Tohoku Mathematical Journal

Deformation and applicability of surfaces in Lie sphere geometry

Emilio Musso and Lorenzo Nicolodi

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The theory of surfaces in Euclidean space can be naturally formulated in the more general context of Legendre surfaces into the space of contact elements. We address the question of deformability of Legendre surfaces with respect to the symmetry group of Lie sphere contact transformations from the point of view of the deformation theory of submanifolds in homogeneous spaces. Necessary and sufficient conditions are provided for a Legendre surface to admit non-trivial deformations, and the corresponding existence problem is discussed.

Article information

Tohoku Math. J. (2), Volume 58, Number 2 (2006), 161-187.

First available in Project Euclid: 22 August 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A40: Other special differential geometries
Secondary: 53C24: Rigidity results

Legendre surfaces deformation of surfaces Lie-applicable surfaces Lie sphere geometry rigidity


Musso, Emilio; Nicolodi, Lorenzo. Deformation and applicability of surfaces in Lie sphere geometry. Tohoku Math. J. (2) 58 (2006), no. 2, 161--187. doi:10.2748/tmj/1156256399.

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  • R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt and P. A. Griffiths, Exterior differential systems, Math. Sci. Res. Inst. Publ. 18, Springer-Verlag, New York, 1991.
  • H. Bernstein, Non-special, non-canal isothermic tori with spherical lines of curvature, Trans. Amer. Math. Soc. 353 (2001), 2245--2274.
  • L. Bianchi, Ricerche sulle superficie isoterme e sulla deformazione delle quadriche, Ann. Mat. Pura Appl. 11 (1905), 93--157.
  • L. Bianchi, Complementi alle ricerche sulle superficie isoterme, Ann. Mat. Pura Appl. 12 (1905), 19--54.
  • W. Blaschke, Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie, B. 3, bearbeitet von G. Thomsen, J. Springer, Berlin, 1929.
  • F. Burstall, Isothermic surfaces: conformal geometry, Clifford algebras and integrable systems, arXiv. math.DG/0003096 (2000).
  • P. Calapso, Sulle superficie a linee di curvatura isoterme, Rend. Circ. Mat. Palermo 17 (1903), 275--286.
  • P. Calapso, Sulle trasformazioni delle superficie isoterme, Ann. Mat. Pura Appl. 24 (1915), 11--48.
  • É. Cartan, Sur le problème général de la déformation, C. R. Congrés Strasbourg (1920), 397--406; Oeuvres Complètes, III 1, 539--548.
  • É. Cartan, Sur la déformation projective des surfaces, Ann. Sci. École Norm. Sup. (3) 37 (1920), 259--356; Oeuvres Complètes, III 1, 441--538.
  • T. E. Cecil, Lie sphere geometry, with applications to submanifolds, Universitext, Springer-Verlag, New York, 1992.
  • E. Ferapontov, Lie sphere geometry and integrable systems, Tohoku Math. J. (2) 52 (2000), 199--233.
  • E. Ferapontov, Integrable systems in projective differential geometry, Kyushu J. Math. 54 (2000), 183--215.
  • E. Ferapontov, Analogue of Wilczynski's projective frame in Lie sphere geometry: Lie-applicable surfaces and commuting Schrödinger operators with magnetic fields, Internat. J. Math. 13 (2002), 956--985.
  • S. P. Finikov, Projective differential geometry, Moscow, Leningrad, 1937.
  • G. Fubini, Applicabilità proiettiva di due superficie, Rend. Circ. Mat. Palermo 41 (1916), 135--162.
  • P. A. Griffiths, On Cartan's method of Lie groups and moving frames as applied to uniqueness and existence questions in differential geometry, Duke Math. J. 41 (1974), 775--814.
  • G. R. Jensen, Deformation of submanifolds of homogeneous spaces, J. Differential Geom. 16 (1981), 213--246.
  • S. Lie and G. Scheffers, Geometrie der Berührungstransformationen, Teubner, Leipzig, 1896.
  • E. Musso, Deformazione di superfici nello spazio di Möbius, Rend. Istit. Mat. Univ. Trieste 27 (1995), 25--45.
  • E. Musso and L. Nicolodi, On the equation defining isothermic surfaces in Laguerre geometry, New developments in differential geometry, Budapest 1996, 285--294, Kluwer Acad. Publ., Dordrecht, 1999.
  • E. Musso and L. Nicolodi, Isothermal surfaces in Laguerre geometry, Boll. Un. Mat. Ital. B (7) 11 (1997), 125--144.
  • E. Musso and L. Nicolodi, Laguerre geometry of surfaces with plane lines of curvature, Abh. Math. Sem. Univ. Hamburg 69 (1999), 123--138.
  • E. Musso and L. Nicolodi, The Bianchi-Darboux transform of $L$-isothermic surfaces, Internat. J. Math. 11 (2000), 911--924.
  • E. Musso and L. Nicolodi, Darboux transforms of Dupin surfaces, PDES, submanifolds and affine diffeential geometry (Warsaw, 2000), 135--154, Banach Center Publ. 57, Polish Acad. Sci., Warsaw, 2002.
  • U. Pinkall, Dupin hypersurfaces, Math. Ann. 270 (1985), 427--440.