinga na prostranstvakh postoyannoi krivizny”, Funkts. analiz i ego pril., 28:2 (1994), 60–63 | MR | Zbl

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

[16] Dorfman I., Dirac structures and integrability of nonlinear evolution equations, John Wiley Sons, Chichester, England, 1993 | MR

[17] Mokhov O. I., “Lokalnye skobki Puassona tretego poryadka”, UMN, 40:5 (1985), 257–258 | MR | Zbl

[18] Mokhov O. I., “Gamiltonovy differentsialnye operatory i kontaktnaya geometriya”, Funkts. analiz i ego pril., 21:3 (1987), 53–60 | MR | Zbl

[19] Cooke D. B., “Classification results and the Darboux theorem for low-order Hamiltonian operators”, J. Math. Phys., 32:1 (1991), 109–119 | DOI | MR | Zbl

[20] Cooke D. B., “Compatibility conditions for Hamiltonian pairs”, J. Math. Phys., 32:11 (1991), 3071–3076 | DOI | MR | Zbl

[21] Nutku Y., “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

[22] Nijenhuis A., “$X_{n-1}$-forming sets of eigenvectors”, Indag. Math., 13:2 (1951), 200–212 | DOI | MR | Zbl

[23] Darboux G., Leçons sur les systèmes orthogonaux et les coordonnées curvilignes. 2nd ed., Gauthier-Villars, Paris, 1910 | MR | Zbl

[24] Zakharov V. E., “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

[25] Mokhov O. I., “Simplekticheskie i puassonovy struktury na prostranstvakh petel gladkikh mnogoobrazii i integriruemye sistemy”, UMN, 53:3 (1998), 85–192 | DOI | MR | Zbl

[26] Alekseev V. L., “O nelokalnykh gamiltonovykh operatorakh gidrodinamicheskogo tipa, svyazannykh s uravneniyami Uizema”, UMN, 50:6 (1995), 165–166 | MR | Zbl