[proofplan]
The proof uses two standard structural inputs from Cartan's equivalence method: the functoriality of the canonical Cartan connection for strictly pseudoconvex real-analytic CR hypersurfaces, and the analytic local equivalence theorem for Cartan geometries viewed as absolute parallelisms. A local CR equivalence lifts uniquely to a principal bundle isomorphism preserving the Cartan connection, so it preserves curvature and all invariant derivatives. Conversely, equality of all invariant derivatives of the Cartan curvature at chosen frames gives, by the analytic Cartan equivalence theorem, a local Cartan-connection isomorphism between the canonical bundles. This isomorphism descends to the base and, by canonicity of the CR Cartan construction, gives the required real-analytic CR equivalence of germs.
[/proofplan]
[step:Define the invariant curvature jet attached to a frame]
Let
\begin{align*}
\mathfrak g:=\mathfrak{su}(2,1)
\end{align*}
be the [Lie algebra](/page/Lie%20Algebra) of $G=SU(2,1)$. For a Cartan geometry $(\mathcal G,\omega)$ of type $(G,P)$ and a point $a\in\mathcal G$, the Cartan connection gives a linear isomorphism
\begin{align*}
\omega_a:T_a\mathcal G\to\mathfrak g.
\end{align*}
For each element $A\in\mathfrak g$, define the $\omega$-constant vector field $\widetilde A\in\mathfrak X(\mathcal G)$ by the condition
\begin{align*}
\omega_b(\widetilde A_b)=A
\end{align*}
for every $b\in\mathcal G$.
If $F:\mathcal G\to W$ is a real-analytic map into a finite-dimensional real [vector space](/page/Vector%20Space) $W$, define
\begin{align*}
\nabla^\omega F:\mathcal G&\to \operatorname{Hom}(\mathfrak g,W)
\end{align*}
by
\begin{align*}
(\nabla^\omega F)(b)(A):=dF_b(\widetilde A_b)
\end{align*}
for $b\in\mathcal G$ and $A\in\mathfrak g$. Iterating this construction defines
\begin{align*}
(\nabla^\omega)^mF:\mathcal G\to \operatorname{Hom}(\mathfrak g^{\otimes m},W)
\end{align*}
for every integer $m\ge 0$, with $(\nabla^\omega)^0F:=F$.
Applying this to $F=\kappa_M$ and $F=\kappa_N$ gives the infinite curvature jets
\begin{align*}
J^\infty_{\omega_M}\kappa_M(u):=\bigl((\nabla^{\omega_M})^m\kappa_M(u)\bigr)_{m=0}^{\infty}
\end{align*}
and
\begin{align*}
J^\infty_{\omega_N}\kappa_N(v):=\bigl((\nabla^{\omega_N})^m\kappa_N(v)\bigr)_{m=0}^{\infty}.
\end{align*}
Both sequences take values in the same fixed model spaces
\begin{align*}
\operatorname{Hom}(\mathfrak g^{\otimes m},\mathbb K),
\end{align*}
so equality of the two infinite jets is a well-defined condition.
[/step]
[step:Lift a local CR equivalence and preserve all invariant derivatives]
Assume that there is a real-analytic CR diffeomorphism of germs
\begin{align*}
f:(M,p)\to(N,q)
\end{align*}
with $f(p)=q$. By the functoriality and uniqueness of the canonical Cartan CR connection for real-analytic strictly pseudoconvex hypersurfaces, the germ $f$ has a unique lift to a real-analytic principal $P$-bundle isomorphism
\begin{align*}
\Phi:(\mathcal G_M,u_0)\to(\mathcal G_N,\Phi(u_0))
\end{align*}
over $f$, for every chosen frame $u_0\in(\mathcal G_M)_p$, satisfying
\begin{align*}
\Phi^*\omega_N=\omega_M.
\end{align*}
Here we are using the standard Cartan-Chern-Moser canonical CR Cartan connection theorem for strictly pseudoconvex hypersurfaces: the canonical Cartan bundle and connection are natural for real-analytic CR diffeomorphisms and the lift is unique once its value at one frame is fixed.
Set
\begin{align*}
u:=u_0,\qquad v:=\Phi(u_0).
\end{align*}
Since $\Phi$ preserves the Cartan connections, it preserves their curvature two-forms. In terms of curvature functions this means
\begin{align*}
\kappa_M=\kappa_N\circ\Phi
\end{align*}
on the common germ of the source bundle.
We prove by induction on $m\ge 0$ that
\begin{align*}
(\nabla^{\omega_M})^m\kappa_M(u)=(\nabla^{\omega_N})^m\kappa_N(v).
\end{align*}
The case $m=0$ is the curvature identity at $u$.
Assume the identity has been proved up to order $m$ in the stronger functional form
\begin{align*}
(\nabla^{\omega_M})^m\kappa_M=(\nabla^{\omega_N})^m\kappa_N\circ\Phi.
\end{align*}
Let $A\in\mathfrak g$. Let $\widetilde A^M\in\mathfrak X(\mathcal G_M)$ and $\widetilde A^N\in\mathfrak X(\mathcal G_N)$ be the $\omega_M$-constant and $\omega_N$-constant vector fields determined by $A$. Since $\Phi^*\omega_N=\omega_M$, the differential $d\Phi$ sends $\widetilde A^M$ to $\widetilde A^N$ along $\Phi$. Differentiating the order-$m$ identity along $\widetilde A^M$ gives the order-$m+1$ identity. Evaluating at $u$ yields the required equality at the frames $u$ and $v$ for all $m$.
[guided]
We start with a real-analytic CR equivalence
\begin{align*}
f:(M,p)\to(N,q)
\end{align*}
satisfying $f(p)=q$. The canonical CR Cartan geometry is functorial: a CR map between strictly pseudoconvex real-analytic hypersurface germs lifts to a principal $P$-bundle map between the canonical Cartan bundles, and this lift preserves the Cartan connection. Thus, after choosing a frame $u_0\in(\mathcal G_M)_p$, there is a real-analytic principal $P$-bundle isomorphism
\begin{align*}
\Phi:(\mathcal G_M,u_0)\to(\mathcal G_N,\Phi(u_0))
\end{align*}
over $f$ such that
\begin{align*}
\Phi^*\omega_N=\omega_M.
\end{align*}
This is precisely the naturality part of the standard Cartan-Chern-Moser canonical connection theorem: the construction is functorial for real-analytic CR diffeomorphisms of strictly pseudoconvex hypersurface germs.
Define
\begin{align*}
u:=u_0,\qquad v:=\Phi(u_0).
\end{align*}
Because $\Phi$ preserves the Cartan connection, it preserves the curvature two-form. The curvature function is just the curvature two-form written in the fixed model representation $\mathbb K$, so this preservation becomes
\begin{align*}
\kappa_M=\kappa_N\circ\Phi.
\end{align*}
Now we explain why all invariant derivatives are preserved, not only the curvature itself. Fix an element $A\in\mathfrak g$. Let $\widetilde A^M$ be the unique vector field on $\mathcal G_M$ satisfying
\begin{align*}
\omega_M(\widetilde A^M)=A,
\end{align*}
and let $\widetilde A^N$ be the unique vector field on $\mathcal G_N$ satisfying
\begin{align*}
\omega_N(\widetilde A^N)=A.
\end{align*}
The identity $\Phi^*\omega_N=\omega_M$ says that, for each point $b\in\mathcal G_M$,
\begin{align*}
\omega_N(d\Phi_b(\widetilde A^M_b))=\omega_M(\widetilde A^M_b)=A.
\end{align*}
By uniqueness of the vector with $\omega_N$-value $A$, this gives
\begin{align*}
d\Phi_b(\widetilde A^M_b)=\widetilde A^N_{\Phi(b)}.
\end{align*}
This is the mechanism behind invariant-derivative preservation. Differentiating
\begin{align*}
\kappa_M=\kappa_N\circ\Phi
\end{align*}
in the direction $\widetilde A^M$ gives
\begin{align*}
d(\kappa_M)_b(\widetilde A^M_b)=d(\kappa_N)_{\Phi(b)}(\widetilde A^N_{\Phi(b)}).
\end{align*}
By the definition of invariant derivative, this is
\begin{align*}
(\nabla^{\omega_M}\kappa_M)(b)(A)=(\nabla^{\omega_N}\kappa_N)(\Phi(b))(A).
\end{align*}
Repeating the same argument for each successive invariant derivative proves inductively that
\begin{align*}
(\nabla^{\omega_M})^m\kappa_M=(\nabla^{\omega_N})^m\kappa_N\circ\Phi
\end{align*}
for every integer $m\ge 0$. Evaluating at $b=u$ and using $\Phi(u)=v$ gives
\begin{align*}
(\nabla^{\omega_M})^m\kappa_M(u)=(\nabla^{\omega_N})^m\kappa_N(v)
\end{align*}
for every integer $m\ge 0$.
[/guided]
[/step]
[step:Integrate matching analytic curvature jets to a local Cartan isomorphism]
Conversely, assume that there exist frames
\begin{align*}
u\in(\mathcal G_M)_p
\end{align*}
and
\begin{align*}
v\in(\mathcal G_N)_q
\end{align*}
such that
\begin{align*}
(\nabla^{\omega_M})^m\kappa_M(u)=(\nabla^{\omega_N})^m\kappa_N(v)
\end{align*}
for every integer $m\ge 0$.
The Cartan connections $\omega_M$ and $\omega_N$ are real-analytic absolute parallelisms on the real-analytic manifolds $\mathcal G_M$ and $\mathcal G_N$. For a Cartan connection $\omega$ of type $(G,P)$, the Cartan structure equation expresses the [exterior derivative](/theorems/1525) of the coframing components as the sum of the fixed algebraic Lie bracket term on $\mathfrak g$ and the variable curvature term $\kappa$. Hence the curvature function, together with this fixed Lie bracket term, is precisely the complete structure function of the absolute parallelism. Since the bracket term is the same for both Cartan geometries, equality of all invariant derivatives of $\kappa_M$ at $u$ and $\kappa_N$ at $v$ is exactly equality of all invariant derivatives of the corresponding structure functions at the two frames.
The analytic Cartan local equivalence theorem for absolute parallelisms says that if two real-analytic coframings have identical structure functions to all invariant orders at two points, then there are neighbourhoods of those points and a real-analytic diffeomorphism between them carrying one coframing to the other. Applying this theorem to the coframings $\omega_M$ and $\omega_N$ gives open neighbourhoods
\begin{align*}
\mathcal U\subset\mathcal G_M
\end{align*}
and
\begin{align*}
\mathcal V\subset\mathcal G_N
\end{align*}
with $u\in\mathcal U$ and $v\in\mathcal V$, and a real-analytic diffeomorphism
\begin{align*}
\Phi:\mathcal U\to\mathcal V
\end{align*}
such that
\begin{align*}
\Phi(u)=v
\end{align*}
and
\begin{align*}
\Phi^*\omega_N=\omega_M.
\end{align*}
The real-analytic hypothesis is used here: equality of infinite formal jets is being promoted to an actual local analytic solution of the first-order equivalence system.
[guided]
The converse direction is a local equivalence problem for analytic coframings. The Cartan connection $\omega_M$ is a real-analytic absolute parallelism on $\mathcal G_M$, meaning that each map
\begin{align*}
(\omega_M)_a:T_a\mathcal G_M\to\mathfrak g
\end{align*}
is a linear isomorphism, and the same holds for $\omega_N$ on $\mathcal G_N$. For an absolute parallelism, the structure functions are the coefficients that express the exterior derivatives of the coframing components in the coframing itself.
For a Cartan connection of fixed type $(G,P)$, these structure functions split into two parts. One part is fixed once the Lie bracket on $\mathfrak g$ is fixed. The other part is the Cartan curvature function $\kappa$. Therefore, in this fixed type, knowing every invariant derivative of $\kappa$ at a frame is the same as knowing every invariant derivative of the full structure functions at that frame. The hypothesis gives exactly this equality at $u$ and $v$.
We now apply the analytic Cartan local equivalence theorem for absolute parallelisms. Its hypothesis is equality of all invariant derivatives of the structure functions at the two base frames, and its conclusion is a real-analytic local diffeomorphism carrying one coframing to the other. Thus there are open neighbourhoods
\begin{align*}
\mathcal U\subset\mathcal G_M
\end{align*}
and
\begin{align*}
\mathcal V\subset\mathcal G_N
\end{align*}
with $u\in\mathcal U$ and $v\in\mathcal V$, and a real-analytic diffeomorphism
\begin{align*}
\Phi:\mathcal U\to\mathcal V
\end{align*}
such that
\begin{align*}
\Phi(u)=v
\end{align*}
and
\begin{align*}
\Phi^*\omega_N=\omega_M.
\end{align*}
This is the step where real analyticity is essential: the infinite formal agreement of the invariant jets is integrated to an actual local analytic isomorphism of coframed manifolds.
[/guided]
[/step]
[step:Descend the Cartan isomorphism to a CR equivalence of hypersurface germs]
Let $\mathfrak p:=\operatorname{Lie}(P)$. Since $\omega_M$ and $\omega_N$ are Cartan connections of the same type $(G,P)$ and $\Phi^*\omega_N=\omega_M$, the map $\Phi$ respects the fundamental vertical vector fields generated by $\mathfrak p$. Choose sufficiently small connected open neighbourhoods
\begin{align*}
U_M\subset M
\end{align*}
and
\begin{align*}
U_N\subset N
\end{align*}
of $p$ and $q$ such that the local $P$-saturations of the chosen neighbourhoods of $u$ and $v$ over $U_M$ and $U_N$ remain inside the domain and image of $\Phi$. After replacing $\mathcal U$ and $\mathcal V$ by these smaller local fibre-saturated neighbourhoods, we have $\pi_M(\mathcal U)=U_M$ and $\pi_N(\mathcal V)=U_N$, and $\Phi$ maps local $P$-orbits in $\mathcal G_M$ to local $P$-orbits in $\mathcal G_N$.
Consequently $\Phi$ descends to a real-analytic diffeomorphism
\begin{align*}
f:U_M\to U_N
\end{align*}
defined by the relation
\begin{align*}
f(\pi_M(a))=\pi_N(\Phi(a))
\end{align*}
for all $a\in\mathcal U$. This definition is independent of the chosen frame $a$ over a base point because points in the same fibre differ by the local $P$-action and $\Phi$ carries those fibre directions to fibre directions.
Since $\Phi(u)=v$, we have
\begin{align*}
f(p)=q.
\end{align*}
Finally, the canonical Cartan CR construction is faithful: a local isomorphism of the canonical Cartan geometries of strictly pseudoconvex CR hypersurfaces descends exactly to a local CR diffeomorphism of the underlying CR manifolds. Hence
\begin{align*}
f:(M,p)\to(N,q)
\end{align*}
is a real-analytic CR equivalence of germs.
[/step]
[step:Identify frame changes with the residual $P$-orbit condition]
It remains to justify the equivalent orbit formulation. For a frame $u\in\mathcal G_M$ and an element $r\in P$, define the translated frame
\begin{align*}
u\cdot r\in\mathcal G_M
\end{align*}
using the right principal action of $P$ on $\mathcal G_M$. The $P$-equivariance of the canonical Cartan connection and of its curvature function induces, for each integer $m\ge 0$, a representation of $P$ on
\begin{align*}
\operatorname{Hom}(\mathfrak g^{\otimes m},\mathbb K)
\end{align*}
such that
\begin{align*}
(\nabla^{\omega_M})^m\kappa_M(u\cdot r)
\end{align*}
is obtained from
\begin{align*}
(\nabla^{\omega_M})^m\kappa_M(u)
\end{align*}
by this induced $P$-action. The same statement holds for $N$.
Therefore changing the chosen frames over $p$ and $q$ changes the two infinite curvature jets only by the residual $P$-action on the product of model spaces
\begin{align*}
\prod_{m=0}^{\infty}\operatorname{Hom}(\mathfrak g^{\otimes m},\mathbb K).
\end{align*}
Thus there exist frames $u\in(\mathcal G_M)_p$ and $v\in(\mathcal G_N)_q$ with equal infinite curvature jets if and only if the full infinite Cartan curvature jet at $p$ and the full infinite Cartan curvature jet at $q$ lie in the same residual $P$-orbit. Combining this with the two implications proved above completes the proof.
[/step]
