CR embedded submanifolds of CR manifolds

TL;DR

This paper develops a comprehensive local theory for CR embedded submanifolds within CR manifolds, establishing relationships between tractor bundles and invariants, and introduces a CR analogue of the Bonnet theorem for submanifolds.

Contribution

It introduces a new CR invariant tractor calculus for embeddings, relating submanifold and ambient structures, and provides explicit formulas for CR invariants, including a CR Bonnet theorem.

Findings

Established a CR Gauss formula relating tractor connections.

Developed explicit CR invariant calculations using Tanaka-Webster calculus.

Proved a CR analogue of the classical Bonnet theorem.

Abstract

We develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. In particular, we establish the subtle relationship between the submanifold and ambient standard tractor bundles, allowing us to relate the respective normal Cartan (or tractor) connections via a CR Gauss formula. The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation of the pseudohermitian invariants of the embedding which are also CR invariant. These are extremely difficult to find and compute by more na\"ive methods. We conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.