Canonical Cartan Connections on Maximally Minimal Generic Submanifolds M^5 in C^4

TL;DR

This paper constructs a canonical Cartan connection for a specific class of 5-dimensional CR-generic submanifolds in C^4, advancing the understanding of their local biholomorphic equivalence and differential invariants.

Contribution

It develops a canonical Cartan connection for maximally minimal CR submanifolds, highlighting the limitations of Cartan geometries compared to full normalizations of torsion.

Findings

Constructed a Cartan connection for the class III-1 CR structures.

Showed the limitations of Cartan geometries in capturing all invariants.

Provided insights into the structure of differential invariants for these submanifolds.

Abstract

On a real analytic 5-dimensional CR-generic submanifold M^5 in C^4 of codimension 3, hence of CR dimension 1, which enjoys the generically satisfied nondegeneracy condition that Lie brackets up to length 3 of T^{1,0}M generate CTM, a canonical Cartan connection is constructed after reduction to a certain partially explicit e-structure of the concerned local biholomorphic equivalence problem. More advanced explorations of the incoming differential invariants due to the first and to the third authors already appeared in January 2014, hence the purpose is to show, while studying this specific Class III-1 of 5-dimensional CR structures, why and how the construction of Cartan geometries usually provides less information than a complete ramified discussion of potentially normalizable essential torsion coefficients.