The local equivalence problem for $7$-dimensional, $2$-nondegenerate $\operatorname{CR}$ manifolds

We apply E. Cartan’s method of equivalence to classify $7$-dimensional, $2$-nondegenerate $\operatorname{CR}$ manifolds $M$ up to local $\operatorname{CR}$ equivalence in the case that the cubic form of $M$ satisfies a certain symmetry property with respect to the Levi form of $M$. The solution to the equivalence problem is given by a parallelism on a principal bundle over $M$. When the nondegenerate part of the Levi form has definite signature, the parallelism takes values in $\mathfrak{su}(2,2)$. When this signature is split and an additional “isotropy-switching” hypothesis holds, the parallelism takes values in $\mathfrak{su}(3,1)$. Differentiating the parallelism provides a complete set of local invariants of $M$. We exhibit an explicit example of a real hypersurface in $\mathbb{C}^4$ whose invariants are nontrivial.

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Received 29 January 2016

Accepted 5 July 2017

Published 30 December 2019