[proofplan] The proof compares $M$ with the flat Chern-Moser model, namely the Heisenberg quadric, which is locally CR-equivalent to the unit sphere. Local sphericity implies flatness because, in Chern-Moser [normal coordinates](/theorems/2713), the sphere has zero normal-form remainder and hence all curvature invariants vanish. Conversely, vanishing of the full curvature system is precisely the Chern-Moser-Cartan flatness condition: in CR dimension at least $2$ this is expressed by the Chern-Moser tensor and its covariant derivatives, while in CR dimension $1$ it is expressed by Cartan's scalar invariant and its covariant derivatives. Real analyticity converts the resulting formal flat normal form into an actual local real-analytic CR equivalence with the spherical model. [/proofplan] [step:Identify the spherical model with the Heisenberg quadric] Let \begin{align*} Q:=\{(z,w)\in\mathbb C^m\times\mathbb C:\operatorname{Im}(w)=|z|^2\} \end{align*} be the Heisenberg quadric, equipped with the CR structure induced by its embedding in $\mathbb C^{m+1}$. Define the boundary Cayley map \begin{align*} C:S^{2m+1}\setminus\{(0,\dots,0,-1)\}\to Q \end{align*} by \begin{align*} C(\zeta',\zeta_{m+1})=\left(\frac{\zeta'}{1+\zeta_{m+1}},i\frac{1-\zeta_{m+1}}{1+\zeta_{m+1}}\right), \end{align*} where $\zeta'=(\zeta_1,\dots,\zeta_m)\in\mathbb C^m$. The denominator is non-zero exactly away from the south pole $(0,\dots,0,-1)$, and [citetheorem:9201] states that this map is a real-analytic CR diffeomorphism from that punctured sphere onto $Q$. For a local statement near an arbitrary point of $S^{2m+1}$, first shrink the spherical target neighbourhood so that it avoids the pole, or precompose with a unitary automorphism of $S^{2m+1}$ moving the point away from the pole. Therefore local CR equivalence to an open subset of $S^{2m+1}$ is equivalent to local CR equivalence to an open subset of $Q$. [guided] The flat model for strictly pseudoconvex hypersurface CR geometry has two coordinate realizations. The spherical realization is \begin{align*} S^{2m+1}:=\{\zeta=(\zeta',\zeta_{m+1})\in\mathbb C^m\times\mathbb C:|\zeta|=1\}. \end{align*} The affine realization is the Heisenberg quadric \begin{align*} Q:=\{(z,w)\in\mathbb C^m\times\mathbb C:\operatorname{Im}(w)=|z|^2\}. \end{align*} Both CR structures are the hypersurface CR structures induced by their embeddings in complex Euclidean space. The map connecting them is the boundary Cayley transform. Define \begin{align*} C:S^{2m+1}\setminus\{(0,\dots,0,-1)\}\to\mathbb C^m\times\mathbb C \end{align*} by \begin{align*} C(\zeta',\zeta_{m+1})=\left(\frac{\zeta'}{1+\zeta_{m+1}},i\frac{1-\zeta_{m+1}}{1+\zeta_{m+1}}\right). \end{align*} This formula is defined precisely where $1+\zeta_{m+1}\ne 0$, namely away from the south pole. Let $C(\zeta',\zeta_{m+1})=(z,w)$. Since $|\zeta'|^2+|\zeta_{m+1}|^2=1$, direct computation gives \begin{align*} \operatorname{Im}(w)=\operatorname{Re}\left(\frac{1-\zeta_{m+1}}{1+\zeta_{m+1}}\right)=\frac{1-|\zeta_{m+1}|^2}{|1+\zeta_{m+1}|^2}=\frac{|\zeta'|^2}{|1+\zeta_{m+1}|^2}=|z|^2. \end{align*} Thus $C$ maps the punctured sphere into $Q$. By [citetheorem:9201], this boundary Cayley transform is in fact a real-analytic CR diffeomorphism from the punctured sphere onto the boundary of the Siegel upper half-space \begin{align*} \mathcal U^m:=\{(z,w)\in\mathbb C^m\times\mathbb C:\operatorname{Im}(w)>|z|^2\}. \end{align*} That boundary is exactly $Q$. There is one local technical point: the Cayley transform has a pole. If the spherical neighbourhood under consideration contains the pole, we first compose with a unitary automorphism of $S^{2m+1}$ so that the point and a sufficiently small neighbourhood avoid the pole. Such automorphisms are real-analytic CR diffeomorphisms of the sphere. Hence replacing the sphere by $Q$ loses no local information, and proving local CR equivalence to $Q$ is equivalent to proving local CR equivalence to $S^{2m+1}$. [/guided] [/step] [step:Show that local sphericity forces all Chern-Moser-Cartan invariants to vanish] Assume that there are open neighbourhoods $U_0\subset M$ of $p$ and $V_0\subset S^{2m+1}$ and a real-analytic CR diffeomorphism \begin{align*} F:U_0\to V_0. \end{align*} By the preceding step, after restricting $U_0$ and composing with the boundary Cayley transform, we may regard $F$ as a real-analytic CR diffeomorphism from $U_0$ onto an open subset of $Q$. The quadric $Q$ is already in Chern-Moser normal form with vanishing normal-form remainder: \begin{align*} \operatorname{Im}(w)-|z|^2=0. \end{align*} Thus every normal-form coefficient which defines a Chern-Moser curvature invariant is zero on $Q$. Since the Chern-Moser tensor, its covariant derivatives, Cartan's scalar invariant in CR dimension $1$, and its covariant derivatives are CR invariants, their pullbacks under the CR diffeomorphism $F$ vanish on $U_0$. Therefore the appropriate curvature system vanishes identically on a neighbourhood of $p$. [/step] [step:Embed locally and identify the curvature system with the normal Cartan curvature] Assume now that the curvature system vanishes on an open neighbourhood $U\subset M$ of $p$, and first suppose $m\ge 2$. We apply the local argument at the base point $p$. Since $M$ is a real-analytic CR manifold of hypersurface type, the [Andreotti-Fredricks local embedding theorem](/theorems/9196) [citetheorem:9196] gives, after shrinking $U$ around $p$, a real-analytic CR embedding \begin{align*} \iota:U\to\mathbb C^{m+1} \end{align*} whose image $\iota(U)$ is a real-analytic embedded hypersurface. Strict pseudoconvexity is intrinsic through the Levi form, so $\iota(U)$ is a strictly pseudoconvex real-analytic hypersurface. Hence the Chern-Moser normal form theorem [citetheorem:9218] applies to $\iota(U)$ at $p$. The external input used here is the classical Chern-Moser equivalence theorem in CR dimension $m\ge 2$. Let $G_m := SU(m+1,1)$ and let $P_m \subset G_m$ denote the parabolic subgroup stabilizing a complex null line in the standard Hermitian model; then $G_m/P_m$ is the homogeneous flat CR model. For a real-analytic strictly pseudoconvex hypersurface, the curvature function of the associated normal parabolic Cartan connection of type $(G_m,P_m)$ is represented in Chern-Moser normal coordinates by the trace-free Chern-Moser curvature tensor and, after applying the Tanaka-Webster connection, by its full covariant derivative tower. Thus the vanishing of the tensor and all of these covariant derivatives is precisely the vanishing of the normal Cartan curvature function on the corresponding Cartan bundle. The same equivalence theorem further says that vanishing of this Cartan curvature on an [open set](/page/Open%20Set) is equivalent to local isomorphism of the Cartan geometry with the flat model $G_m/P_m$. The hypotheses of this theorem are satisfied because, after the embedding above, we have a real-analytic strictly pseudoconvex hypersurface of CR dimension $m\ge 2$, and the present hypothesis says exactly that the Chern-Moser tensor and all of its Tanaka-Webster covariant derivatives vanish on $U$. Therefore the normal Cartan curvature of $M$ vanishes on a possibly smaller neighbourhood of $p$. [/step] [step:Apply Cartan flatness to obtain an actual spherical equivalence] By the flatness conclusion of the classical Chern-Moser-Cartan local equivalence theorem stated in the preceding step, there exist open neighbourhoods $U_p\subset M$ of $p$ and $O_p\subset SU(m+1,1)/P_m$ of a point in the homogeneous model, together with a real-analytic CR diffeomorphism \begin{align*} H:U_p\to O_p. \end{align*} The homogeneous model $SU(m+1,1)/P_m$ is the standard CR sphere $S^{2m+1}$; in the affine chart that avoids one point, it is the Heisenberg quadric $Q$. Using the boundary Cayley transform from [citetheorem:9201] and shrinking $U_p$ if the image meets the Cayley pole, we obtain open sets $U_0\subset M$ and $V_0\subset S^{2m+1}$ and a real-analytic CR diffeomorphism \begin{align*} F:U_0\to V_0. \end{align*} This proves local sphericity in the case $m\ge 2$. [/step] [step:Handle CR dimension $1$ through Cartan's scalar invariant] It remains to consider $m=1$. By the Andreotti-Fredricks local embedding theorem [citetheorem:9196], after shrinking the given neighbourhood of $p$, the abstract real-analytic hypersurface-type CR three-manifold $M$ is real-analytically CR embedded as a hypersurface in $\mathbb C^2$. Strict pseudoconvexity is preserved under this CR embedding, so the embedded image satisfies the hypotheses of Cartan's local equivalence theory for real-analytic strictly pseudoconvex CR three-manifolds. The external input in dimension $1$ is Cartan's flatness theorem. Let $G_1 := SU(2,1)$ and let $P_1 \subset G_1$ denote the parabolic subgroup stabilizing a complex null line; then $G_1/P_1$ is the homogeneous flat model. For a real-analytic strictly pseudoconvex CR three-manifold, Cartan's scalar invariant and all of its covariant derivatives encode the curvature function of the associated normal Cartan connection of type $(G_1,P_1)$, and this Cartan curvature vanishes on a neighbourhood if and only if the Cartan geometry is locally isomorphic to the flat model $G_1/P_1$. The Cartan local equivalence principle [citetheorem:9217] supplies the corresponding real-analytic local CR equivalence. Its hypotheses are satisfied here because $M$ is real-analytic, strictly pseudoconvex, and three-dimensional, and the theorem hypothesis says that Cartan's scalar invariant and all of its covariant derivatives vanish on a neighbourhood of $p$. Therefore there are open neighbourhoods $U_p\subset M$ of $p$ and $V_p\subset S^3$ and a real-analytic CR diffeomorphism \begin{align*} F_p:U_p\to V_p. \end{align*} Thus the conclusion also holds in CR dimension $1$. [/step] [step:Combine the two implications] The forward implication was proved by transporting the flat normal form of the sphere, equivalently the Heisenberg quadric, through a local CR diffeomorphism. The converse was proved for $m\ge 2$ by the classical Chern-Moser identification of the listed curvature system with the normal Cartan curvature and the Cartan flatness theorem for the model $SU(m+1,1)/P$, and for $m=1$ by Cartan's corresponding scalar-curvature flatness theorem for $SU(2,1)/P$. Hence the complete Chern-Moser-Cartan curvature system vanishes in a neighbourhood of $p$ if and only if $M$ is locally CR-equivalent near $p$ to an open subset of the unit sphere. [/step]