The equivalence problem for higher-codimensional CR structures
A CR-structure of codimension \(c\) on the \((2n+c)-\)dimensional real smooth manifold \(M\) is a rank \(n\) complex subbundle \(H\subset {\mathbb{C}}\otimes TM\) such that: (a) \(H\cap \overline H\) is the zero subbundle, (b) \([H,H]\subset H\). The authors apply Cartan's method, in the study of the equivalence problem for \(G\)-structures, in the case of CR-structures of codimension \(c\), where \(c\geq 3\) and \(n>c^2\). They assume that the considered CR-structure is tractable, i.e., its Levi form satisfies a certain ``genericity'' condition. They show that the structure bundle can be reduced twice, obtaining a bundle with the structure group isomorphic to \({\mathbb{C}}^*\). Then the canonical connection defined on this reduced bundle is used in order to obtain a canonical affine connection adapted to the considered CR-structure. It follows that the automorphisms of this CR-structure constitute a Lie group.
scientific article; zbMATH DE number 1054061