CR regular embeddings of $S^{4n-1}$ in $\mathbb{C}^{2n+1}$

TL;DR

This paper constructs explicit CR regular embeddings of odd-dimensional spheres into complex Euclidean spaces, generalizing previous work and establishing conditions for such embeddings based on sphere dimension parity.

Contribution

It provides explicit examples of CR regular embeddings of spheres $S^{4n-1}$ into $C^{2n+1}$ and characterizes when odd-dimensional spheres admit such embeddings.

Findings

Explicit CR regular embeddings of $S^{4n-1}$ in $C^{2n+1}$

Odd spheres $S^{2m-1}$ with $m>1$ embed CR regularly in $C^{m+1}$ iff $m$ is even

Generalization of Ahern and Rudin's $S^3$ embedding to higher dimensions

Abstract

Ahern and Rudin have given an explicit construction of a totally real embedding of $S^3$ in $\mathbb{C}^3$. As a generalization of their example, we give an explicit example of a CR regular embedding of $S^{4n-1}$ in $\mathbb{C}^{2n+1}$. Consequently, we show that the odd dimensional sphere $S^{2m-1}$ with $m>1$ admits a CR regular embedding in $\mathbb{C}^{m+1}$ if and only if $m$ is even.