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

• Autorzy: Joël Merker , Masoud Sabzevari
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Cartan connection; Heisenberg sphere; Cohomology of Lie algebras; Infinitesimal CR automorphisms; Differential algebra; Curvature function; Bianchi identities

Kategorie tematyczne

• 17B56: Cohomology of Lie (super)algebras
• 32V25: Extension of functions and other analytic objects from CR manifolds
• 58A15: Exterior differential systems (Cartan theory)
• 16W25: Derivations, actions of Lie algebras
• 32V40: Real submanifolds in complex manifolds

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2012
• Tom: 10
• Numer: 5
• Strony: 1801-1835

Daty

wydano

2012-10-01

online

2012-07-24

Twórcy

autor

Joël Merker

• Université Paris XI — Orsay

autor

Masoud Sabzevari

• Shahrekord University

Bibliografia

• [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
• [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
• [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
• [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
• [7] Boggess A., CR manifolds and the tangential Cauchy-Riemann complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, 1991
• [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
• [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
• [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
• [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
• [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
• [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
• [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
• [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
• [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
• [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
• [21] Merker J., Porten E., Holomorphic extension of CR functions, envelopes of holomorphy and removable singularities, IMRS Int. Math. Res. Surv., 2006, #28295
• [22] Merker J., Sabzevari M., Canonical Cartan connection for maximally minimal 5-dimensional generic M 5 ⊂ ℂ4 (in preparation)
• [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
• [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
• [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
• [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

Identyfikatory

DOI

10.2478/s11533-012-0052-4