Rigidity of pseudoconformal connections

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 ${{\mathbf{C}}^N}$.