On the existence of local frames of CR vector bundles

TL;DR

This paper establishes the equivalence between the existence of CR local frames and local embeddability of CR manifolds, demonstrating that some CR bundles lack local frames and constructing examples on non-embeddable manifolds.

Contribution

It proves the equivalence between CR local frames and embeddability, and constructs CR line bundles without local frames on non-embeddable 3D CR manifolds.

Findings

Existence of CR local frames is equivalent to local embeddability.

There exist CR bundles without local frames.

Constructed CR line bundles on non-embeddable manifolds without local frames.

Abstract

Given a CR manifold D, we shall show that existence of a CR local frame of a certain CR bundle over D is equivalent to the local imbeddability of D. This will imply that there exists a CR vector bundle which doesn't have CR local frames. Using this bundle, we shall construct CR line bundles over 3-dimensional non-imbeddable CR manifolds which don't have CR local frames.