Let $m\ge 1$, and let $M$ be a real-analytic strictly pseudoconvex abstract CR manifold of hypersurface type with CR bundle $T^{1,0}M\subset \mathbb C TM$ of complex rank $m$. Let $p\in M$. For $m\ge 2$, let the Chern-Moser-Cartan curvature system mean the curvature function of the normal parabolic Cartan connection of type $(SU(m+1,1),P_m)$, equivalently the Chern-Moser curvature tensor and all of its Tanaka-Webster covariant derivatives in Chern-Moser [normal coordinates](/theorems/2713). For $m=1$, let the Chern-Moser-Cartan curvature system mean Cartan's scalar CR invariant and all of its covariant derivatives, equivalently the curvature function of Cartan's normal connection of type $(SU(2,1),P_1)$. Here $P_m\subset SU(m+1,1)$ denotes the parabolic subgroup stabilizing a complex null line in the standard Hermitian model.

Then the following are equivalent.

1. There exists an open neighbourhood $U\subset M$ of $p$ on which the Chern-Moser-Cartan curvature system appropriate to the CR dimension $m$ vanishes identically. 2. There exist an open neighbourhood $U_0\subset M$ of $p$, an open subset $V_0\subset S^{2m+1}$ of the unit sphere

\begin{align*} S^{2m+1}:=\{\zeta\in\mathbb C^{m+1}:|\zeta|=1\}, \end{align*}

and a real-analytic CR diffeomorphism

\begin{align*} F:U_0\to V_0. \end{align*}