Explicit expression of Cartan’s connection for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere

Joël Merker ; Masoud Sabzevari

Open Mathematics, Tome 10 (2012), p. 1801-1835 / Harvested from The Polish Digital Mathematics Library

Résumé

We study effectively the Cartan geometry of Levi-nondegenerate C 6-smooth hypersurfaces M 3 in ℂ2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M 3 ⊂ ℂ2 to the Heisenberg sphere ℍ3, such M’s being necessarily real analytic.

Détail

Publié le : 2012-01-01

Zbl 1262.32040

EUDML-ID : urn:eudml:doc:269577

Bibliographie

[1] Aghasi M., Alizadeh B.M., Merker J., Sabzevari M., A Gröbner-bases algorithm for the computation of the cohomology of Lie (super) algebras, Adv. Appl. Clifford Algebr. (in press), DOI: 10.1007/s00006-011-0319-z | Zbl 1323.17018

[2] Aghasi M., Merker J., Sabzevari M., Effective Cartan-Tanaka connections for C 6-smooth strongly pseudoconvex hypersurfaces M 3 ⊂ ℂ2, C. R. Math. Acad. Sci. Paris, 2011, 349(15–16), 845–848 http://dx.doi.org/10.1016/j.crma.2011.07.020 | Zbl 1233.32024

[3] Aghasi M., Merker J., Sabzevari M., Effective Cartan-Tanaka connections on C 6 strongly pseudoconvex hypersurfaces M 3 ⊂ ℂ2, preprint available at http://arxiv.org/abs/1104.1509 | Zbl 1233.32024

[4] Aghasi M., Merker J., Sabzevari M., Some Maple worksheets accompanying the present publication, preprint available on demand

[5] Beloshapka V.K., A universal model for a real submanifold, Math. Notes, 2004, 75(3–4), 475–488 http://dx.doi.org/10.1023/B:MATN.0000023331.50692.87

[6] Beloshapka V., Ezhov V., Schmalz G., Canonical Cartan connection and holomorphic invariants on Engel CR manifolds, Russ. J. Math. Phys., 2007, 14(2), 121–133 http://dx.doi.org/10.1134/S106192080702001X | Zbl 1140.32024

[7] Boggess A., CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1991 | Zbl 0760.32001

[8] Čap A., Schichl H., Parabolic geometries and canonical Cartan connections, Hokkaido Math. J., 2000, 29(3), 453–505

[9] Čap A., Slovák J., Parabolic Geometries. I, Math. Surveys Monogr., 154, American Mathematical Society, Providence, 2009 | Zbl 1183.53002

[10] Cartan É., Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes. I, Ann. Math. Pures Appl., 1932, 11, 17–90 http://dx.doi.org/10.1007/BF02417822 | Zbl 58.1256.03

[11] Cartan É., Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes. II, Ann. Sc. Norm. Super. Pisa Cl. Sci., 1932, 2(1), 333–354 | Zbl 0005.37401

[12] Chern S.S., Moser J.K., Real hypersurfaces in complex manifolds, Acta Math., 1975, 133, 219–271 http://dx.doi.org/10.1007/BF02392146 | Zbl 0302.32015

[13] Crampin M., Cartan connections and Lie algebroids, SIGMA Symmetry Integrability Geom. Methods Appl., 2009, 5, #061

[14] Ezhov V., McLaughlin B., Schmalz G., From Cartan to Tanaka: getting real in the complex world, Notices Amer. Math. Soc., 2011, 58(1), 20–27 | Zbl 1244.32017

[15] Gaussier H., Merker J., Nonalgebraizable real analytic tubes in ℂn, Math. Z., 2004, 247(2), 337–383 http://dx.doi.org/10.1007/s00209-003-0617-9 | Zbl 1082.32026

[16] Isaev A., Spherical Tube Hypersurfaces, Lecture Notes in Math., 2020, Springer, Heidelberg, 2011

[17] Jacobowitz H., An introduction to CR structures, Math. Surveys Monogr., 32, American Mathematical Society, Providence, 1990 | Zbl 0712.32001

[18] Le A., Cartan connections for CR manifolds, Manuscripta Math., 2007, 122(2), 245–264 http://dx.doi.org/10.1007/s00229-006-0070-2 | Zbl 1145.32018

[19] Merker J., Lie symmetries and CR geometry, J. Math. Sciences (N.Y.), 2008, 154(6), 817–922 http://dx.doi.org/10.1007/s10958-008-9201-5 | Zbl 06406043

[20] Merker J., Nonrigid spherical real analytic hypersurfaces in ℂ2, Complex Var. Elliptic Equ., 2010, 55(12), 1155–1182 http://dx.doi.org/10.1080/17476931003728370 | Zbl 1208.32031

[21] Merker J., Porten E., Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities, IMRS Int. Math. Res. Surv., 2006, #28295 | Zbl 1149.32019

[22] Merker J., Sabzevari M., Canonical Cartan connection for maximally minimal 5-dimensional generic M 5 ⊂ ℂ4 (in preparation) | Zbl 1306.32027

[23] Nurowski P., Sparling G.A.J., Three-dimensional Cauchy-Riemann structures and second-order ordinary differential equations, Class. Quant. Gravity, 2003, 20(23), 4995–5016 http://dx.doi.org/10.1088/0264-9381/20/23/004 | Zbl 1051.32019

[24] Olver P.J., Equivalence, Invariants and Symmetry, Cambridge, Cambridge University Press, 1995

[25] Sharpe R.W., Differential Geometry, Grad. Texts in Math., 166, Springer, New York, 1997

[26] Tanaka N., On differential systems, graded Lie algebras and pseudogroups, J. Math. Kyoto Univ., 1970, 10, 1–82

[27] Tanaka N., On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan J. Math. (N.S.), 1976, 2(1), 131–190 | Zbl 0346.32010

[28] Webster S.M., Holomorphic differential invariants for an ellipsoidal real hypersurface, Duke Math. J., 2000, 104(3), 463–475 http://dx.doi.org/10.1215/S0012-7094-00-10435-8 | Zbl 0971.32019

[29] Yamaguchi K., Differential systems associated with simple graded Lie algebras, In: Progress in Differential Geometry, Adv. Stud. Pure Math., 22, Kinokuniya, Tokyo, 1993, 413–494 | Zbl 0812.17018