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Cartan equivalence problem for 5-dimensional bracket-generating CR manifolds in \(\mathbb C^4\). (English) Zbl 1354.32013

Summary: We reduce to various absolute parallelisms, namely to certain \(\{e\}\)-structures on manifolds of dimensions 7, 6, 5, the biholomorphic equivalence problem or the intrinsic CR equivalence problem for 5-dimensional CR-generic submanifolds \(M^5\subset\mathbb C^4\) of CR dimension 1 and of codimension 3 whose CR bundle \(T^{1,0}M\) satisfies the specific Lie-bracket generating property: \[ \begin{aligned} \mathbb C\otimes_{\mathbb R}TM=\Gamma\big(T^{1,0}M\big)\oplus\Gamma\big(\overline{T^{1,0}M}\big ) & \oplus\big[\Gamma\big(T^{1,0}M\big),\, \Gamma \big(\overline{T^{1,0}M}\big)\big] \\ &\oplus\big[\Gamma\big (T^{1,0}M\big),\, \big[\Gamma\big (T^{1,0}M\big),\, \Gamma\big(\overline{T^{1,0}M}\big)\big]\big] \\ &\oplus\big[\Gamma\big(\overline{T^{1,0}M}\big),\, \big[\Gamma\big (T^{1,0}M\big),\, \Gamma\big(\overline{T^{1,0}M}\big)\big]\big], \end{aligned} \] and which are known to be geometry-preserving deformations of the natural cubic model \(M_{\mathsf c}^5\subset\mathbb C^4\) of Beloshapka having, in coordinates \((z,w_1,w_2,w_3)\in\mathbb C^4\), the three graphed equations: \[ \begin{aligned} \mathrm{Im }w_1 & = z\overline{z}, \\ \mathrm{Im }w_2 & =z\overline{z}\,\big(z+\overline{z}\big), \\ \mathrm{Im }w_3 & =-\,i\,z\overline{z}\,\big(z-\overline{z}\big).\end{aligned} \] On the way, we develop a new “Differential Algebra Calculus” that enables us to explore in depth some nonlinear branching features while inspecting incoming essential torsions and intermediate Cartan curvatures.

MSC:

32M05 Complex Lie groups, group actions on complex spaces
32V40 Real submanifolds in complex manifolds
53C10 \(G\)-structures
58A15 Exterior differential systems (Cartan theory)

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