CR regular embeddings of 𝑆⁴ⁿ⁻¹ in ℂ²ⁿ⁺¹

Abstract

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