Canonical Cartan connections on maximally minimal generic submanifolds \mathbf{M^5 \subset \mathbb{C}^4}

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2014, 21: 153-166. doi: 10.3934/era.2014.21.153

Canonical Cartan connections on maximally minimal generic submanifolds $\mathbf{M^5 \subset \mathbb{C}^4}$

Masoud Sabzevari ^1, , Joël Merker ^2, and Samuel Pocchiola ^2,

1.	Department of Pure Mathematics, University of Shahrekord, 88186-34141 Shahrekord, Iran
2.	Département de Mathématiques d'Orsay, Bâtiment 425, Faculté des Sciences, F-91405 Orsay Cedex, France, France

Received May 2014 Revised September 2014 Published November 2014

Abstract
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On a real analytic $5$-dimensional CR-generic submanifold $M^5 \subset \mathbb{C}^4$ of codimension $3$ hence of CR dimension $1$, which enjoys the generically satisfied nondegeneracy condition \begin{align*} {\bf 5} &= \text{rank}_\mathbb{C} \big( T^{1,0}M+T^{0,1}M + \big[T^{1,0}M,\,T^{0,1}M\big] \,+ \\&\qquad + \big[T^{1,0}M,\,[T^{1,0}M,T^{0,1}M]\big] + \big[T^{0,1}M,\,[T^{1,0}M,T^{0,1}M]\big] \big), \end{align*} a canonical Cartan connection is constructed after reduction to a certain partially explicit $\{ e\}$-structure of the concerned local biholomorphic equivalence problem.

Keywords: Cartan equivalence problem, Absorption of torsion, Nilpotent Lie algebra, Normalization of group parameters, Prolongation of G-structures., Cartan connection.

Mathematics Subject Classification: Primary 53B15, Secondary 32V4.

Citation: Masoud Sabzevari, Joël Merker, Samuel Pocchiola. Canonical Cartan connections on maximally minimal generic submanifolds $\mathbf{M^5 \subset \mathbb{C}^4}$. Electronic Research Announcements, 2014, 21: 153-166. doi: 10.3934/era.2014.21.153

References:

[1]	V. Beloshapka, V. Ezhov and G. Schmalz, Canonical Cartan connection and holomorphic invariants on Engel CR manifolds, Russian J. Mathematical Physics, 14 (2007), 121-133. doi: 10.1134/S106192080702001X.
[2]	É. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), 1 (1932), 333-354.
[3]	J. Merker, Rationality in differential algebraic geometry, to appear in Proceedings of the Abel Symposium 2013, Springer Verlag, arXiv:1405.7625, 2013, 47 pp.
[4]	J. Merker, S. Pocchiola and M. Sabzevari, Equivalences of $5$-dimensional CR manifolds, I: General introduction, overview of results, and nonlinear computational aspects, http://arXiv.org, to appear.
[5]	J. Merker, S. Pocchiola and M. Sabzevari, Equivalences of $5$-dimensional CR manifolds, II: General classes $\sfI$, $\sf{II}$, $\sf{III}_{\sf{1}}$, $\sf{III}_{\sf{2}}$, $\sf{IV}_{\sf{1}}$, $\sf{IV}_{\sf{2}}$, arXiv:1311.5669, 95 pp.
[6]	J. Merker and M. Sabzevari, Explicit expression of Cartan's connections for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere, Cent. Eur. J. Math., 10 (2012), 1801-1835. doi: 10.2478/s11533-012-0052-4.
[7]	J. Merker and M. Sabzevari, Cartan equivalences for $5$-dimensional CR-manifolds in $\mathbbC^4$ belonging to general class III, arXiv:1401.4297, 172 pp.
[8]	P. J. Olver, Equivalence, Invariants and Symmetry, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511609565.
[9]	S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964.

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References:

[1]	V. Beloshapka, V. Ezhov and G. Schmalz, Canonical Cartan connection and holomorphic invariants on Engel CR manifolds, Russian J. Mathematical Physics, 14 (2007), 121-133. doi: 10.1134/S106192080702001X.
[2]	É. Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes, II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2), 1 (1932), 333-354.
[3]	J. Merker, Rationality in differential algebraic geometry, to appear in Proceedings of the Abel Symposium 2013, Springer Verlag, arXiv:1405.7625, 2013, 47 pp.
[4]	J. Merker, S. Pocchiola and M. Sabzevari, Equivalences of $5$-dimensional CR manifolds, I: General introduction, overview of results, and nonlinear computational aspects, http://arXiv.org, to appear.
[5]	J. Merker, S. Pocchiola and M. Sabzevari, Equivalences of $5$-dimensional CR manifolds, II: General classes $\sfI$, $\sf{II}$, $\sf{III}_{\sf{1}}$, $\sf{III}_{\sf{2}}$, $\sf{IV}_{\sf{1}}$, $\sf{IV}_{\sf{2}}$, arXiv:1311.5669, 95 pp.
[6]	J. Merker and M. Sabzevari, Explicit expression of Cartan's connections for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere, Cent. Eur. J. Math., 10 (2012), 1801-1835. doi: 10.2478/s11533-012-0052-4.
[7]	J. Merker and M. Sabzevari, Cartan equivalences for $5$-dimensional CR-manifolds in $\mathbbC^4$ belonging to general class III, arXiv:1401.4297, 172 pp.
[8]	P. J. Olver, Equivalence, Invariants and Symmetry, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511609565.
[9]	S. Sternberg, Lectures on Differential Geometry, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1964.

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