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

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Central European Journal of Mathematics

Abstract

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.

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Authors and Affiliations

  1. Départment de Mathématiques d’Orsay, Bâtiment 425, Faculté des Sciences, Université Paris XI — Orsay, 91405, Orsay Cedex, France

    Joël Merker

  2. Department of Pure Mathematics, Faculty of Sciences, Shahrekord University, 115, Shahrekord, Iran

    Masoud Sabzevari

Authors
  1. Joël Merker
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  2. Masoud Sabzevari
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Correspondence to Joël Merker.

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Merker, J., Sabzevari, M. Explicit expression of Cartan’s connection for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere. centr.eur.j.math. 10, 1801–1835 (2012). https://doi.org/10.2478/s11533-012-0052-4

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  • DOI: https://doi.org/10.2478/s11533-012-0052-4

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