CR Embeddings, Chains, and the Fefferman Bundle

TL;DR

This paper characterizes when CR embeddings between strictly pseudoconvex hypersurfaces preserve chains, linking this to lifts to Fefferman bundles with special geometric properties, especially when the target is a sphere.

Contribution

It establishes a precise criterion connecting CR embeddings, chain preservation, and lifts to Fefferman bundles with vanishing second fundamental form.

Findings

CR embeddings preserve chains iff they lift to conformal isometries with vanishing second fundamental form.

Conditions simplify significantly when the target hypersurface is a sphere.

Provides geometric characterization of chain-preserving CR embeddings.

Abstract

In this paper it is shown that a CR embedding from one strictly pseudoconvex hypersurface into another (of strictly larger dimension) sends chains on the source to chains on the target if and only if the embedding has a lift to a conformal isometry of the associated Fefferman bundles with vanishing pseudo-Riemannian second fundamental form. When the target is a sphere these conditions simplify nicely.