# American Institute of Mathematical Sciences

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

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

 1 Department of Pure Mathematics, University of Shahrekord, 88186-34141 Shahrekord, Iran 2 Département de Mathématiques d'Orsay, Bâtiment 425, Faculté des Sciences, F-91405 Orsay Cedex, France, France

Received  May 2014 Revised  September 2014 Published  November 2014

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.
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. doi: 10.1134/S106192080702001X. Google Scholar [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. Google Scholar [3] J. Merker, Rationality in differential algebraic geometry,, to appear in Proceedings of the Abel Symposium 2013, (2013). Google Scholar [4] J. Merker, S. Pocchiola and M. Sabzevari, Equivalences of $5$-dimensional CR manifolds, I: General introduction, overview of results, and nonlinear computational aspects,, , (). Google Scholar [5] J. Merker, S. Pocchiola and M. Sabzevari, Equivalences of $5$-dimensional CR manifolds, II: General classes $\sfI$, $\sf{II}$, $\sf{III}_{\sf{1}}$, $\sf{III}_{\sf{2}}$, $\sf{IV}_{\sf{1}}$, $\sf{IV}_{\sf{2}}$,, , (). Google Scholar [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. doi: 10.2478/s11533-012-0052-4. Google Scholar [7] J. Merker and M. Sabzevari, Cartan equivalences for $5$-dimensional CR-manifolds in $\mathbbC^4$ belonging to general class III,, , (). Google Scholar [8] P. J. Olver, Equivalence, Invariants and Symmetry,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511609565. Google Scholar [9] S. Sternberg, Lectures on Differential Geometry,, Prentice-Hall, (1964). Google Scholar

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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. doi: 10.1134/S106192080702001X. Google Scholar [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. Google Scholar [3] J. Merker, Rationality in differential algebraic geometry,, to appear in Proceedings of the Abel Symposium 2013, (2013). Google Scholar [4] J. Merker, S. Pocchiola and M. Sabzevari, Equivalences of $5$-dimensional CR manifolds, I: General introduction, overview of results, and nonlinear computational aspects,, , (). Google Scholar [5] J. Merker, S. Pocchiola and M. Sabzevari, Equivalences of $5$-dimensional CR manifolds, II: General classes $\sfI$, $\sf{II}$, $\sf{III}_{\sf{1}}$, $\sf{III}_{\sf{2}}$, $\sf{IV}_{\sf{1}}$, $\sf{IV}_{\sf{2}}$,, , (). Google Scholar [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. doi: 10.2478/s11533-012-0052-4. Google Scholar [7] J. Merker and M. Sabzevari, Cartan equivalences for $5$-dimensional CR-manifolds in $\mathbbC^4$ belonging to general class III,, , (). Google Scholar [8] P. J. Olver, Equivalence, Invariants and Symmetry,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511609565. Google Scholar [9] S. Sternberg, Lectures on Differential Geometry,, Prentice-Hall, (1964). Google Scholar
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