The Local Equivalence Problem for 7-Dimensional, 2-Nondegenerate CR Manifolds whose Cubic Form is of Conformal Unitary Type

Abstract

We apply E. Cartan's method of equivalence to classify 7-dimensional, 2-nondegenerate CR manifolds $M$ up to local 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 is satisfied, 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.

This version accounts for a previous oversight in section 3.2. There is branching in the case that the nondegenerate part of the Levi form has split signature. The geometric distinction between the two split-signature sub-cases is explained in section 2.3, while the corresponding differences in normalization of the 2-adapted coframe bundle are shown in Lemma 3.1

References in corpus (3)

• Exterior Differential Systems and Euler-Lagrange Partial Differential Equations
• Homogeneous models for Levi-degenerate CR manifolds
• Reduction of five-dimensional uniformly Levi degenerate CR structures to absolute parallelisms

Cited by in corpus (2)

• Homogeneous models for Levi-degenerate CR manifolds
• Absolute parallelism for 2-nondegenerate CR structures via bigraded Tanaka prolongation