Quantum integrability for the Beltrami--Laplace operators of projectively equivalent metrics of arbitrary signatures

V. S. Matveev

Čebyševskij sbornik, Tome 21 (2020) no. 2, pp. 275-289.

Résumé

We generalize the result of [31] to all signatures.

Détail

Keywords: integrable systems, Killing tensors, quantum integrable systems, Carter quantisation, commutative operators, projectively equivalent metrics, geodesically equivalent metrics, separation of variables, normal forms, geometric theory of PDE, c-projectively equivalent metrics.

Bibliographie

[1] E. Beltrami, “Resoluzione del problema: riportari i punti di una superficie sopra un piano in modo che le linee geodetische vengano rappresentante da linee rette”, Ann. Mat., 1:7 (1865)

[2] S. Benenti, C. Chanu, G. Rastelli, “Remarks on the connection between the additive separation of the Hamilton-Jacobi equation and the multiplicative separation of the Schrödinger equation. II. First integrals and symmetry operators”, J. Math. Phys., 43:11 (2002), 5223–5253 | DOI | MR | Zbl

[3] A. V. Bolsinov, V. S. Matveev, “Geometrical interpretation of Benenti systems”, Journal of Geometry and Physics, 44 (2003), 489–506 | DOI | MR | Zbl

[4] A. V. Bolsinov, V. S. Matveev, G. Pucacco, “Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta”, J. Geom. Phys., 59:7 (2009), 1048–1062 | DOI | MR | Zbl

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

[6] A. V. Bolsinov, V. S. Matveev, “Local normal forms for geodesically equivalent pseudo-Riemannian metrics”, Trans. Amer. Math. Soc., 367 (2015), 6719–6749 | DOI | MR | Zbl

[7] A. V. Bolsinov, V. S. Matveev, Th. Mettler, S. Rosemann, “Four-dimensional Kähler metrics admitting c-projective vector fields”, J. Math. Pures Appl. (9), 103:3 (2015), 619–657 | DOI | MR | Zbl

[8] A. V. Bolsinov, V. S. Matveev, S. Rosemann, 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, arXiv: 1510.00275 | MR

[9] A. Bolsinov, V. S. Matveev, E. Miranda, S. Tabachnikov, “Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems”, Philosophical Transactions A, 376 (2018), 20170430 | DOI | MR | Zbl

[10] A. V. Bolsinov, A. Yu. Konyaev, V. S. Matveev, Nijenhuis Geometry, arXiv: 1903.04603

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

[12] R. Bryant, G. Manno, V. S. Matveev, “A solution of S. Lie Problem: Normal forms of 2-dim metrics admitting two projective vector fields”, Math. Ann., 340:2 (2008), 437–463 | DOI | MR | Zbl

[13] D. Calderbank, M. Eastwood, V. S. Matveev, K. Neusser, “C-projective geometry”, Mem. AMS, 1512, arXiv: 1512.04516 | MR

[14] D. Calderbank, V. S. Matveev, S. Rosemann, “Curvature and the c-projective mobility of Kaehler metrics with hamiltonian 2-forms”, Compositio Math., 152 (2016), 1555–1575 | DOI | MR | Zbl

[15] B. Carter, “Killing tensor quantum numbers and conserved currents in curved space”, Phys. Rev. D (3), 16:12 (1977), 3395–3414 | DOI | MR

[16] C. Chanu, L. Degiovanni, G. Rastelli, “Modified Laplace-Beltrami quantization of natural Hamiltonian systems with quadratic constants of motion”, J. Math. Phys., 58:3 (2017), 033509 | DOI | MR | Zbl

[17] Th. Daudé, N. Kamran, F. Nicolea, “Separability and Symmetry Operators for Painlevé Metrics and their Conformal Deformations”, SIGMA, 15 (2019), 069, 42 pp., arXiv: 1903.10573 | MR | Zbl

[18] C. Duval, G. Valent, “Quantum integrability of quadratic Killing tensors”, J. Math. Phys., 46:5 (2005), 053516, 22 pp. | DOI | MR | Zbl

[19] M. Eastwood, V. S. Matveev, “Metric connections in projective differential geometry”, Symmetries and Overdetermined Systems of Partial Differential Equations (Minneapolis, MN, 2006), IMA Vol. Math. Appl., 144, Springer, New York, 2007, 339–351 | MR

[20] A. Fedorova, V. Kiosak, V. Matveev, S. Rosemann, “The only closed Kähler manifold with degree of mobility $>2$ is $ (CP (n) g_{Fubini-Study})$”, Proc. London Math. Soc., 105:1 (2012), 153–188 | DOI | MR | Zbl

[21] A. R. Gover, V. S. Matveev, “Projectively related metrics, Weyl nullity and metric projectively invariant equations”, Proc. Lond. Math. Soc. (3), 114:2 (2017), 242–292 | DOI | MR | Zbl

[22] V. Kiosak, V. S. Matveev, “Complete Einstein metrics are geodesically rigid”, Comm. Math. Phys., 289:1 (2009), 383–400 | DOI | MR | Zbl

[23] V. Kiosak, V. S. Matveev, “Proof Of The Projective Lichnerowicz Conjecture For Pseudo-Riemannian Metrics With Degree Of Mobility Greater Than Two”, Comm. Mat. Phys., 297:2 (2010), 401–426 | DOI | MR | Zbl

[24] B. Kruglikov, V. S. Matveev, “On vanishing of topological entropy for certain integrable systems”, Electron. Res. Announc. Amer. Math. Soc., 12 (2006), 19–28 | DOI | MR | Zbl

[25] B. Kruglikov, V. S. Matveev, “The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta”, Nonlinearity, 29 (2016), 1755–1768 | DOI | MR | Zbl

[26] T. Levi-Civita, “Sulle trasformazioni delle equazioni dinamiche”, Ann. di Mat., serie $2^a$, 24 (1896), 255–300 | Zbl

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

[28] V. S. Matveev, P. Topalov, “Metric with ergodic geodesic flow is completely determined by unparameterized geodesics”, Electron. Res. Announc. Amer. Math. Soc., 6 (2000), 98–104 | DOI | MR | Zbl

[29] V. S. Matveev, P. Topalov, “Integrability in the theory of geodesically equivalent metrics”, Kowalevski Workshop on Mathematical Methods of Regular Dynamics (Leeds, 2000), J. Phys. A, 34, no. 11, 2001, 2415–2433 | DOI | MR | Zbl

[30] V. S. Matveev, “Commuting operators and separation of variables for Laplacians of projectively equivalent metrics”, Let. Math. Phys., 54 (2000), 193–201 | DOI | MR | Zbl

[31] V. S. Matveev, P. Topalov, “Quantum integrability for the Beltrami-Laplace operator as geodesic equivalence”, Math. Z., 238:4 (2001), 833–866 | DOI | MR | Zbl

[32] V. S. Matveev, “Three-dimensional manifolds having metrics with the same geodesics”, Topology, 42:6 (2003), 1371–1395 | DOI | MR | Zbl

[33] V. S. Matveev, “Beltrami problem, Lichnerowicz-Obata conjecture and applications of integrable systems in differential geometry”, Tr. Semin. Vektorn. Tenzorn. Anal., 26, 2005, 214–238 | MR

[34] V. S. Matveev, “Hyperbolic manifolds are geodesically rigid”, Invent. math., 151 (2003), 579–609 | DOI | MR | Zbl

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

[36] Matveev V. S., “The eigenvalues of the Sinjukov mapping are globally ordered”, Math. Notes, 77:3–4 (2005), 380–390 | DOI | MR | Zbl

[37] V. S. Matveev, “Proof of the projective Lichnerowicz-Obata conjecture”, J. Differential Geom., 75:3 (2007), 459–502 | DOI | MR | Zbl

[38] V. S. Matveev, P. Mounoud, “Gallot-Tanno theorem for closed incomplete pseudo-Riemannian manifolds and applications”, Ann. Global Anal. Geom., 38:3 (2010), 259–271 | DOI | MR | Zbl

[39] V. S. Matveev, “Geodesically equivalent metrics in general relativity”, J. Geom. Phys., 62 (2012), 675–691 | DOI | MR | Zbl

[40] V. S. Matveev, “Two-dimensional metrics admitting precisely one projective vector field. This paper has an Appendix Dini theorem for pseudoriemannian metrics (joint with A. Bolsinov and G. Pucacco)”, Math. Ann., 352:4 (2012), 865–909 | DOI | MR | Zbl

[41] V. S. Matveev, S. Rosemann, “Proof of the Yano-Obata Conjecture for holomorph-projective transformations”, J. Diff. Geom., 92 (2012), 221–261 | DOI | MR | Zbl

[42] V. S. Matveev, S. Rosemann, “Conification construction for Kähler manifolds and its application in c-projective geometry”, Adv. Math., 274 (2015), 1–38 | DOI | MR | Zbl

[43] Matveev V. S., “On the number of nontrivial projective transformations of closed manifolds”, Journal of Mathematical Sciences, 223 (2015), 2017, 734–738 | DOI | MR

[44] V. S. Matveev, “Projectively invariant objects and the index of the group of affine transformations in the group of projective transformations”, Bull. Iran. Math. Soc., 44 (2018), 341–375 | DOI | MR | Zbl

[45] T. Otsuki, Y. Tashiro, “On curves in Kaehlerian spaces”, Math. J. Okayama Univ., 4 (1954), 57–78 | MR | Zbl

[46] Sinjukov N. S., Geodesic mappings of Riemannian spaces, Nauka, M., 1979 (in Russian) | MR | Zbl

[47] Y. Tashiro, “On a holomorphically projective correspondence in an almost complex space”, Math. J. Okayama Univ., 6 (1957), 147–152 | MR | Zbl

[48] G. Thompson, “Killing tensors in spaces of constant curvature”, J. Math. Phys., 27:11 (1986), 2693–2699 | DOI | MR | Zbl

[49] P. Topalov, “Geodesic hierarchies and involutivity”, J. Math. Phys., 42:8 (2001), 3898–3914 | DOI | MR | Zbl

[50] Th. Wolf, “Structural equations for Killing tensors of arbitrary rank”, Comput. Phys. Comm., 115 (1998), 316–329 | DOI | MR | Zbl

[51] A. Zeghib, “On discrete projective transformation groups of Riemannian manifolds”, Adv. Math., 297 (2016), 26–53 | DOI | MR | Zbl