Geodesic
Symplectic Applicability of Lagrangian Surfaces
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: Lagrangian surfaces; affine symplectic geometry; moving frames; differential invariants; applicability. 
@article{SIGMA_2009_5_a66,
     author = {Emilio Musso and Lorenzo Nicolodi},
     title = {Symplectic {Applicability} of {Lagrangian} {Surfaces}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2009},
     volume = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a66/}
}
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Emilio Musso; Lorenzo Nicolodi. Symplectic Applicability of Lagrangian Surfaces. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a66/

[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. III. Differentialgeometrie der Kreise und Kugeln, bearbeitet von G. Thomsen, J. Springer, Berlin, 1929 | Zbl

[5] Bryant R. L., “A duality theorem for Willmore surfaces”, J. Differential Geom., 20 (1984), 23–53 | MR | Zbl

[6] Burstall F., Hertrich-Jeromin U., Pedit F., Pinkall U., “Curved flats and isothermic surfaces”, Math. Z., 225 (1997), 199–209 ; dg-ga/9411010 | DOI | MR | Zbl

[7] Calapso P., “Sulle superficie a linee di curvatura isoterme”, Rend. Circ. Mat. Palermo, 17 (1903), 273–286 | DOI

[8] Cartan É., “Sur le problème général de la déformation”, C. R. Congrés Strasbourg, 1920, 397–406 ; Oeuvres Complètes, III 1, 539–548 | Zbl

[9] Cartan É., “Sur la déformation projective des surfaces”, Ann. Scient. Éc. Norm. Sup. (3), 37 (1920), 259–356 ; Oeuvres Complètes, III 1, 441–538 | MR | Zbl

[10] Chern S.-S., Wang H.-C., “Differential geometry in symplectic space. I”, Sci. Rep. Nat. Tsing Hua Univ., 4 (1947), 453–477 | MR

[11] Deconchy V., “Hypersurfaces in symplectic affine geometry”, Differential Geom. Appl., 17 (2002), 1–13 | DOI | MR | Zbl

[12] Fubini G., “Applicabilità proiettiva di due superficie”, Rend. Circ. Mat. Palermo, 41 (1916), 135–162 | DOI | Zbl

[13] Gálvez J. A., Martínez A., Milán F., “Flat surfaces in the hyperbolic 3-space”, Math. Ann., 316 (2000), 419–435 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[15] Guillemin V., Sternberg S., Variations on a theme by Kepler, American Mathematical Society Colloquium Publications, 42, American Mathematical Society, Providence, RI, 1990 | MR | Zbl

[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 | DOI | MR | Zbl

[17] Jensen G. R., “Deformation of submanifolds of homogeneous spaces”, J. Differential Geom., 16 (1981), 213–246 | MR

[18] Jensen G. R., Musso E., “Rigidity of hypersurfaces in complex projective space”, Ann. Sci. École Norm. Sup. (4), 27 (1994), 227–248 | MR | Zbl

[19] Kamran N., Olver P., Tenenblat K., “Local symplectic invariants for curves”, Commun. Contemp. Math., 11:2 (2009), 165–183 | DOI | MR | Zbl

[20] Kokubu M., Umehara M., Yamada K., “Flat fronts in hyperbolic 3-space”, Pacific J. Math., 216 (2004), 149–175 ; math.DG/0301224 | DOI | MR | Zbl

[21] McKay B., “Lagrangian submanifolds in affine symplectic geometry”, Differential Geom. Appl., 24 (2006), 670–689 ; math.DG/0508118 | DOI | MR | Zbl

[22] Musso E., “Deformazione di superfici nello spazio di Möbius”, Rend. Istit. Mat. Univ. Trieste, 27 (1995), 25–45 | MR | Zbl

[23] Musso E., Nicolodi L., “Deformation and applicability of surfaces in Lie sphere geometry”, Tohoku Math. J., 58 (2006), 161–187 ; math.DG/0408009 | DOI | MR | Zbl

[24] Musso E., Nicolodi L., Conformal deformation of spacelike surfaces in Minkowski space, Houston J. Math., to appear | MR