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Cartan Equivalence Problem for 5-Dimensional Bracket-Generating CR Manifolds in \(\mathbb {C}^4\)
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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: $$\begin{aligned} \begin{aligned} \mathbb {C}\otimes _\mathbb {R}TM \,=\, \Gamma \big (T^{1,0}M\big ) \oplus \Gamma \big (\overline{T^{1,0}M}\big )&\oplus \big [ \Gamma \big (T^{1,0}M\big ),\, \Gamma \big (\overline{T^{1,0}M}\big ) \big ] \\&\oplus \big [ \Gamma \big (T^{1,0}M\big ),\, \big [ \Gamma \big (T^{1,0}M\big ),\, \Gamma \big (\overline{T^{1,0}M}\big ) \big ]\big ] \\&\oplus \big [ \Gamma \big (\overline{T^{1,0}M}\big ),\, \big [ \Gamma \big (T^{1,0}M\big ),\, \Gamma \big (\overline{T^{1,0}M}\big ) \big ]\big ], \end{aligned} \end{aligned}$$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: $$\begin{aligned} \begin{aligned} \mathrm{Im}\,w_1&= z\overline{ z}, \\ \mathrm{Im}\,w_2&= z\overline{z}\, \big (z+\overline{z}\big ), \\ \mathrm{Im}\,w_3&= -\,i\,z\overline{z}\, \big (z-\overline{z}\big ). \end{aligned} \end{aligned}$$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.KeywordsCR manifoldsCartan equivalence problemInfinitesimal CR automorphismsDifferential algebraTorsion and curvatureMathematics Subject Classification32M0532V4053C1058A15References1.Aghasi, M., Merker, J., Sabzevari, M.: Effective Cartan-Tanaka connections for \({\cal {C}}^{6}\)-smooth strongly pseudoconvex hypersurfaces \(M^3 \subset \mathbb{C}^2\). 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