Argument shift method and sectional operators: applications to differential geometry

A. V. Bolsinov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 20 (2015) no. 3, pp. 5-31.

Résumé

This paper is an attempt to present, in a systematic way, a construction that establishes an interesting relationship between some ideas and notions well-known in the theory of integrable systems on Lie algebras and a rather different area of mathematics studying projectively equivalent Riemannian and pseudo-Riemannian metrics.

Bibliographie

[1] Alekseevskii D. V., “Rimanovy prostranstva s neobychnymi gruppami golonomii”, Funkts. analiz i ego pril., 2:2 (1968), 1–10 | MR | Zbl

[2] Bogoyavlenskii O. I., “Integriruemye uravneniya Eilera na algebrakh Li, voznikayuschie v zadachakh matematicheskoi fiziki”, Izv. AN SSSR. Ser. matem., 48:5 (1984), 883–938 | MR | Zbl

[3] Bolsinov A. V., Borisov A. V., “Soglasovannye skobki Puassona na algebrakh Li”, Matem. zametki, 72:1 (2002), 11–34 | DOI | MR | Zbl

[4] Bolsinov A. V., Konyaev A. Yu., “Algebraicheskie i geometricheskie svoistva kvadratichnykh gamiltonianov, zadavaemykh sektsionnymi operatorami”, Matem. zametki, 90:5 (2011), 689–702 | DOI | MR | Zbl

[5] Golikov V. I., “Geodezicheskie otobrazheniya gravitatsionnykh polei obschego vida”, Tr. semin. po vekt. i tenz. analizu, 12, 1963, 79–129 | MR

[6] Domashev V. V., Mikesh I., “K teorii golomorfno-proektivnykh otobrazhenii kelerovykh prostranstv”, Matem. zametki, 23:2 (1978), 297–303 | MR | Zbl

[7] Konyaev A. Yu., “Bifurkatsionnaya diagramma i diskriminant spektralnoi krivoi integriruemykh sistem na algebrakh Li”, Matem. sb., 201:9 (2010), 27–60 | DOI | MR | Zbl

[8] Konyaev A. Yu., “Odnoznachnost vosstanovleniya parametrov sektsionnykh operatorov na prostykh kompleksnykh algebrakh Li”, Matem. zametki, 90:3 (2011), 384–393 | DOI | MR | Zbl

[9] Manakov S. V., “Zamechanie ob integrirovanii uravnenii Eilera dinamiki $n$-mernogo tvërdogo tela”, Funkts. analiz i ego pril., 10:4 (1976), 93–94 | MR | Zbl

[10] Mischenko A. S., Fomenko A. T., “Uravneniya Eilera na konechnomernykh gruppakh Li”, Izv. AN SSSR. Ser. matem., 42:2 (1978), 396–415 | MR | Zbl

[11] Mischenko A. S., Fomenko A. T., “Integriruemost uravnenii Eilera na poluprostykh algebrakh Li”, Tr. semin. po vekt. i tenz. analizu, 19, 1979, 3–94

[12] Petrov A. Z., “O geodezicheskikh otobrazheniyakh rimanovykh prostranstv neopredelënnoi metriki”, Uchën. zap. Kazansk. un-ta., 109, 1949, 4–36

[13] Reiman A. G., Semënov-tyan-Shanskii M. A., Integriruemye sistemy: teoretiko-gruppovoi podkhod, In-t kompyut. issled., M.–Izhevsk, 2003

[14] Sinyukov N. S., Geodezicheskie otobrazheniya rimanovykh prostranstv, Nauka, M., 1979 | MR

[15] Solodovnikov A. S., “Proektivnye preobrazovaniya rimanovykh prostranstv”, UMN, 11:4 (1956), 45–116 | MR | Zbl

[16] Trofimov V. V., Fomenko A. T., “Gruppovye neinvariantnye simplekticheskie struktury i gamiltonovy potoki na simmetricheskikh prostranstvakh”, Tr. semin. po vekt. i tenz. analizu, 21, 1983, 23–83 | MR | Zbl

[17] Trofimov V. V., Fomenko A. T., “Dinamicheskie sistemy na orbitakh lineinykh predstavlenii grupp Li i polnaya integriruemost nekotorykh gidrodinamicheskikh sistem”, Funkts. analiz i ego pril., 17:1 (1983), 31–39 | MR | Zbl

[18] Trofimov V. V., Fomenko A. T., Algebra i geometriya integriruemykh gamiltonovykh differentsialnykh uravnenii, Faktorial, M., 1995 | MR

[19] Shirokov A. P., “Ob odnom svoistve kovariantno postoyannykh affinorov”, Dokl. AN SSSR, 102 (1955), 464–467

[20] Aminova A. V., “Projective transformations of pseudo-Riemannian manifolds”, J. Math. Sci., 113:3 (2003), 367–470 | DOI | MR | Zbl

[21] Bérard-Bergery L., Ikemakhen A., “On the holonomy of Lorentzian manifolds”, Differential Geometry: Geometry in Mathematical Physics and Related Topics (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., 54, Amer. Math. Soc., Providence, 1993, 27–39 | DOI | MR

[22] Berger M., “Sur les groupes d'holonomie des variétés à connexion affine et des variétés Riemanniennes”, Bull. Soc. Math. France, 83 (1955), 279–330 | MR | Zbl

[23] Bloch A. M., Brinzanescu V., Iserles A., Marsden J. E., Ratiu T. S., “A class of integrable flows on the space of symmetric matrices”, Commun. Math. Phys., 290 (2009), 399–435 | DOI | MR | Zbl

[24] Bolsinov A. V., “Commutative families of functions related to consistent Poisson brackets”, Acta Appl. Math., 24 (1991), 253–274 | DOI | MR | Zbl

[25] Bolsinov A. V., Kiosak V., Matveev V. S., “A Fubini theorem for pseudo-Riemannian geodesically equivalent metrics”, J. London Math. Soc. (2), 80 (2009), 341–356 | DOI | MR | Zbl

[26] Bolsinov A. V., Matveev V. S., “Geometrical interpretation of Benenti systems”, J. Geom. Phys., 44 (2003), 489–506 | DOI | MR | Zbl

[27] Bolsinov A. V., Matveev V. S., “Splitting and gluing lemmas for geodesically equivalent pseudo-Riemannian metrics”, Trans. Amer. Math. Soc., 363:8 (2011), 4081–4107 | DOI | MR | Zbl

[28] Bolsinov A. V., Matveev V. S., Local normal forms for geodesically equivalent pseudo-Riemannian metrics, 2013, arXiv: 1301.2492v2

[29] Bolsinov A. V., Matveev V. S., Rosemann S., Local normal forms for c-projectively equivalent metrics and proof of the Yano–Obata conjecture in arbitrary signature. Proof of the projective Lichnerowicz conjecture for Lorentzian metrics, 2015, arXiv: 1510.00275v1

[30] Bolsinov A., Tsonev D., “On a new class of holonomy groups in pseudo-Riemannian geometry”, J. Diff. Geom., 97 (2014), 377–394 | MR | Zbl

[31] Boubel C., “On the holonomy of Lorentzian metrics”, Ann. Fac. Sci. Toulouse Math., 16:3 (2007), 427–475 ; ENS Lyon preprint No. 323, 2004 | DOI | MR | Zbl

[32] Boubel C., The algebra of the parallel endomorphisms of a germ of pseudo-Riemannian metric, 2013, arXiv: 1207.6544v5 | MR

[33] Boubel C., “On the algebra of parallel endomorphisms of a pseudo-Riemannian metric”, J. Differential Geom., 99:1 (2015), 77–123 | MR | Zbl

[34] Bryant R., “A survey of Riemannian metrics with special holonomy groups”, Proc. ICM Berkeley, Amer. Math. Soc., 1987, 505–514 | MR

[35] Bryant R., “Metrics with exceptional holonomy”, Ann. Math., 126 (1987), 525–576 | DOI | MR | Zbl

[36] Bryant R., “Classical, exceptional, and exotic holonomies: A status report”, Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger (Luminy, France, 12–18 juillet 1992), Sémin. Congr., 1, ed. A. L. Besse, Soc. Math. France, Paris, 1996, 93–165 | MR | Zbl

[37] Cartan É., “Les groupes d'holonomie des espaces généralisés”, Acta Math., 48 (1926), 1–42 ; Oeuvres complétes, Part III, v. 2, 1955, 997–1038 | DOI | Zbl

[38] Cartan É., “Sur une classe remarquable d'espaces de Riemann. I”, Bull. Soc. Math. France, 54 (1926), 214–264 ; “Sur une classe remarquable d'espaces de Riemann. II”, Bull. Soc. Math. France, 55 (1927), 114–134 ; Oeuvres complétes, Part I, v. 2, 1952, 587–659 | MR | Zbl | MR | Zbl

[39] Fomenko A. T., Symplectic Geometry. Methods and Applications, Gordon and Breach, Amsterdam, 1988, 1995 | MR

[40] Fomenko A. T., Trofimov V. V., Integrable Systems on Lie Algebras and Symmetric Spaces, Gordon and Breach, Amsterdam, 1988 | MR

[41] Fubini G., “Sui gruppi transformazioni geodetiche”, Mem. Acc. Torino, 53 (1903), 261–313 | Zbl

[42] Fubini G., “Sulle coppie di varieta geodeticamente applicabili. $1^\circ$ Sem.”, Rend. Acc. Lincei, 14:1 (1905), 678–683 | Zbl

[43] Galaev A. S., Classification of connected holonomy groups of pseudo-Kählerian manifolds of index 2, 2005, arXiv: math/0405098v2[math.DG]

[44] Galaev A., “The space of curvature tensors for holonomy algebras of Lorentzian manifolds”, Differential Geom. Appl., 22:1 (2005), 1–18 | DOI | MR | Zbl

[45] Galaev A., “Metrics that realize all Lorentzian holonomy algebras”, Int. J. Geom. Methods Mod. Phys., 3:5–6 (2006), 1025–1045 | DOI | MR | Zbl

[46] Galaev A. S., Leistner T., “Holonomy groups of Lorentzian manifolds: classification, examples, and applications”, Recent Developments in Pseudo-Riemannian Geometry, ESI Lect. Math. Phys., eds. D. V. Alekseevsky, H. Baum, Eur. Math. Soc., Zürich, 2008, 53–96 | MR | Zbl

[47] Ikemakhen A., “Examples of indecomposable non-irreducible Lorentzian manifolds”, Ann. Sci. Math. Québec, 20:1 (1996), 53–66 | MR | Zbl

[48] Joyce D., “Compact Riemannian $7$-manifolds with holonomy $G_2$. I”, J. Differential Geom., 43 (1996), 291–328 | MR | Zbl

[49] Joyce D., “Compact Riemannian $7$-manifolds with holonomy $G_2$. II”, J. Differential Geom., 43 (1996), 329–375 | MR | Zbl

[50] Joyce D., “A new construction of compact $8$-manifolds with holonomy $\mathrm{Spin}(7)$”, J. Differential Geom., 53 (1999), 89–130 | MR | Zbl

[51] Lancaster P., Rodman L., “Canonical forms for hermitian matrix pairs under strict equivalence and congruence”, SIAM Rev., 47 (2005), 407–443 | DOI | MR | Zbl

[52] Leep D. B., Schueller L. M., “Classication of pairs of symmetric and alternating bilinear forms”, Exposition. Math., 17:5 (1999), 395–414 | MR

[53] Leistner T., “On the classification of Lorentzian holonomy groups”, J. Differential Geom., 76:3 (2007), 423–484 | MR | Zbl

[54] Levi-Civita T., “Sulle trasformazioni delle equazioni dinamiche”, Ann. Mat. Pura Appl., Ser. 2a, 24 (1896), 255–300 | DOI | Zbl

[55] Marsden J. E., Ratiu T. S., Introduction to Mechanics and Symmetry, Springer, New York, 1999 | MR | Zbl

[56] Matveev V. S., Topalov P. J., “Trajectory equivalence and corresponding integrals”, Regular and Chaotic Dynamics, 3:2 (1998), 30–45 | DOI | MR | Zbl

[57] Matveev V. S., Topalov P. J., “Geodesic equivalence via integrability”, Geom. Dedicata, 96 (2003), 91–115 | DOI | MR | Zbl

[58] Merkulov S., Schwachhöfer L., “Classification of irreducible holonomies of torsion-free affine connections”, Ann. Math., 150 (1999), 77–149 | DOI | MR | Zbl

[59] Perelomov A. M., Integrable systems of classical mechanics and Lie algebras, v. I, Birkhäuser, Basel, 1990 | MR | Zbl

[60] Schwachhöfer L., “Connections with irreducible holonomy representations”, Adv. Math., 160:1 (2001), 1–80 | DOI | MR | Zbl