Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs
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Sergey I. Agafonov
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Thaís G. P. Alves
Abstract
We prove that if the geodesic flow on a surface has an integral which is fractional-linear in momenta, then the dimension of the space of such integrals is either 3 or 5, the latter case corresponding to constant gaussian curvature. We give also a geometric criterion for the existence of fractional-linear integrals: such an integral exists if and only if the surface carries a geodesic 4-web with constant cross-ratio of the four directions tangent to the web leaves.
Funding statement: This research was supported by the FAPESP grant #2022/12813-5 (S.I.A) and by the CAPES grant #88882.434346/2019-01 (T.G.P.A).
Communicated by: P. Eberlein
References
[1] S. I. Agafonov, Hexagonal geodesic 3-webs. Int. Math. Res. Not. IMRN (2021) no. 15, 11585–11617. MR4294127 Zbl 1496.3707310.1093/imrn/rnz172Suche in Google Scholar
[2] S. I. Agafonov, Quadratic integrals of geodesic flow, webs, and integrable billiards. J. Geom. Phys. 161 (2021), Paper No. 104041, 7 pages. MR4188322 Zbl 1478.5301810.1016/j.geomphys.2020.104041Suche in Google Scholar
[3] S. Agapov, V. Shubin, Rational integrals of 2-dimensional geodesic flows: new examples. J. Geom. Phys. 170 (2021), Paper No. 104389, 8 pages. MR4321657 Zbl 1484.5311410.1016/j.geomphys.2021.104389Suche in Google Scholar
[4] S. V. Agapov, Rational integrals of a natural mechanical system on a two-dimensional torus. (Russian) Sibirsk. Mat. Zh. 61 (2020), 255–265. English translation: Sib. Math. J. 61 (2020), 199–207. MR4188387 Zbl 1473.7003210.1134/S0037446620020020Suche in Google Scholar
[5] S. V. Agapov, On the first integrals of two-dimensional geodesic flows. (Russian) Sibirsk. Mat. Zh. 61 (2020), 721–734. English translation: Sib. Math. J. 61 (2020), 563–574. MR4195239 Zbl 1453.3705610.1134/S0037446620040011Suche in Google Scholar
[6] M. Bialy, A. E. Mironov, Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete Contin. Dyn. Syst. 29 (2011), 81–90. MR2725282 Zbl 1232.3703510.3934/dcds.2011.29.81Suche in Google Scholar
[7] W. Blaschke, Einführung in die Geometrie der Waben. Basel–Stuttgart: Birkhäuser Verlag 1955. MR75630 Zbl 0068.3650110.1007/978-3-0348-6952-2Suche in Google Scholar
[8] G. Darboux, Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal, I–IV. Paris: Gautier–Villars 1887, 1889. Bronx, NY: Chelsea Publ. Co. 1972. Sceaux: Éditions Jacques Gabay 1993. MR396214 MR1324110 MR1365962 JFM 19.0746.02 JFM 25.1159.02Suche in Google Scholar
[9] U. Dini, Sopra un problema che si presenta nella teoria generale delle rappresentazione geografiche di una superficie su di un’altra. Brioschi Ann. (2) 3 (1869), 269–294. JFM 02.0622.0210.1007/BF02422982Suche in Google Scholar
[10] J. P. Dufour, D. Lehmann, Le rang des tissus de Nakai. Preprint 2022, arXiv:2204.10077Suche in Google Scholar
[11] S. Finsterwalder, Mechanische Beziehungen bei der Flächendeformation. Deutsche Math.-Ver. 6 (1899), 45–90. JFM 30.0623.02Suche in Google Scholar
[12] H. Graf, R. Sauer, Über dreifache Geradensysteme in der Ebene, welche Dreiecksnetze bilden. Münchener Sitzungsb. Math.-Naturw. Abt. (1924), 119–156. JFM 50.0396.02Suche in Google Scholar
[13] C. G. J. Jacobi, Vorlesungen über Dynamik. In: Gesammelte Werke, Supplementband. Edited by E. Lottner. Berlin: Reimer 1884, 1891. English translation: Jacobi’s lectures on dynamics. New Delhi: Hindustan Book Agency 2009. JFM 16.0028.01 JFM 18.0016.03 MR2569315 Zbl 1200.01019Suche in Google Scholar
[14] G. Koenigs, Sur les géodesiques a intégrales quadratiques. Bull. Soc. Philom. Paris (8) 5 (1893), 26–28. Also in [8], volume IV, as note II. JFM 25.1196.03 JFM 24.0729.01Suche in Google Scholar
[15] V. N. Kolokol’ tsov, Geodesic flows on two-dimensional manifolds with an additional first integral that is polynomial with respect to velocities. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 994–1010, 1135. English translation: Math. USSR, Izv. 21 (1983), 291–306. MR675528 Zbl 0518.5803310.1070/IM1983v021n02ABEH001792Suche in Google Scholar
[16] V. V. Kozlov, On rational integrals of geodesic flows. Regul. Chaotic Dyn. 19 (2014), 601–606. MR3284603 Zbl 1343.3704710.1134/S156035471406001XSuche in Google Scholar
[17] B. Kruglikov, Invariant characterization of Liouville metrics and polynomial integrals. J. Geom. Phys. 58 (2008), 979–995. MR2441213 Zbl 1145.5306710.1016/j.geomphys.2008.03.005Suche in Google Scholar
[18] S. Lie, Untersuchungen über geodätische Curven. Math. Ann. 20 (1882), 357–454. MR151017310.1007/BF01443601Suche in Google Scholar
[19] G. Manno, M. V. Pavlov, Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows. J. Geom. Phys. 113 (2017), 197–205. MR3603765 Zbl 1358.5308810.1016/j.geomphys.2016.10.023Suche in Google Scholar
[20] I. Nakai, Curvature of curvilinear 4-webs and pencils of one forms: variation on a theorem of Poincaré, Mayrhofer and Reidemeister. Comment. Math. Helv. 73 (1998), 177–205. MR1611766 Zbl 0926.5301110.1007/s000140050051Suche in Google Scholar
[21] L. V. Ovsiannikov, Group analysis of differential equations. Academic Press 1982. MR668703 Zbl 0485.5800210.1016/B978-0-12-531680-4.50012-5Suche in Google Scholar
[22] R. Sauer, Flächen mit drei ausgezeichneten Systemen geodätischer Linien, die sich zu einem Dreiecksnetz verknüpfen lassen. Sitzungsberichte München (1926), 353–397. Zbl 52.0700.02Suche in Google Scholar
[23] R. W. Sharpe, Differential geometry. Springer 1997. MR1453120 Zbl 0876.53001Suche in Google Scholar
[24] P. Stäckel, Lineare Scharen geodätischer Linien. Math. Ann. 56 (1902), 501–506. MR1511185 Zbl 0266071210.1007/BF01444172Suche in Google Scholar
[25] V. V. Ten, Local integrals of geodesic flows. (Russian) Regul. Khaoticheskaya Din. 2 (1997), 87–89. MR1652149 Zbl 1083.37510Suche in Google Scholar
[26] S. P. Tsarëv, Poisson brackets and one-dimensional Hamiltonian systems of hydrodynamic type. (Russian) Dokl. Akad. Nauk SSSR 282 (1985), 534–537. English translation: Sov. Math., Dokl. 31 (1985), 488–491. MR796577 Zbl 0605.35075Suche in Google Scholar
[27] E. Vessiot, Sur l’intégration des systèmes différentiels qui admettent des groupes continus de transformations. Acta Math. 28 (1904), 307–349. MR1555005 JFM 35.0343.0410.1007/BF02418390Suche in Google Scholar
[28] O. Volk, Über Flächen mit geodätischen Dreiecknetzen. Sitzungsber. Heidelb. Akad. Wiss., Math.-Naturwiss. Kl. (1929), 2–32. JFM 55.0399.0510.1515/9783111409665Suche in Google Scholar
© 2024 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Inequalities for f*-vectors of lattice polytopes
- Lower bound on the translative covering density of octahedra
- Variations on the Weak Bounded Negativity Conjecture
- Poisson Structures on moduli spaces of Higgs bundles over stacky curves
- Generalized Shioda–Inose structures of order 3
- Deformation cones of Tesler polytopes
- Some observations on conformal symmetries of G2-structures
- Characterization of the sphere and of bodies of revolution by means of Larman points
- Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs
- The feet of orthogonal Buekenhout–Metz unitals
Artikel in diesem Heft
- Frontmatter
- Inequalities for f*-vectors of lattice polytopes
- Lower bound on the translative covering density of octahedra
- Variations on the Weak Bounded Negativity Conjecture
- Poisson Structures on moduli spaces of Higgs bundles over stacky curves
- Generalized Shioda–Inose structures of order 3
- Deformation cones of Tesler polytopes
- Some observations on conformal symmetries of G2-structures
- Characterization of the sphere and of bodies of revolution by means of Larman points
- Fractional-linear integrals of geodesic flows on surfaces and Nakai’s geodesic 4-webs
- The feet of orthogonal Buekenhout–Metz unitals
