Tohoku Mathematical Journal

Lie sphere geometry and integrable systems

Eugene V. Ferapontov

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Two basic Lie-invariant forms uniquely defining a generic (hyper)surface in Lie sphere geometry are introduced. Particularly interesting classes of surfaces associated with these invariants are considered. These are the diagonally cyclidic surfaces and the Lie-minimal surfaces, the latter being the extremals of the simplest Lie-invariant functional generalizing the Willmore functional in conformal geometry. Equations of motion of a special Lie sphere frame are derived, providing a convenient unified treatment of surfaces in Lie sphere geometry. In particular, for diagonally cyclidic surfaces this approach immediately implies the stationary modified Veselov-Novikov equation, while the case of Lie-minimal surfaces reduces in a certain limit to the integrable coupled Tzitzeica system. In the framework of the canonical correspondence between Hamiltonian systms of hydrodynamic type and hypersurfaces in Lie sphere geometry, it is pointed out that invariants of Lie-geometric hypersurfaces coincide with the reciprocal invariants of hydrodynamic type systems. Integrable evolutions of surfaces in Lie sphere geometry are introduced. This provides an interpretation of the simplest Lie-invariant functional as the first local conservation law of the (2+1)-dimensional modified Veselov-Novikov hierarchy. Parallels between Lie sphere geometry and projective differential geometry of surfaces are drawn in the conclusion.

Article information

Tohoku Math. J. (2), Volume 52, Number 2 (2000), 199-233.

First available in Project Euclid: 3 May 2007

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Zentralblatt MATH identifier

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) 37K25: Relations with differential geometry 53A05: Surfaces in Euclidean space


Ferapontov, Eugene V. Lie sphere geometry and integrable systems. Tohoku Math. J. (2) 52 (2000), no. 2, 199--233. doi:10.2748/tmj/1178224607.

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  • [1] M A. AKIVIS AND V. V. GOLDBERG, Projective differential geometry of submanifolds, Math Library, Vol 49, North-Holland, 1993
  • [2] W BLASCHKE, Vorlesungen uber Differentialgeometrie, vol 3, Springer-Verlag, Berlin, 192
  • [3] W. BLASCHKE AND G. BOL, Geometric der Gewebe, Springer-Verlag, Berlin, 1938
  • [4] W BLASCHKE, Einfhrung in die Geometrie der Waben, Birkhauser-Verlag, Basel, Switzerland, 1955
  • [5] L. V. BOGDANOV, Veselov-Novikov equation as a natural two-dimensional generalizatrion of the Korteweg de Vries equation, Theoret and Math. Phys. 70 (1987), 309-314
  • [6] G. BOL, Projektive Differentialgeometrie, 2 Teil, Gttingen, 1954
  • [7] T. CECIL, Lie sphere geometry, Springer-Verlag, 199
  • [8] T. CECIL, On the Lie curvature of Dupin hypersurfaces, Kodai Math. J 13 (1990), 143-15
  • [9] A. DEMOULIN, Sur deux transformations des surfaces dont les quadriques de Lie n'ont que deux ou troi points caracteristiques, Bull, de 1'Acad Belgique 19 (1933), 479-502, 579-592, 1352-1363.
  • [10] E V FERAPONTOV, Reciprocal transformations and their invariants, Differ. Uravn. 25 (1989), no 7, 1256 1265 (English translation in Differential Equations 25 (1989), no 7, 898-905).
  • [11] E. V. FERAPONTOV, Reciprocal autotransformations andhydrodynamic symmetries, Differ. Uravn 27 (1991), no 7, 1250-1263 (English translation in Differential Equations 27 (1989) no 7, 885-895)
  • [12] E. V. FERAPONTOV, Dupin hypersurfaces and integrable Hamiltonian systems of hydrodynamic type whic do not possess Riemann invariants, Differential Geom. Appl. 5 (1995), 121-152.
  • [13] E V. FERAPONTOV, Nonlocal Hamiltonian operators of hydrodynamic type: Differential geometry and Ap plications, Amer. Math. Soc. Transl. Ser. 2, 170 (1995), 33-58.
  • [14] E V FERAPONTOV, Hamiltonian systems of hydrodynamic type and their realization on hypersurfaces of pseudo-euclidean space, Soviet!. Math 55 (1991), 1970-1995.
  • [15] E. V. FERAPONTOV, On integrability of 3x3 semihamiltonian hydrodynamic type systems which do no possess Riemann invariants, Phys D 63 (1993), 50-70.
  • [16] E. V. FERAPONTOV, Surfaces in Lie sphere geometry andthe stationary Davey-Stewartson hierarchy, Preprin SFB 288 (1997), no 287, Berlin
  • [17] E. V. FERAPONTOV AND W K. SCHIEF, Surfaces of Demoulin: Differntial geometry, Backhand transforma tion and Integrability, J. Geom. Phys. 30 (1999). 343-363
  • [18] E. V. FERAPONTOV, Stationary Veselov-Novikov equation and isothermally asymptotic surfaces in projectiv differential geometry, Differential Geom. Appl. 11 (1999), 117-128.
  • [19] E. V FERAPONTOV, Surfaces with flat normal bundle: an explicit construction, Preprint SFB 288 (1998), no 320, Berlin.
  • [20] S. P. FINIKOV, Projective Differential Geometry, Moscow-Leningrad, 193
  • [21] S. P. FINIKOV, Theory of congruences, Moscow-Leningrad, 1950
  • [22] L. GODEAUX, La theorie des surfaces et espace regie (Geometrie projective differentielle), Actualites scien tifiques et industrielles, N138, Paris, Hermann, 1934
  • [23] B G KONOPELCHENKO, Induced surfaces and their integrable dynamics, Stud Appl Math. (1996), 9-5
  • [24] B G. KONOPELCHENKO, Nets in R3, their integrable evolutions and the DS hierarchy, Physics lett. A 18 (1993), 153-159
  • [25] B G. KONOPELCHEKNO AND U. PiNKALL, Integrable deformations of affine surfaces via Nizhnik-Veselov Novikov equation, Preprint SFB 288 (1998), no 318, Berlin
  • [26] E LANE, A Treatise on Projective Differential Geometry, The Univ. of Chicago Press, 194
  • [27] S LIE, Uber Komplexe, inbesondere Linen- und Kugelkomplexe, mit Anwendung auf der Theorie der par tieller Differentialgleichungen, Math Ann 5(1872), 145-208, 209-256
  • [28] A. V MIKHAILOV, The reduction problem and the inverse scattering method, Phys D 3 (1981), 73-117
  • [29] R MIYAOKA, Dupin hypersurfaces and a Lie invariant, Kodai Math. J. 12 (1989), 228-25
  • [30] U PINKALL, Dupinische Hyperflachen in E4, Manuscripta Math 51 (1985), 89-11
  • [31] U. PINKALL, Dupin hypersurfaces, Math Ann 270 (1985), 427-44
  • [32] C ROGERS AND W F SHADWICK, Backlund transformations and their applications, Academic Press, Ne York, 1982
  • [33] C. ROGERS, Reciprocal transformations and their applications, Nonlinear Evolutions, Proc of the 5th Work shop on Nonlinear Evolution Equations and Dynamical systems, France, 1987, 109-123
  • [34] O. ROZET, Sur certaines congruences W attachee aux surfaces dont les quadriques de Lie n'ont que deu points caracteristiques, Bull. Sci.Math II 58 (1934), 141-151
  • [35] W K. SCHIEF, On the geometry of an integrable 2 + 1-dimensional sine-Gordon system, Proc Roy So London Ser A 453 (1997), 1671-1688
  • [36] D SERRE, Oscillations non lineaires des systemes hyperboliques: methodes et resultats qualitatifs, Ann. Ins H Poincare, Ann Non Lineaire 8 (1991), no 3-4, 351-417
  • [37] I A. TAIMANOV, Modified Novikov-Veselov equation and differential geometry of surfaces, in Solitons, Geometry and Topology (eds V. M Buchstaber and S P Novikov) Amer Math Soc.Transl Ser 2, 179 (1997), 133-155
  • [38] I A TAIMANOV, Surfaces of revolution in terms of solitons, Ann Global Anal Geom. 15 (1997), no 5, 419-435
  • [39] S. P. TSAREV, The geometry of Hamiltonian systems of hydrodynamic type The generalized hodograp transform, Math USSR Izv 37 (1991), 397-419
  • [40] G TZITZEICA, Sur une nouvelle classe de surfaces, C R. Acad Sci Paris 150 (1910), 955-95
  • [41] E. I. WILCZYNSKI, Projective-differential geometry of curved surfaces, Trans. AMS 8 (1907), 233-260; (1908), 79-120, 293-315.