Rigidity of Pseudoconformal Connections

Abstract

Let ${M^{2n - 1}}(n \geqslant 3)$ be a strictly pseudoconvex abstract ${\text {CR}}$-hypersurface ${\text {CR}}$-immersed in the unit sphere in ${{\mathbf {C}}^N}$. We show that the pseudoconformal connection induced on $M$ by the standard flat connection agrees with the intrinsic normal connection of Cartan-Chern-Tanaka if and only if $M$ is pseudoconformally flat. In this case $M$ is a piece of the transverse intersection of ${S^{2N - 1}}$ with a complex $n$-plane in ${{\mathbf {C}}^N}$.