Invariants and Umbilical Points on Three Dimensional CR Manifolds embedded in $\mathbb C^2$

TL;DR

This paper introduces a new sequence of CR invariant determinants for 3D CR manifolds in C^2, linking them to umbilical points and demonstrating their behavior under perturbations.

Contribution

It presents a novel sequence of CR invariants, including a determinant representing Cartan's 6th order invariant, and applies this to study umbilical points under perturbations.

Findings

The lowest order invariant corresponds to Cartan's umbilical tensor.

Generic perturbations of the sphere contain curves or surfaces of umbilical points.

The new invariants provide a tool for analyzing umbilical points on CR manifolds.

Abstract

We introduce a new sequence of CR invariant determinants on a three dimensional CR manifold $M$ embedded in $\mathbb C^2$. The lowest order invariant $\det A_3$ represents E. Cartan's 6th order invariant (the umbilical "tensor"), whose zero locus yields the set of umbilical points on $M$. As an application of this new presentation of the umbilical invariant, we show that generic, almost circular perturbations of the unit sphere always contain curves or surfaces of umbilical points.