Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 067, 18 pages      arXiv:0906.5607
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Symplectic Applicability of Lagrangian Surfaces

Emilio Musso a and Lorenzo Nicolodi b
a) Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy
b) Dipartimento di Matematica, Università degli Studi di Parma, Viale G.P. Usberti 53/A, I-43100 Parma, Italy

Received February 25, 2009, in final form June 15, 2009; Published online June 30, 2009

We develop an approach to affine symplectic invariant geometry of Lagrangian surfaces by the method of moving frames. The fundamental invariants of elliptic Lagrangian immersions in affine symplectic four-space are derived together with their integrability equations. The invariant setup is applied to discuss the question of symplectic applicability for elliptic Lagrangian immersions. Explicit examples are considered.

Key words: Lagrangian surfaces; affine symplectic geometry; moving frames; differential invariants; applicability.

pdf (297 kb)   ps (202 kb)   tex (20 kb)


  1. Álvarez Paiva J.C., Durán C.E., Geometric invariants of fanning curves, math.SG/0502481.
  2. Barbot T., Charette V., Drumm T., Goldman W.M., Melnik K., A primer on the (2+1) Einstein universe, arXiv:0706.3055.
  3. Bianchi L., Complementi alle ricerche sulle superficie isoterme, Ann. Mat. Pura Appl. 12 (1905), 20-54.
  4. Blaschke W., Vorlesungen über Differentialgeometrie und geometrische Grundlagen von Einsteins Relativitätstheorie, Bd. 3, bearbeitet von G. Thomsen, J. Springer, Berlin, 1929.
  5. Bryant R.L., A duality theorem for Willmore surfaces, J. Differential Geom. 20 (1984), 23-53.
  6. Burstall F., Hertrich-Jeromin U., Pedit F., Pinkall U., Curved flats and isothermic surfaces, Math. Z. 225 (1997), 199-209, dg-ga/9411010.
  7. Calapso P., Sulle superficie a linee di curvatura isoterme, Rend. Circ. Mat. Palermo 17 (1903), 273-286.
  8. Cartan É., Sur le problème général de la déformation, C. R. Congrés Strasbourg (1920), 397-406 (or Oeuvres Complètes, III 1, 539-548).
  9. Cartan É., Sur la déformation projective des surfaces, Ann. Scient. Éc. Norm. Sup. (3) 37 (1920), 259-356 (or Oeuvres Complètes, III 1, 441-538).
  10. Chern S.-S., Wang H.-C., Differential geometry in symplectic space. I, Sci. Rep. Nat. Tsing Hua Univ. 4 (1947), 453-477.
  11. Deconchy V., Hypersurfaces in symplectic affine geometry, Differential Geom. Appl. 17 (2002), 1-13.
  12. Fubini G., Applicabilità proiettiva di due superficie, Rend. Circ. Mat. Palermo 41 (1916), 135-162.
  13. Gálvez J.A., Martínez A., Milán F., Flat surfaces in the hyperbolic 3-space, Math. Ann. 316 (2000), 419-435.
  14. Griffiths P.A., 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.
  15. Guillemin V., Sternberg S., Variations on a theme by Kepler, American Mathematical Society Colloquium Publications, Vol. 42, American Mathematical Society, Providence, RI, 1990.
  16. Hertrich-Jeromin U., Musso E., Nicolodi L., Möbius geometry of surfaces of constant mean curvature 1 in hyperbolic space, Ann. Global Anal. Geom. 19 (2001), 185-205, math.DG/9810157.
  17. Jensen G.R., Deformation of submanifolds of homogeneous spaces, J. Differential Geom. 16 (1981), 213-246.
  18. Jensen G.R., Musso E., Rigidity of hypersurfaces in complex projective space, Ann. Sci. École Norm. Sup. (4) 27 (1994), 227-248.
  19. Kamran N., Olver P., Tenenblat K., Local symplectic invariants for curves, Commun. Contemp. Math., to appear.
  20. Kokubu M., Umehara M., Yamada K., Flat fronts in hyperbolic 3-space, Pacific J. Math. 216 (2004), 149-175, math.DG/0301224.
  21. McKay B., Lagrangian submanifolds in affine symplectic geometry, Differential Geom. Appl. 24 (2006), 670-689, math.DG/0508118.
  22. Musso E., Deformazione di superfici nello spazio di Möbius, Rend. Istit. Mat. Univ. Trieste 27 (1995), 25-45.
  23. Musso E., Nicolodi L., Deformation and applicability of surfaces in Lie sphere geometry, Tohoku Math. J. 58 (2006), 161-187, math.DG/0408009.
  24. Musso E., Nicolodi L., Conformal deformation of spacelike surfaces in Minkowski space, Houston J. Math., to appear, arXiv:0712.0807.

Previous article   Next article   Contents of Volume 5 (2009)