Finite jet determination of constantly degenerate CR embeddings
Peter Ebenfelt
We prove finite jet determination results for smooth CR embeddings which are of constant degeneracy, using the method of complete systems. As an application, we derive a reflection principle for mappings between a Levi-nondegenerate hypersurface in $ \CN$ and a Levi-nondegenerate hypersurface in $\CNplus$. We also give an independent proof of the reflection principle for mappings between strictly pseudoconvex hypersurfaces in any codimension due to Forstneric \cite{FO1}.
EL_finite_submission.tex


\documentclass{amsart}
\usepackage{amssymb,amsxtra,enumerate}

\setlength{\textwidth}{16truecm}\oddsidemargin=-0.1truecm\evensidemargin=-0.1truecm
\setlength{\textheight}{20cm}

\newcommand{\dbar}{\bar\partial}
\newcommand{\Cn}{\mathbb{C}^n}
\newcommand{\Cnh}{\mathbb{C}^{\hat n}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\bC}{\mathbb{C}}
\newcommand{\CN}{\mathbb{C}^N}
\newcommand{\CNplus}{\mathbb{C}^{N+1}}
\newcommand{\CNh}{\mathbb{C}^{\hat N}}
\newcommand{\CNp}{\mathbb{C}^{N^\prime}}
\newcommand{\Rd}{\mathbb{R}^d}
\newcommand{\Rn}{\mathbb{R}^n}
\newcommand{\RN}{\mathbb{R}^N}
\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\da}{\delta}
\newcommand{\Da}{\Delta}
\newcommand{\io}{\iota}
\newcommand{\al}{\alpha}
\newcommand{\eps}{\epsilon}
\newcommand{\bt}{\beta}
\newcommand{\gm}{\gamma}
\newcommand{\Gm}{\Gamma}
\newcommand{\sg}{\sigma}
\newcommand{\Om}{\Omega}
\newcommand{\om}{\omega}
\newcommand{\ld}{\lambda}
\newcommand{\gth}{\theta}
\newcommand{\Th}{\Theta}
\newcommand{\td}{\tilde}
\newcommand{\dop}[1]{\frac{\partial}{\partial #1}}
\newcommand{\vardop}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\br}[1]{\langle#1 \rangle}
\newcommand{\nequiv}{{\equiv \!\!\!\!\!\!  / \,\,}}
\newcommand{\p}{\prime}
\newcommand{\crb}[1]{\mathcal{V} (#1)}
\newcommand{\proj}{\pi}
\newcommand{\GLk}{{GL(\C^k)}}
\newcommand{\CRder}[1]{D_{#1}}
\newcommand{\atlas}{\mathcal{A}}
\newcommand{\fatA}{\mathbf{A}}
\newcommand{\dblind}{\blacktriangle}
\newcommand{\indexset}{\mathcal{K}}
\newcommand{\inindexset}{\mathbf{k}}
\newcommand{\indexsettwo}{\mathcal{K}_{\hat n}}
\newcommand{\stringA}{\mathbb{A}}
\newcommand{\Span}{\text{\rm span}}
\newlength{\extendaxesby}\setlength{\extendaxesby}{.4cm}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\spanc}{span} \DeclareMathOperator{\imag}{Im}
\DeclareMathOperator{\real}{Re}
\DeclareMathOperator{\class}{class}
\DeclareMathOperator{\pr}{proj}
\DeclareMathOperator{\id}{id}
\newtheorem{thm}{Theorem}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{defin}[thm]{Definition}
\newtheorem{cor}[thm]{Corollary}
\newtheorem{cl}{Claim}
\theoremstyle{definition}
\newtheorem{exple}{Example}
\newtheorem{rem}{Remark}


\begin{document}
\title[Finite jet determination of CR embeddings]
{Finite jet determination of constantly degenerate CR embeddings}%
\author{Peter Ebenfelt}%
\address{Department of Mathematics, University of California at
San Diego, La Jolla, CA 92093}%
\email{pebenfel@math.ucsd.edu}%
\author{Bernhard Lamel}
\address{Department of Mathematics, University of Illinois,
Urbana}
\email{lamelb@member.ams.org}%
\thanks{The first
  author is a Royal Swedish Academy of Sciences Research Fellow
  supported by a grant from the Knut and Alice Wallenberg
Foundation and also by DMS-0100110. The second author was
supported by the ANACOGA research network.}
\subjclass{32H02}%
\keywords{}%
\begin{abstract}
We prove finite jet determination results for smooth CR
embeddings which are of constant degeneracy, using the method of
complete systems. As an application, we derive a reflection
principle for mappings between a Levi-nondegenerate hypersurface
in $ \CN$ and  a Levi-nondegenerate hypersurface in $\CNplus$. We
also give an independent proof of the reflection principle
for mappings between strictly pseudoconvex hypersurfaces
in any codimension due to Forstneric \cite{FO1}.
\end{abstract}
\maketitle



\section{Introduction} In this paper, we shall derive a complete
system of differential equations for CR mappings between a smooth
hypersurface in $\CN$ and a smooth hypersurface in $\CNh$ (see
e.g.\ Theorem \ref {T:complsys}). This system of differential
equations will then be used to prove finite determination results
(Theorems \ref{T:codimension1}, \ref{T:strpcx}, and
\ref{T:nondegenerate}) for such mappings. For historical
information and background on the finite determination problem,
we refer the reader to the introduction of \cite{E5} or the
survey article \cite{BERbull}. As an application of our main
results, we also prove a new reflection principle for smooth
mappings between real-analytic Levi nondegenerate hypersurfaces
(Corollary \ref{C:reflection1}).


To formulate our results precisely, we shall need to introduce
some notation. Let $M$ be a real hypersurface in $\bC^N$, $N\geq
2$, and let us denote by $\mathcal V\subset \bC TM$ its CR bundle;
the reader is referred e.g.\ to \cite{BERbook} for basic notions
and facts about CR manifolds. Recall that a mapping $f\colon M\to
\bC^k$ is called CR if $f_*(\mathcal V)\subset T^{0,1}\bC^k$,
where $f_*$ denotes the induced mapping of the tangent spaces (the
push forward). This is equivalent to saying that, near every point
$p\in M$, there is a neighborhood $U\subset M$ of $p$ in which
\begin{equation}\label{eq-CR}
L_{\bar{a}}f_j=0,\quad a=1,\ldots,n,\ j=1,\ldots k,
\end{equation}
where $n=N-1$ and $L_{\bar 1},\ldots,L_{\bar n}$ is a basis for
the CR vector fields in $U$. We use here the convention that e.g.\
$L_{\bar{a}}=\overline{L_a}$ so that the $L_1,\ldots, L_n$ form a
basis for the vector fields valued in $\overline{\mathcal V}$ near
$p$ on $M$.


Suppose that $\hat M\subset \bC^{\hat N}$ is a real hypersurface
and $f\colon M\to \bC^{\hat N}$ is a CR mapping sending $M$ into
$\hat M$. Let $p$ be a  point in $M$ and $\hat\rho$ a local
defining function for $\hat M$ near $\hat p:=f(p)\in \hat M$. The
mapping $f$ is called {\it transversal} at $p$ if $f_*(\bC T_pM)$
is not contained in the subspace (of codimension 1) $\hat
{\mathcal V}_{\hat p}+\overline{\hat {\mathcal V}}_{\hat
p}\subset\bC T_{\hat p}\hat M$. One can show (see section
\ref{SS:pullback}) that if $f$ is transversal and $M$ is Levi
nondegenerate at $p\in M$, then $f$ is in fact a local embedding
near $p$.


Following Lamel \cite{L1}, we define an increasing sequence of
subspaces $E_k(q)\subset\bC^{\hat N}$, for $q\in M$ near $p$,
associated to the germ of the mapping $f\colon (M,q)\to \hat M$ as
follows. Let $L_{\bar 1},\ldots, L_{\bar n}$ be a basis for the CR
vector fields on $M$ near $p$ as above and define
\begin{equation}\label{eq-ksdeg1}
E_k(q):=\Span\{L^{\bar J}(\hat \rho_{Z'}\circ f)(q) : J\in
(\mathbb Z_+)^n, |J|\leq k\} \subset \bC^{\hat N},
\end{equation}
where $\hat \rho_Z=\big({\partial\hat \rho}/{\partial
Z'_j}\big)_{1\leq j\leq \hat N}$ in some local coordinate system
$Z'$ near $\hat p$; we use here standard multi-index notation $
L^{\bar J}:=L_{\bar 1}^{\bar J_1}\ldots L_{\bar n}^{\bar J_n}$ and
$|J|=J_1+\ldots+J_n$. One can show (cf.\ \cite{L1}) that the
increasing sequence of numbers $d_k(q):=\dim E_k(q)$ is
independent of the choice of local defining function $\hat \rho$
and coordinates $Z'$, as well as of the choice of basis of the CR
vector fields $L_{\bar 1},\ldots, L_{\bar n}$. We shall say, again
following Lamel ({\it loc.\ cit.}) that $f\colon (M,q)\to \hat M$
is $(k_0,s)$-degenerate at $q$ if $s$ is the minimum of the
decreasing sequence of codimensions of $E_k$, i.e.\
$$s:=\min_k(\hat N-d_k),$$ and $k_0$ is the smallest integer such that this
minimum is attained. The case where $s=0$ is actually a
nondegenerate situation, and we say that $f$ is
$k_0$-nondegenerate if it is $(k_0,0)$-degenerate. If $f$ is
$(k_0,s)$-degenerate at a point $q$ and the degeneracy $s$ is
constant in a nieghborhood of $q$, then we shall say that $f$ is
constantly $(k_0,s)$-degenerate at $q$. The notion of
$(k_0,s)$-degeneracy will be presented in a more intrinsic form in
section \ref{SS: conditions}.



The following are the main results in this paper.


\begin{thm}\label{T:codimension1}
Let $M\subset\CN$ and $\hat M \subset \CNplus$ be smooth
hypersurfaces which are Levi-nondegenerate at $p_0\in M$ and $\hat
p_0\in \hat M$, and $f:M\to\hat M$ a smooth CR mapping with
$f(p_0)= \hat p_0$ which is constantly $(k_0,s)$-degenerate and
transversal at $p_0$. Then $f$ is uniquely determined by its
$2k_0$-jet $j^{2k_0}_{p_0} f$ at $p_0$ in a neighbourhood of
$p_0$. That is, there exists a neighbourhood $U$ of $p_0$ in $M$
such that if $g: M\to \hat M$ is another smooth CR map of constant
degeneracy $s$ with $j^{2k_0}_{p_0} f = j^{2k_0}_{p_0} g$, then
$f|_{U} = g|_U$.
\end{thm}


\begin{thm}\label{T:strpcx}
Let $M\subset\CN$ and $\hat M \subset \CNh$ be smooth
hypersurfaces, $p_0\in M$, $\hat p_0\in \hat M$, $\hat M$ strictly
pseudoconvex at $\hat p_0$, and $f:M\to\hat M$ a smooth CR
mapping with $f(p_0)= \hat p_0$ which is constantly
$(k_0,s)$-degenerate at $p_0$. Then $f$ is uniquely determined by
its $2k_0$-jet $j^{2k_0}_{p_0} f$ at $p_0$ in a neighbourhood of
$p_0$. That is, there exists a neighbourhood $U$ of $p_0$ in $M$
such that if $g: M\to \hat M$ is another smooth CR map of constant
degeneracy $s$ with $j^{2k_0}_{p_0} f = j^{2k_0}_{p_0} g$, then
$f|_{U} = g|_U$.
\end{thm}


Let us note here that in the setting of Theorem~\ref{T:strpcx},
the mapping $f$ is automatically transversal (see e.g. the paper
\cite{FO1} by Forstneric). The reader should also notice that in
Theorem \ref{T:strpcx}, where the hypersurfaces are required to be
strictly pseudoconvex, there is no restriction on the codimension
$\hat N-N$ of the mapping $f$ whereas in Theorem
\ref{T:codimension1}, where no pseudoconvexity is assumed, the
codimension is restricted to be one. The reason for this is easily
seen by considering the case where $M\subset \bC^N$ is given by
the equation
$$
\Im z_{1}=\sum_{k=2}^{N}|z_k|^2
$$
and $\hat M\subset \bC^{N+2}$ by
$$
\Im z_{1}=\sum_{k=2}^{N}|z_k|^2+|z_{N+1}|^2-|z_{N+2}|^2.
$$
The mapping
\begin{equation}\label{E:h}
(z_1,\ldots,z_N)\mapsto (z_1,\ldots,z_{N},f(z),-f(z)),
\end{equation}
where $f$ is an arbitrary holomorphic function with $f(0)=0$,
sends $(M,0)$ into $(\hat M,0)$. It is not difficult to see that
if $k_0$ is the smallest integer $\geq 2$ for which there is a
multi-index $\alpha\in\mathbb Z_+^{N-1}$ with
$\partial^\alpha_{z'} f(0)\neq 0$, where $z'=(z_2,\ldots,z_N)$,
then the mapping \eqref{E:h} is constantly $(k_0,1)$-degenerate at
$0$. Clearly, you can find two different such functions $f_1,f_2$
whose jets at $0$ agree up to arbitrary order and, hence, Theorem
\ref{T:codimension1} is false in codimension $2$.

We shall see that in fact, given a mapping $f$ as in the preceding
theorems, there exists a dense, open subset $M_f$ of $M$ such
that for each $p\in M_f$, $f$ is constantly $(k_0,s)$- degenerate
for some $s$ and furthermore, that $k_0 \leq \hat N -N+1 - s\leq
\hat N - N + 1$ (see Lemma~\ref{L:bounds}). Hence, we have the
following corollaries.

\begin{cor}\label{C:codimension1}
Let $M\subset\CN$ and $\hat M \subset \CNplus$ be smooth
hypersurfaces which are Levi-nondegenerate,
and $f:M\to\hat M$ and $g: M\to \hat M$  smooth, transversal CR
mappings.
 If for any $p_0$ in the dense open subset
$M_f\cap M_g$ of $M$ we have that $j^{2(\hat N - N+1)}_{p_0} f =
j^{2(\hat N - N+1)}_{p_0} g$, then
$f = g$.
\end{cor}


\begin{cor}\label{C:strpcx}
Let $M\subset\CN$ and $\hat M \subset \CNh$ be smooth strictly
pseudoconvex hypersurfaces and $f:M\to\hat M$ and $g: M \to \hat
M$ smooth CR mappings. If for any $p_0$ in the dense open subset
$M_f\cap M_g$ of $M$ we have that $j^{2(\hat
N - N+1)}_{p_0} f = j^{2(\hat N - N+1)}_{p_0} g$, then $f = g$.
\end{cor}

As an application of Theorem \ref{T:codimension1}, we obtain the
following reflection principle.


\begin{cor}\label{C:reflection1}
Let $M\subset\CN$ and $\hat M \subset \CNplus$ be real-analytic
hypersurfaces which are Levi-nondegenerate, and $f:M\to\hat M$ \ a
smooth transversal CR mapping. Then there exists an open dense
subset $M_0\subset M$ such that for any $p\in M_0$, there exists
an open neighbourhood $V$ of $p$ in $\CN$ and a holomorphic map
$F\colon V\to \CNh$ such that $F|_{V\cap M} = f$.
\end{cor}


Let us give the (short) proof here. We first point out that given
a mapping $f$ as in Corollary \ref{C:reflection1}, there exists a
dense, open subset $M_0$ of $M$ such that for each $p\in M_0$, $f$
is constantly $(k_0,s)$-degenerate, for some $s$, in an open
neighborhood of $p$ (see Lemma \ref{L:global}); in fact, one may
choose $M_0$ such that $f$ is locally constantly
$(k_0,s)$-degenerate on $M_0$ with $s\leq \hat N-N$ and $k_0 \leq
\hat N -N+1 - s\leq \hat N - N + 1$ (see Lemma~\ref{L:bounds}).
For points $p_0\in M_0$, we can use a result of the second author
(\cite{L2}, Theorem 6) to conclude that the formal Taylor series
of $f$ at $p_0$ converges (in some neighbourhood V of $p_0$) to a
holomorphic mapping $F: V \to \CNh$ with $F(M\cap V)\subset \hat
M$. By Theorem~\ref{T:codimension1}, $F|_M = f$, which proves the
corollary.


The same argument (using Lemma~\ref{L:global} and Theorem
\ref{T:strpcx}) also gives an independent proof of the following
theorem due to Forstneric \cite{FO1}.


\begin{cor}\label{C:reflection2}
Let $M\subset\CN$ and $\hat M \subset \CNplus$ be real-analytic
strictly pseudoconvex hypersurfaces and $f:M\to\hat M$ a smooth CR
mapping. Then there exists an open dense subset $M_0\subset M$
such that for any $p\in M_0$, there exists an open neighbourhood
$V$ of $p$ in $\CN$ and a holomorphic map $F\colon V\to \CNh$ such
that $F|_{V\cap M} = f$.
\end{cor}


A special case for which we also derive finite determination
results  without any conditions on $\hat N$ is the case of {\em
nondegenerate} mappings. These have already been investigated by
Kim and Zaitsev \cite{KZ1} in the general codimension case.
However, our method is a bit different, and we have an easy,
independent proof of the following result for nondegenerate
mappings between hypersurfaces.


\begin{thm}\label{T:nondegenerate}
Let $M\subset\CN$ and $\hat M \subset \CNh$ be smooth
hypersurfaces, $p_0\in M$ a point of finite type, and let
$f:M\to\hat M$ be a smooth CR mapping which is $k_0$-nondegenerate
at $p_0$. Then $f$ is uniquely determined by its $2 k_0$-jet
$j^{2k_0}_{p_0} f$ at $p_0$ in a neighbourhood of $p_0$. That is,
there exists a neighbourhood $U$ of $p_0$ in $M$ such that if $g:
M\to \hat M$ is a smooth CR map with $j^{2k_0}_{p_0} f =
j^{2k_0}_{p_0} g$, then $f|_{U} = g|_U$.
\end{thm}


In order to prove Theorems \ref{T:codimension1}, \ref{T:strpcx},
and \ref{T:nondegenerate} we will construct complete systems of
differential equations for all the cases considered. To formulate
the result more precisely, we shall use the following notation.
For an open subset $U\subset M$, we shall denote by $J^{m} ( U,
\hat M )$ the space of $m$-jets of smooth (not necessarily CR)
mappings $F\colon U\to \hat M$ and by $j^m_p F$ the $m$-jet  of
such a mapping at $p\in U$ (which we shall think of as the
collection of all derivatives of $F$, in some local coordinate
system near $p$, up to order $m$). The result needed to prove
Theorems \ref{T:codimension1} and \ref{T:strpcx} can now be
formulated as follows.


\begin{thm}\label{T:complsys} Let $M\subset\CN$, $\hat M \subset \CNh$ be
smooth
hypersurfaces, $p_0\in M$, $\hat p_0\in\hat M$, $f: M\to \hat M$ a
transversal smooth CR map with $f(p_0)= \hat p_0$ of constant
degeneracy $(k_0, s)$ at $p_0$, and assume that either $M\subset
\CN$ and $\hat M\subset \C^{N+1}$ are Levi nondegenerate at $p_0$
and $\hat p_0$ respectively, or that $M$ and $\hat M$ are strictly
pseudoconvex at $p_0$ and $\hat p_0$ respectively (with no
restrictions on $\hat N$). Then there exists a neighbourhood $U$
of $p_0$ in $M$ and a smooth function $\phi : U \times J^{2k_0 +
2} ( U, \hat M )\to  J^{2k_0 + 2} ( U, \hat M )$ which only
depends on $M$ and $\hat M$ such that
\begin{equation}\label{E:complsys}
  j^{2k_0 + 3}_x f = \phi (x, j^{2k_0+2}_x f), \quad x\in U.
\end{equation}
Furthermore, there exists a neigbourhood $V$ of $j^{2k_0 +2}_{p_0}
f$ such that if $g: M\to \hat M$ a transversal smooth CR map with
$g(p_0)= \hat p_0$ of constant degeneracy $(k_0, s)$ at $p_0$ with
$j^{2 k_0 + 2}_{p_0} g \in V$, then (for some neighbourhood $U^\p$
of $p_0$)
\begin{equation}\label{E:complsys2}
  j^{2k_0 + 3}_x g = \phi (x, j^{2k_0+2}_x g), \quad x\in U^\p.
\end{equation}
\end{thm}


Theorems \ref{T:codimension1} and \ref{T:strpcx} follow from
Theorem \ref{T:complsys} using standard arguments about uniqueness
of solutions of complete systems of differential equations (see
e.g.\ \cite{BCG}) and the fact that the $m$-jet of any mapping
$f\colon M\to \hat M$, as in Theorems \ref{T:codimension1} and
\ref{T:strpcx}, at $p_0$, for any $m$ (and hence in particular the
$2k_0+2$-jet) is determined by its $2k_0$ jet at $p_0$ (\cite{L2},
Theorem 8). Thus, to prove Theorems \ref{T:codimension1} and
\ref{T:strpcx} it suffices to prove Theorem \ref{T:complsys}.
Also, by a remark given earlier, we could drop the assumption of
transversality in the strictly pseudoconvex case, since it is
satisfied automatically.


The organization of the paper is as follows: In section
\ref{S:CRvbundles}, we review the definition of a CR vector
bundle, give definitions of the nondegeneracy conditions
introduced above in this context, and introduce some essential
notation and tools. Following that, we give the proof of Theorem
\ref{T:nondegenerate}. In the next two sections, we derive some
jet reflection identities which we use in section
\ref{S:completesystem} to prove Theorem~\ref{T:complsys}.


\section{CR vector bundles}\label{S:CRvbundles} In this section, we
define the notion of a CR vector bundle. This notion is not new
and has been extensively used in the literature, but we give a
thorough definition nonetheless. We also give some examples, and
introduce the canonical derivative operator $ \CRder{}$ (which is
the same as $\bar
\partial_b$ in most of the literature). For our purposes, the
local formalism of this derivative operator is more important than
the global formalism developed alongside it.
\subsection{Definition} Let $M$ be a CR manifold. Let us first recall some
basic definitions. We are given a subbundle $\crb{M} \subset \C T
M$ (which we will refer to as the {\em CR bundle} of $M$) which
satisfies
\begin{equation}\label{E:CRbundle}
  [ \crb{M} , \crb{M} ] \subset \crb{M}, \quad \text{and } \crb{M}
  \cap \overline{\crb{M}} = \{ 0 \}.
\end{equation}
In \eqref{E:CRbundle}, $[ \crb{M} , \crb{M} ] \subset \crb{M}$
means that for all sections $X,Y$ of $\crb{M}$, the section
$[X,Y]$ of $\C T M$ takes values in $\crb{M}$, and $0$ stands for
the $0$ section of $\C T M$. If $M$ is a CR manifold, we also say
that $M$ has a CR structure, and sometimes we even refer to
$\crb{M}$ as a CR structure on $M$. If $U\subset M$ is open, a
section $\bar L \in \Gm(U,\crb{M})$ is called a CR vector field
over $U$. A CR function on $U$ is one which is annihilated by all
CR vector fields on $U$. If $M$, $\hat M$ are two CR manifolds
and $f:M \to \hat M$ is a map, say of class $C^1$, we say that
$f$ is CR provided $f_* (\crb{M}) \subset \crb{\hat M}$. Let us
also recall that if $X$ is a complex manifold, then $X$ has a
natural CR structure given by $\crb{X} = T^{(0,1)} X$.


We consider a complex vector bundle $E$ over $M$ with projection
$\proj:E\to M$. A vector bundle atlas $\atlas$ consists of pairs
$(U_\al,\phi_\al)$ where $\{U_\al\colon \al\in A\}$  is an open
cover of $M$, and $\phi_\al : \proj^{-1} (U_\al)\to  U_\al \times
\C^k$ is a diffeomorphism such that $\proj|_{\proj^{-1}(U_\al)} =
\pr_1 \circ \phi_\al$, where $\pr_l$ denotes the
projection on the $l$th component in a product. The
chart change $\phi_\al \circ \phi_\bt^{-1}\colon (U_\al \cap
U_\bt) \times \C^k\to (U_\al \cap U_\bt) \times \C^k$,
is of the form $\phi_\al \circ \phi_\bt^{-1} (x,v) = (x,\phi_{\al
\bt} (x) v)$, where the {\em transition functions} $\phi_{\al
\bt}$ are smooth functions (on $U_\al\cap U_\bt$) valued in $\GLk$
satisfying the usual cocyle conditions (cf.\ e.g\ \cite{GH}).


We say that $(E,\atlas)$ is a {\em CR vector bundle} if for $\al,
\bt \in A$  the transition functions $\phi_{\al \bt}$  are CR. We
refer to $\atlas$ as a {\em CR atlas} for $E$, and if a CR atlas
is fixed, we simply say that $E$ is a CR vector bundle.


A vector bundle chart $(U, \phi)$  for $E$ is said to be {\em
compatible} with $\atlas$ if $\phi_\al \circ \phi^{-1} (x,v) = (x,
\psi_\al (x) v)$ where $\psi_\al : U\cap U_\al\to \GLk$ is CR for
all $\al \in A$; we say that $(U,\phi)$ is a {\em CR chart} (for
$(E,\atlas)$), and that $U$ is a {\em trivializing neighbourhood}.
If $\atlas$ has the property that every CR chart $(U,\phi)$ for
$(E,\atlas)$ belongs to $\atlas$, we say that $\atlas$ is a {\em
maximal CR atlas} (for E). By including all compatible charts we
can associate to $\atlas$ a unique maximal atlas $\atlas_{max}$
with the property that $\atlas \subset \atlas_{max}$. Two CR
vector bundles $(E,\atlas)$ and $(E,\atlas^\p)$  are compatible if
$\atlas_{max} = \atlas_{max}^\p$. Equivalently, the CR vector
bundles $(E,\atlas)$ and $(E,\atlas^\p)$ are compatible if every
chart $(U_\al,\phi_\al)\in \atlas$ is a CR chart for
$(E,\atlas^\p)$ and vice versa. If a CR atlas $\atlas$ for $E$ is
fixed, we drop the atlas from the notation, understanding that if
we refer to $E$ we really refer to $(E,\atlas)$.


We should point out that a given vector bundle $E$ may have many
incompatible CR vector bundle structures. For instance, the
trivial line bundle $E=M\times \C$ with the trivial CR atlas
$\atlas$ consisting of the single chart $(M,\phi)$ with
$\phi(x,u)=(x,u)$ is compatible with the CR atlas $\atlas'$
consisting of the single chart $(M,\phi')$ with
$\phi'(x,u)=(x,f(x)u)$, where $f\colon M\to \C$ is some
nonvanishing function, if and only if $f$ is CR.






\subsection{The canonical CR structure} In what follows, $(E,\atlas)$
will denote a CR vector bundle over the CR manifold $M$. We shall
keep the CR atlas $\atlas$ fixed and simply refer to the CR vector
bundle $(E,\atlas)$ as $E$. In this section, we shall define a CR
structure on $E$. The main point is that the CR structures induced
on $\proj^{-1}(U_\al)$ by the product CR structures on $U_\al
\times \C^k$, via  the diffeomorphisms $\phi_\al$, fit together
since the transition functions are CR. The CR structure induced on
$E$ in this way can also be characterized by a property of
sections of $E$.


\begin{lem}\label{L:CRstruc} Assume that at each point $p\in M$
there exists a CR function $g$ with ${\bar L} \bar g (p) \neq 0$
for some CR vector field ${\bar L}$. Then there exists a unique CR
structure on $E$ such that a section $\sg\in\Gm(U,E)$ on an open
subset $U\subset M$ is CR if and only if for each CR chart
$(V,\phi)$ of $E$ the components of $\pr_2\circ\phi \circ \sg$ are
CR functions on $U\cap V$. Furthermore, relative to this CR
structure on $E$, the projection $\proj$ is CR, and for each $p\in
M$, there exist a neighbourhood $U$ of $p$ and a local basis of CR
sections for $E$ over $U$.
\end{lem}


Before we start with the proof, let us explain some of the notions
used in the lemma. $M$ is of finite type if the Lie algebra
generated by the CR vector fields and their conjugates, evaluated
at $p\in M$, is all of $\C T_p M$.  A CR section on $U$ is a
section $\sg \in \Gm(U,E)$ which is CR as a map $U\to E$, i.e.
$\sg_* \crb{M }_p \subset \crb{E}_{\sg(p)}$ for all $p\in U$. A
local basis of CR sections (on $U$) is a collection of CR sections
$\sg_1,\dots ,\sg_k\in \Gm(U,E)$ such that $\sg_1 (p), \dots,
\sg_k (p)$ is a basis of $E_p$ for $p\in U$.


We will use that a bundle chart $(U_\al,\phi_\al)$ induces a
splitting of the tangent space $T_P E$ for $P\in \pi^{-1}
(U_\al)$: If we write $\phi_\al(P) = (p,v)$, then $\C T_P E \cong
\C T_{(p,v)} (U_\al \times \C^k) \cong \C T_p M \oplus \C T_v
\C^k$. The former isomorphism is given by the push forward
$(\phi_\al)_*$. To make the latter isomorphism explicit, let
$\io_1$ denote the natural inclusion $\C T_p M\hookrightarrow \C
T_p M \oplus \C T_v \C^k$ given by $\iota_1(X)=X\oplus 0$, and
similarly $\io_2: \C T_v \C^k \hookrightarrow \C T_p M \oplus \C
T_v \C^k $. A tangent vector $\io_1 X + \io_2 Y\in\C T_p M \oplus
T_v \C^k$ with $X\in \C T_{p_0} M$ and $ Y \in \C T_{v_0} \C^k$
acts on the germ of a smooth function $f(p,v)$ at $(p_0,v_0)$ by
$(\io_1 X + \io_2 Y) f = X(f(p,v_0)) + Y(f(p_0,v))$.


\begin{proof}[Proof of Lemma $\ref{L:CRstruc}$]  We define $\crb{E}_P =
(\phi_\al^{-1})_* (\crb{M}_p \oplus T^{(0,1)}_v \C^k)$. We will
show that $\crb{E}$ defined this way is the unique CR structure
claimed in Lemma~\ref{L:CRstruc}. To start, let us first show that
this definition is independent of the chart used. Consider a chart
change $\phi = \phi_\al \circ \phi_\bt^{-1}$. We denote the fiber
coordinate in the chart $\phi_\al$ by $u=(u^1,\dots,u^k)$, the
fiber coordinate in the chart $\phi_\bt$ by $v= (v^1,\dots ,v^k)$,
and let $P_0\in \proj^{-1}(U_\al \cap U_\bt)$, $\phi_\al (P_0) =
(p_0, u_0)$, $\phi_\bt (P_0) = (p_0, v_0)$; thus,
$$
(p,u)=\phi(p,v)=(p,\phi_{\al\bt}(p) v). $$ We must show that
$\phi_*(\crb{M}_{p_0} \oplus T^{(0,1)}_{v_0} \C^k)=\crb{M}_{p_0}
\oplus T^{(0,1)}_{v_0} \C^k$, or equivalently
$\phi_*(\io_1\crb{M}_{p_0} )\subset\crb{M}_p \oplus
T^{(0,1)}_{v_0} \C^k$ and $\phi_*(\io_2T^{(0,1)}_{v_0}
\C^k)\subset\crb{M}_{p_0} \oplus T^{(0,1)}_{v_0} \C^k$. For this
purpose, let $\bar L_{p_0}\in \crb{M}_{p_0}$. Then
\begin{align*}(\phi_{*} {\io_1 \bar L}_{p_0}) f(p,u)
& = {\bar L}_{p_0} f(p,\phi_{\al\bt}(p) v_0) \\
& = {\bar L}_{p_0} f(p, \phi_{\al\bt} (p_0) v_0) + \vardop{f}{u}
{\bar L}_{p_0} (\phi_{\al\bt} (p)) v_0 + \vardop{f}{\bar u} {\bar
L}_{p_0} ( \overline{
\phi_{\al\bt} (p)} ) \bar v_0 \\
&= \io_1 {\bar L}_{p_0} f +  \left( \io_2 \vardop{}{\bar
u}\biggr|_{u_0} f \right)  {\bar L}_{p_0} ( \overline{
\phi_{\al\bt} (p)} ) \bar v_0;
\end{align*}
the last equality holds since $\phi_{\al \bt}$ is CR. Also,
\begin{align*}
(\phi_{*} \io_2 \vardop{}{\bar v^j}\biggr|_{v_0} ) f(p,u) & =
    \io_2 \vardop{}{\bar v^j} \biggr|_{v_0} f(p, \phi_{\al\bt} (p) v) \\
    &= \vardop{}{\bar v^j} \biggr|_{v_0} f(p_0, \phi_{\al\bt} (p_0) v)
    \\
    &= \left( \io_2 \vardop{}{\bar u} \biggr|_{u_0} f \right) \phi_{\al\bt}
(p_0),
\end{align*}
so that $\phi_{*} {\io_1 \bar L}_{p_0} \in \crb{M}_{p_0} \oplus
T^{(0,1)}_{u_0} \C^k$ and $\phi_{*} \io_2 \vardop{}{\bar
v^j}\bigr|_{v_0}\in T^{(0,1)}_{u_0} \C^k$. Hence $\phi_{*}
(\crb{M}_{p_0} \oplus T^{(0,1)}_{v_0} \C^k) = \crb{M}_{p_0} \oplus
T^{(0,1)}_{u_0} \C^k$, as claimed.


It is easy to check that $[\crb{E}, \crb{E}] \subset \crb{E}$
(since the derivatives in the fiber directions and the
derivatives in the base directions commute). It also follows
immediately that $\crb{E} \cap \overline{\crb{E}} = \{ 0 \}$. We
conclude that $\crb{E}$  defines a CR structure on $E$;
furthermore, $\proj_{*} (\crb{E}) = \crb{M}$, so that $\proj$ is
CR as claimed.


We next verify that $\crb{E}$ has the properties claimed in
Lemma~\ref{L:CRstruc}. Let $\sg: U\to E|_U$ be a section, and
$(V,\phi)$ a CR chart with $U\cap V$ nonempty. By the construction
of the CR structure on $E$, $\sigma$ is CR if and only if
$\phi\circ\sigma$ is CR. Since $\sigma$ is a section, we have
$\phi\circ\sigma(p)=(p,\tilde\sigma(p))$, where $\td \sg=
\pr_2\circ\phi \circ \sg = (\sg^1, \dots , \sg^k)^t$. We compute
$(\phi\circ\sigma)_{*} {\bar L}_{p_0}$, where, for ease of
notation, we drop the inclusions $\io_1, \io_2$:
\begin{align*}
((\phi\circ\sigma)_{*} {\bar L}_{p_0}) f(p,u) &= {\bar L}_{p_0} f(p, \td
\sg(p)) \\
 &= {\bar L}_{p_0} f(p, \td \sg(p_0))\, + \, \vardop{f(p_0,u)}{\bar u}
  \Biggr|_{\td \sg(p_0)} {\bar L}_{p_0} \bar { \td \sg }\, +
\,\vardop{f(p_0,u)}{u}
 \Biggr|_{\td \sg(p_0)} {\bar L}_{p_0} \td \sg.
\end{align*}
From this we see that $\phi\circ\sigma$ (and hence $\sg$) is CR if
and only if the components of $\td \sg$ are CR functions.


Now let $p\in M$ and assume that there exists a CR function $g$
with the property that $\bar L \bar g (p) \neq 0$. Let us show
that the CR structure constructed above is unique with the
properties in Lemma \ref{L:CRstruc}. Pick $P_0\in E$ and a CR
chart $(V,\phi)$ with $P_0\in \pi^{-1}(V)$. Write
$\phi(P_0)=(p_0,u_0)$. There is no loss of generality in assuming
that $g(p_0)=u_0$. Using the computation above, first with $\tilde
\sigma=u_0$ and then with $\td \sg = g e_j$ for standard unit
vectors $e_j$, $j=1,\dots,k$, we conclude, since
$\sigma=\phi^{-1}(p,\tilde\sigma)$ is CR for each of the choices
of $\tilde \sigma$ above, that $\crb{E}$ must be the bundle
$(\phi^{-1})_{*}(\crb{M}_{p_0} \oplus T^{(0,1)}_{u_0} \C^k)$. This
completes the proof of the Lemma.
\end{proof}


\begin{rem} {\rm In what follows, unless specifically stated
otherwise, every CR vector bundle will be equipped with the
canonical CR structure provided by Lemma \ref{L:CRstruc}.}
\end{rem}


\subsection{Constructions with CR vector bundles} All standard vector
bundle constructions work in the category of CR vector bundles.
In particular, the sum, tensor product, and exterior product of
CR vector bundles carry the structure of a CR vector bundle, and
the dual bundle $E^*$ is a CR vector bundle. Also, if $\hat E$ is
a CR vector bundle over another CR manifold $\hat M$, and $f:M\to
\hat M$ is a CR map, the pullback bundle $f^{\sharp} \hat E$
inherits the structure of a CR vector bundle.


Let us discuss the construction of the pullback bundle in detail.
We let $E_x = {\hat E}_{f(x)}$ and define a vector bundle
structure on $E= \cup_{x\in M} E_x$ as follows: If $\hat U \subset
\hat M$ is open and $\hat \phi: (\hat \proj)^{-1} (\hat U) \to
\hat U \times \C^k$ is a CR chart for $\hat E$ over $\hat U$,
$(\hat \phi^{-1} \circ (f\times \id))^{-1}$ is a bundle chart for
$E$ over $U=f^{-1}(\hat U)$. Since the transition functions for
$E$ are just the pullbacks (by $f$) of the transition functions
for $\hat E$, $E$ is a CR vector bundle. The map $f:M\to \hat M$
lifts to a differentiable vector bundle map $F:E\to \hat E$. The
canonical CR structure on $\hat E$ is connected with the
structure on $E$ by $\crb{E}_p \subset \crb{\hat E}_{f(p)}$, so
that $F:E\to \hat E$ is a CR map. Also note that the pullback (by
$f$) $f^\sharp \sg\in \Gm(f^{-1}(\hat U), E)$ of a CR section
$\sg\in\Gm(\hat U , \hat E)$ is a CR section of $E$.


\subsection{The derivative operator} In analogy with complex analytic
vector bundles, we can differentiate sections of a CR vector
bundle $E$ (equipped with the canonical CR structure provided by
Lemma \ref{L:CRstruc}) with respect to CR vector fields. To see
this, let $\sg_1,\dots , \sg_k$ be a local basis of CR sections
over some open set $U$. If $\sg \in \Gm(U,E)$ is given by $\sg =
\sum_{j=1}^{k} a^j \sg_j$ with $a^j\in C^\infty (U)$ and ${\bar
L}$ is a CR vector field on $U$, we define
\[
\CRder{{\bar L}} \sg = \sum_{j=1}^{k} (\bar L a^j) \sg_j.
\]
This definition is invariant under changes of the local CR basis
$\sg_j$: Observe that a section $\sg \in \Gm(U,E)$, $\sg =
\sum_{j=1}^{k} a^j \sg_j$ is CR if and only if all the $a^j$ are
CR functions on $U$. Hence, if $\td \sg_1,\dots,\td \sg_k$ is
another basis of CR sections, $\sg_j = \sum_l c_j^{\;\, l}
\td\sg_l$ where the $c_j^{\;\, l}$ are CR functions on $U$. It
follows that $\sg = \sum_l (\sum_j c_j^{\;\, l} a^j)
\td\sg_l=\sum_l\tilde a^l\tilde\sigma_l$, and
\begin{align*}
\sum_l(\bar L\tilde a^l)\tilde\sigma_l  &= \sum_l {\bar L} (\sum_j
c_j^{\;\,l} a^j)
 \td\sg_l \\& =
 \sum_l  (\sum_j c_j^{\;\,l} \bar L a^j) \td\sg_l \\
 &= \sum_j (\bar L a^j) \sg_j,
\end{align*}
so that $\CRder{{\bar L}} \sg$ is independent of the local basis
of CR sections used. This allows us to define $\CRder{{\bar L}}
\sg$ for a section $\sg \in \Gm(U,E)$ on any open set $U\in M$.
Note that $\sg$ is CR if and only if $\CRder{{\bar L}} \sg = 0$
(the $0$ section) for all CR vector fields ${\bar L}$.


We also note that the operator $\om \mapsto \CRder{{\bar L}}\om$
is local, and since $\CRder{a\bar L} \om = a \CRder{{\bar L}}\om$
for any function $a$, $(\CRder{{\bar L}}\om)_p$ depends only on
${\bar L}_p$. Moreover, $\CRder{}$ satisfies the product rule:
\begin{equation}\label{E:productrule}
  \CRder{\bar L} (a\om) = (\bar L a) \om + a \CRder{\bar L} \om.
\end{equation}
If $\omega=(\om_1,\dots ,\om_k)$ is any local (not necessarily CR)
basis of sections over $U$ and ${\bar L}_1,\dots, {\bar L}_n$ is a
local basis of CR vector fields over $U$, we can write
$\CRder{{\bar L}_i} \om_j = \sum_l a_{ij}^{\;\;\, l} \om_l$, (or,
using matrix notation, $\CRder{{\bar L}_i} \om = A_i \om$) and
$A_i=(a_{i j}^{\;\;\,l})_i$ determines $\CRder{{\bar L}} \sigma$
for all sections $\sigma$ over $U$. If we change basis by $\td
\om_j = \sum_m C^{\;m}_j \om_m$, which we will write in matrix
notation as $\td \om = C \om$, we obtain
\begin{align*} \CRder{{\bar L}_i} \td \om
&= \CRder{{\bar L}_i}  C  \om \\
&= ({\bar L}_i C) \om + C (\CRder{{\bar L}_i} \om) \\
&= (({\bar L}_i C) C^{-1} + C A_i C^{-1}) \td \om.
\end{align*}
This means that under a change of basis, the $A_i$ transform as
follows:
\[ \td \om = C \om \Longrightarrow \td A_i = ({\bar L}_i C) C^{-1} + CA_i
C^{-1}.
\]
We can also think of $\CRder{}$ as an operator from sections of
$E$ to sections of $E\otimes \crb{M}^* \simeq E\otimes (\C T^* M /
\mathcal V(M)^\perp)$.  Since $\CRder{}$ satisfies the product
rule \eqref{E:productrule}, we may consider $\CRder{}$ as a
partial connection in $E$.


If $E$ is the trivial bundle $M\times \C$ or the {\it holomorphic
cotangent bundle} $T^\p M=\mathcal V(M)^\perp$, then $D$ is just
$\bar\partial_b$. If $M$ is a complex manifold and $E$ is an
analytic vector bundle, $D$ is $\bar
\partial$.


\section{The CR vector bundle structure on the holomorphic
cotangent bundle}


\subsection{Definition} Let now $M$ be an {\em integrable} CR manifold of CR
dimension $n = \dim \crb{M}_p$ and CR codimension $d=\dim M-2n$.
Integrability means that at each point $p\in M$ there exist a
neighbourhood $U$ of $p$ and  $N = n+d$ smooth CR functions $Z_j$
on $U$, $j=1,\dots ,N$ such that the differentials $dZ_1, \dots ,
dZ_N$ are linearly independent on $U$. The functions
$Z_1,\dots,Z_N$ are called a {\em family of basic solutions} in
$U$. One can show that $q\mapsto (Z_1(q),\ldots, Z_N(q))$ embeds
$M$, locally near $p\in M$, as a generic submanifold in $\C^N$.
Conversely, any embeddable (in this sense) CR manifold is
integrable.


We recall the definition of the holomorphic cotangent bundle $T^\p
M$: At each $p\in M$, $T^\p_p M = \crb{M}_p^\perp$. Since $\dim \C
T_p M = 2n + d$ and $\dim \crb{M}_p = n$, $\dim T^\p_p M = N$.
Hence, on a neighbourhood $U$ of a fixed point $p$ as above, the
$1$-forms $dZ_1, \dots, dZ_N$ form a basis of the sections of
$T^\p M$ over $U$. We want to define a CR atlas on $T'M$ such that
the bases obtained in this way are CR bases of sections. Assume
that we have another family of basic solutions $\td Z_1, \dots ,
\td Z_N$ on the open set $V\in M$. Then $d\td Z_j = \sum_k
a_j^{\;\,k} \, d Z_k$ on $U\cap V$, and exterior differentiation
yields
\[ \sum_k d a_j^{\;\,k} \wedge d Z_k = 0
\]
in the overlap $U\cap V$. By Cartan's Lemma, we conclude that $d
a_j^{\;\,k} = \sum C^l_k\, dZ_l$, and hence, ${\bar L} a_j^{\;\,k}
= \br{d a_j^{\;\,k}, {\bar L}} = 0$ for all CR vector fields
${\bar L}$, so that the $a_j^{\;\,k}$ are CR functions on $U\cap
V$. Hence, if we define a CR chart for $T^\p M$ over $U$ by
$\phi(\om_p) = (p, \ld^1,\dots,\ld^N)$ where $\om_p \in T^\p_p M$
is written as $\om_p = \sum_j \ld^j \,(dZ_j)_p$, then, by the
computation above, the collection of all such charts defines a CR
atlas on $T^\p M$. We equip $T'M$ with the canonical CR structure,
relative to this CR atlas, given by Lemma \ref{L:CRstruc}.


By the same argument as above using Cartan's lemma, we see that
the following lemma holds.


\begin{lem}\label{L:diffCR}
Let $g$ be a smooth function on an open set $U\subset M$. Then
$g$ is CR if and only if $dg$ is a CR section of $T^\p  M$ over
$U$.
\end{lem}


Sections of $T^\p M$ are called {\em holomorphic forms} (on $M$).
Let now $\om\in\Gm(M,T^\p M)$, and let ${\bar L}$ be a CR vector
field on $M$. The following useful formula holds:
\begin{equation}\label{E:DonTprime}
\CRder{{\bar L}} \om = {\bar L} \lrcorner d\om.
\end{equation}
To verify this identity, let $X\in\Gm (M,\C TM)$ be a tangent
vector field, and $\om = \sum_j a^j\, dZ_j$ locally. Then
\begin{align*}
\br{ {\bar L} \lrcorner d\om , X } &= \br{d\om , {\bar L}\wedge X} \\
&= \sum_j \br{ da^j \wedge dZ_j , {\bar L}\wedge X} \\
&= \sum_j (da^j({\bar L})\, dZ_j(X) - da^j (X)\, dZ_j ({\bar L})) \\
&= \sum_j ({\bar L}a^j)\, dZ_j (X) \\
&= \br{\CRder{{\bar L}}\om,X}.
\end{align*}


\subsection{Nondegeneracy conditions}\label{SS: conditions}
Recall that the {\em characteristic bundle} $T^0 M$ is defined by
\[ T^0 M = \left(\crb{M} \oplus \overline{\crb{M}}\right)^\perp.
\]
A {\em characteristic form} is a real nonvanishing section of $T^0
M$, and we write $\xi^0 (U)$ for the space of characteristic
forms on $U\subset M$. We usually denote a characteristic form by
$\gth$.


By the discussion above, an integrable CR manifold $M$ is finitely
nondegenerate (at $p$) as introduced by Baouendi, Huang and
Rothschild \cite{BHR1} if and only if for some $k$,
\[ \spanc \{ (\CRder{{\bar L}_1} \dots \CRder{{\bar L}_l} \gth) (p)
\colon {\bar L}_j \in \Gm(U,\crb{M}), \gth \in \xi^0 (U), l\leq k
\} = T^\p_p M,
\]
where $U$ is a suitable small neighbourhood of $p$. This follows
from the characterization of finite nondegeneracy found in e.g.
\cite{BERbook} and equation \eqref{E:DonTprime}.


We will use the notion of the pullback bundle introduced before in
order to give an intrinsic characterization of the nondegeneracy
of a CR map $f:M\to \hat M$ as introduced in \cite{L1}. Let $E =
f^\sharp T^\p \hat M$. $f$ is nondegenerate at $p_0$ if and only
if for some $k$,
\[
\spanc \{ \CRder{{\bar L}_1}\dots \CRder{{\bar L}_k} f^\sharp \gth
(p_0) \colon {\bar L}_j \in \Gm(U,\crb{M}), \gth \in \hat \xi
(\hat U), j\leq k \} = E_{p_0},
\]
where again $U$ is some neighbourhood of $p_0$, contained in
$f^{-1}(\hat U)$. More generally, we can define subspaces $E^k_p
\subset E_p$ (for $p\in U$, say) by
\[
E^k_p = \spanc \{ \CRder{{\bar L}_1}\dots \CRder{{\bar L}_l} \gth
(p) \colon {\bar L}_j \in \Gm(U,\crb{M}), \gth \in \xi^\p (U),
l\leq k \}
\]
Then $E^0_p \subset E^1_p \subset \dots \subset E^k_p \subset
\dots$, and for some smallest integer $k_0=k_0(p)$, $E^k_p =
E^{k_0}_p$ for $k \geq k_0 $. We call $s=s(p) = \dim E_{p} - \dim
E^{k_0(p)}_p$ the degeneracy of $f$ at $p$ and say that $f$ is
$(k_0,s)$-degenerate at $p$; we say that $f$ is of constant
degeneracy $s$ at $p_0$ if $s(p)\equiv d$ is constant in a
neighbourhood of $p_0$, and in this case we also say that $f$ is
constantly $(k_0, s)$-degenerate (note that the constancy only
applies to $s$). This characterization is equivalent to the
definition given in \cite{L2},  by a computation in coordinates
which we leave to the reader.


Note that $p\mapsto s(p)$ is upper semicontinuous. Hence, we have
the following:


\begin{lem} \label{L:global}
Let $f\colon M\to \hat M$ be a smooth CR map. Then there exists an
open dense subset $N\subset M$ such that $f$ is of constant
degeneracy at every point of $N$.
\end{lem}


Let us actually observe some more. It is proved in \cite{L2} that
the degeneracy $s$ of a transversal map is bounded by $\hat N -
N$. Furthermore it follows from that paper (and is easy to
prove), that $\dim E^1_p = N$ for any point $p$, if $\hat M$ is
Levi nondegenerate and $f$ is an embedding. Now note that if on
any open subset $U$ we had $E^{k+1}_p \subset E^k_p$ for $p\in
U$, it follows that $E_p = E^k_p$. It follows that if $f$ is of
constant degeneracy $s$ on $U$, then generically in $U$, $k_0
\leq \hat N - N + 1 -s \leq \hat N - N +1$. We state this as a
Lemma:


\begin{lem} \label{L:bounds}
Let $f\colon M\to \hat M$ be a smooth transversal CR embedding
between Levi nondegenerate hypersurfaces $M$ and $\hat M$. Then
there exists an open dense subset $N\subset M$ such that $f$ is
of constant degeneracy $(k_0,s)$ at every point of $N$ for some
$s\leq \hat N - N$ and $k_0\leq \hat N - N +1 - s \leq \hat N - N
+1$.
\end{lem}


\subsection{Some computations} From now on, unless stated otherwise, we assume
that
$M$ and $\hat M$ are of hypersurface type and integrable (so that
the holomorphic cotangent bundles are CR vector bundles), and that
$f:M\to \hat M$ is CR. Let us now fix a point $p_0$ (which we
denote by $0$ from now on) and a local basis ${L}_{\bar 1},\dots
{L}_{\bar n}$ of the CR vector fields on $M$ near $0$. We also
choose a characteristic form $\gth$ for $M$ near 0 and a real
vector field $T$ with $\br{\gth, T} =1$. We denote by $\gth^a$ the
dual form (relative to the basis of vector fields $T,L_1,\ldots,
L_n,L_{\bar 1},\ldots, L_{\bar n}$) to the anti CR vector field
$L_a$. We write $\hat 0 = f(0)$ and, as before, $\hat \gth$
denotes a (local) characteristic form on $\hat M$ near $\hat 0$,
and $\hat T$ a real vector field with $\br{\hat \gth, \hat T} =1$;
we extend $\hat\gth$ to a basis of $T^\p \hat M$ by $\hat n$ forms
$\hat \gth^1,\dots ,\hat \gth^{\hat n}$ which are the dual forms
of a local basis $\hat L_1,\dots \hat L_{\hat n}$ of the anti CR
vector fields on $\hat M$. We work on a small neighbourhood of $0$
which we may shrink, if necessary, without further mentioning it.


Since $f$ is CR, we can write
\[ f_* L_a = \gm_a^A \hat L_A,\quad  f_* L_{\bar a} = \overline{\gm_a^A} \hat
L,
\quad f_* T = \eta^A \hat L_A + \overline{\eta^A} \hat L_{\bar A}
+ \xi \hat T,\] or equivalently, \[ f^* \hat \gth^A = \gm^A_a
\gth^a + \eta^A  \gth, \quad f^* \hat \gth^{\bar A} =
\overline{\gm^A_a} \gth^{\bar a} + \overline{\eta^A}  \gth,\quad
f^* \hat \gth = \xi \gth.\] Here and from now on we use the
summation convention; small indices $a,b$ etc.\ range from $1$ to
$n$, capital indices $A,B$ etc.\ from $1$ to $\hat n$. We observe
here that $f$ is transversal at $p\in M$, as defined in the
introduction, if and only if $\xi(p)\neq 0$.


Let us write $D_{\bar a} = D_{L_{\bar a}}$, $f^\sharp \hat \gth =
\tau$, and $f^\sharp \hat \gth^A = \tau^A$; thus,
$\tau,\tau^1,\ldots,\tau^n$ forms a local basis (not necessarily
CR) for the CR vector bundle $E=f^\sharp T'\hat M$. We define
functions $g_{\bar a_1\dots \bar a_k B}$, $g_{\bar a_1 \dots \bar
a_k}$ on $M$ near $0$ by
\[D_{\bar a_k} D_{\bar a_{k-1}} \dots D_{\bar a_1} \tau =
g_{\bar a_1\dots \bar a_k B} \tau^B + g_{\bar a_1 \dots \bar a_k}
\tau.\]


Note that for elements $\sg_p$ of $E_p$, there is a natural
pairing $\br{\sg_p, X_{f(p)}}$ with elements $X_{f(p)}$ of $\C
T_{f(p)} \hat M$.  With this notation, we can write
\[
g_{\bar a_1\dots \bar a_k B} = \br{D_{\bar a_k} D_{\bar a_{k-1}}
\dots D_{\bar a_1} \tau , \hat L_B}, \quad
g_{\bar a_1 \dots \bar
a_k } = \br{ D_{\bar a_k} D_{\bar a_{k-1}} \dots D_{\bar a_1} \tau
, \hat T}.\]


We introduce similar notation for sections of $T^\p \hat M$ and
$T^\p M$:
\[ D_{\bar a_k} \dots D_{\bar a_1}  \gth = h_{\bar a_1 \dots \bar
a_k b} \gth^b + h_{\bar a_1\dots \bar a_k} \gth,\]
\[ D_{\bar A_k} \dots D_{\bar A_1}  \hat \gth = \hat h_{\bar A_1 \dots \bar
A_k B} \hat \gth^b + \hat h_{\bar A_1\dots \bar A_k} \hat \gth.\]
With $\br{\cdot,\cdot}$ denoting the canonical pairing between
vectors and covectors,
\[h_{\bar a_1\dots \bar a_k b} = \br{D_{\bar a_k} D_{\bar a_{k-1}}
\dots D_{\bar a_1} \gth ,  L_b}, \quad h_{\bar a_1 \dots \bar a_k
} = \br{ D_{\bar a_k} D_{\bar a_{k-1}} \dots D_{\bar a_1} \gth ,
 T},
\]
\[\hat h_{\bar A_1\dots \bar A_k B} = \br{D_{\bar A_k} D_{\bar A_{k-1}}
\dots D_{\bar A_1} \hat \gth , \hat L_B}, \quad \hat h_{\bar A_1
\dots \bar A_k } = \br{ D_{\bar A_k} D_{\bar A_{k-1}} \dots
D_{\bar A_1} \hat \tau , \hat T}.
\]


The $h_{\bar a b}$ relate to the Leviform of $M$ by
\begin{equation}\label{E:habasLeviform}
  h_{\bar a b} = \br{d\gth , L_{\bar a}\wedge L_b} = -\gth \left(
  [L_{\bar a}, L_b]
  \right),
\end{equation}
and similarly for $\hat M$.


%


We need to introduce (but just for a short while) one more
notation for our three bundles:
\[ D_{\bar a} \gth^b = R_{\bar a c}^b \gth^c + R_{\bar a}^b \gth,\]
\[D_{\bar a} \tau^B = {S}_{\bar a C}^B \tau^C + {S}_{\bar a}^B \tau,\]
\[D_{\bar A} \hat \gth^B = {\hat R}_{\bar A C}^B \hat \gth^C+ {\hat R}_{\bar
A}^B \hat \gth.\]


\begin{lem}\label{L:sections}
 Let $\sg$ be a section of $T^\p \hat M$. Then
\begin{equation}\label{E:transformationlaw1}
D_{\bar a} f^\sharp \sg = \overline{\gm_a^A} f^\sharp D_{\bar A}
\sg.
\end{equation}
In particular,
\begin{equation}\label{E:transformationlaw2}
  g_{\bar a B} = \overline{\gm_a^A} \hat h_{\bar A B},
  \quad g_{\bar a} = \overline{\gm_a^A} \hat h_{\bar
  A},
\end{equation}
and
\begin{equation}\label{E:transformaionlaw3}
  {S}_{\bar a C}^B = \overline{\gm_a^A} {\hat R}_{\bar A C}^B,
  \quad {S}_{\bar a}^C = \overline{\gm_a^A} {\hat R}_{\bar A}^a.
\end{equation}
\end{lem}
\begin{proof}
Let us consider a local CR basis $\om_1,\dots \om_{\hat N}$ for
$T^\p \hat M$. We write $\al_j = f^\sharp \om_j$ for the
corresponding local CR basis of $E$. Then locally $\sg = a^j
\om_j$ for smooth functions $a^j$, and we have
\begin{align*}
D_{\bar a} f^\sharp \sg &= D_{\bar a} f^\sharp (a^j \om_j)\\
&= D_{\bar a} (a^j\circ f) \al_j\\
&= (L_{\bar a} (a^j \circ f )) \al_j \\
&= ((f_* L_{\bar a}) a^j) \al_j \\
& = \overline{\gm_a^A} (L_{\bar A} a^j \circ f) \al_j \\
&= \overline{\gm_a^A} f^\sharp D_{\bar A} \sg.
\end{align*}


For the other assertions, we note that
\[ D_{\bar a} \tau = g_{\bar a B} \tau^B + g_{\bar a} \tau \] and
\[ D_{\bar A} \hat \gth = \hat h_{\bar A B} \hat \gth^B + h_{\bar
A} \hat \gth. \]%
Since
\[ \overline{\gm_a^A} f^\sharp (D_{\bar A} \hat \gth) =
\overline{\gm_a^A} \hat h_{\bar A B} \tau^B + \overline{\gm_a^A}
\hat h_{\bar A} \tau,\] comparing coefficients we get
\eqref{E:transformationlaw2}. \eqref{E:transformaionlaw3} follows
by the same argument.
\end{proof}


The following lemma is an immediate consequence of
\eqref{E:productrule} and the definitions above:
\begin{lem}\label{L:derivativrule} With the notations introduced above, the
following hold:
\begin{equation}\label{E:derivativrule1}
  g_{\bar a_1 \dots \bar a_k \bar a B} = L_{\bar a}g_{\bar a_1 \dots \bar a_k
  B} + g_{\bar a_1 \dots \bar a_k  C} S_{\bar a B}^C + g_{\bar a_1 \dots \bar
a_k }g_{ \bar a B}
\end{equation}
\begin{equation}\label{E:derivativrule2}
  {\hat h}_{\bar A_1 \dots \bar A_k \bar A B} = L_{\bar A} {\hat h}_{\bar A_1
\dots \bar A_k
  B} + {\hat h}_{\bar A_1 \dots \bar A_k  C} {\hat R}_{\bar A B}^C + {\hat
h}_{\bar A_1 \dots \bar A_k }{\hat h}_{ \bar A B}
\end{equation}
\begin{equation}\label{E:derivativrule3}
  h_{\bar a_1 \dots \bar a_k \bar a b} = L_{\bar a}h_{\bar a_1 \dots \bar a_k
  b} + h_{\bar a_1 \dots \bar a_k  c} R_{\bar a b}^c + h_{\bar a_1 \dots \bar
a_k }h_{ \bar a
  b}.
\end{equation}
\end{lem}
We also record the following two lemmas for later use. We write
$I= (i_1\dots i_l)$, where $1\leq i_j \leq n$, and we use the
notation $L^I = L_{i_1} \dots L_{i_l}$; for such an $I$, we write
$|I| = l$.
\begin{lem}\label{L:degeneracyincoords} For each $k\in\N$, the two subspaces
of $\C^{\hat n}$ defined by
\[ \spanc \{(g_{\bar a_1 \dots \bar a_l A} (p))_A \colon l\leq k
\}
\]
\[\spanc \{(L^{\bar I} g_{\bar a A} (p))_A \colon |I|\leq k
\}
\]
coincide. Consequently, the following are equivalent definitions
for the degeneracy $d(p)$ of $f$ at $p$ defined in section
\ref{SS: conditions}:
\begin{enumerate}[(i)]
\item $d(p) = \hat n - \max_k \dim \spanc \{(g_{\bar a_1 \dots \bar a_k A}
(p))_{A} \colon l\leq k\}$;
\item $d(p) = \hat n - \max_k \dim \spanc \{ (L^{\bar I} g_{\bar a A} (p))_A
\colon |I|\leq k\}$;
\end{enumerate}
 and in each case, $k_0 (p)$ is the minimal integer
$k$ for which the maximum on the right hand side is attained.
\end{lem}
The proof is by induction using \eqref{E:derivativrule1}.
\begin{lem}\label{L:gisequalrational}
For each $a_1,\dots, a_k$, $1\leq a_l\leq n$, and $B$, $1\leq
B\leq \hat n$ there exist  functions $u_{\bar a_1\dots \bar a_k
B}$ and $u_{\bar a_1\dots \bar a_k}$ polynomial in their arguments
with coefficients smooth on $M\times\hat M$  such that
\begin{equation}\label{E:gisequalrational1}
    g_{\bar a_1\dots \bar a_k B} = u_{\bar a_1\dots \bar a_k B}
    \left( \overline{L^I \gm_a^A} \right),\quad  {|I| \leq k -1}
\end{equation}
\begin{equation}\label{E:gisequalrational2}
    g_{\bar a_1\dots \bar a_k } = u_{\bar a_1\dots \bar a_k}
    \left( \overline{L^I \gm_a^A}  \right), \quad {|I| \leq k -1}.
\end{equation}
Furthermore,  $u_{\bar a_1\dots \bar a_k A}$ and $u_{\bar a_1\dots
\bar a_k}$ depend only on $M$ and $\hat M$ (and not on the
mapping $f$).
\end{lem}
The equalities \eqref{E:gisequalrational1} and
\eqref{E:gisequalrational2} mean the following: if the left hand
side is evaluated at $p$, it equals the right hand side with the
coefficients evaluated at $(p,f(p))$ and the arguments evaluated
at $p$. We will encounter many such equalities, and we will
always use this time-saving notation.


For the proof of Lemma~\ref{L:gisequalrational}, we recall that
for $k=1$, $g_{\bar a B} = \overline{\gm_a^A} \hat h_{\bar A B}$
and $g_{\bar a} = \overline{\gm_a^A} \hat h_{\bar A}$ by
Lemma~\ref{L:sections}. An induction and
Lemma~\ref{L:derivativrule}  finishes the proof of the
Lemma~\ref{L:gisequalrational}; the details are left to the
reader.


\subsection{The pullback operation}\label{SS:pullback}
In this section, we discuss
the relation between our vector bundle $E=f^\sharp T'\hat M$ and
the vector bundle $T^\p M$ over $M$. Recall that we have a pairing
$\br{\sg, X}$ for sections of $E$ and tangent vectors $X$ at
$f(p)$. We define a map $\phi$ by
\[
\br{\phi \sg, X} = \br{\sg,f_* X}.
\]
Since $f$ is CR, $\phi$ maps sections of $E$ to sections of $T^\p
M$. Note that $\phi: \Gm(U,E) \to \Gm(U,T^\p M)$ is linear over
$C^\infty (U)$, since
\[\br{\phi a \sg, X} = (a \sg)(f_*X) = a \sg(f_*X) = a \br{\phi
\sg, X}.
\]
\begin{lem}\label{L:commutativity} For each $j$, $D_{\bar j} \phi = \phi
D_{\bar
j}$. Furthermore, $\phi \circ f^\sharp = f^*$.
\end{lem}
\begin{proof}
 Since the derivative operation is local,
it is actually enough to work on a trivializing open set $U=
f^{-1} (V)$, where  have a family of basic solutions $\hat Z_j$
in $V$. A basis of the CR sections of $E$ is then given by $\sg_j
= f^\sharp d \hat Z_j$. Let $\tau$ be any section of $E$ over $U$.
Then $\tau = a^j \sg_j$, and $\phi \tau = a^j \phi(\sg_j)$. By the
definition of $D_{\bar j}$ we now see it is enough to show that
if $\sg$ is a CR section of $E$, then $\phi\sg$ is a CR section
of $T^\p M$. We compute:
\begin{align*} \br{ (\phi f^\sharp d\hat Z_j)_p , X_p}
&= \br{ (f^\sharp d\hat Z_j)_p , f_* X_p} \\
&= \br{ (d\hat Z_j)_{f(p)}, f_* X_p}  \\
&= \br{ f^* (d \hat Z_j)_p, X_p} \\
&= \br{ d( \hat Z_j\circ f)_p, X_p},
\end{align*}
so that actually $\phi f^\sharp d\hat Z_j = d (\hat Z_j\circ f)$,
which is CR by Lemma \ref{L:diffCR}. The second claim follows from
the same calculation.
\end{proof}
By an appropriate choice of basis $\gth^a$, $\gth$, and similarly
on $\hat M$, we may assume that $R^b_{\bar a c}, R^b_{\bar a},
\hat R^B_{\bar A C}, R^B_{\bar A}$ are all $0$ (cf.\ \cite{E5},
Lemma 1.33). By Lemma \ref{L:sections}, it follows that $S^B_{\bar
a c}=S^B_{\bar a}=0$. Lemma \ref{L:commutativity} implies the
following relations:
\begin{equation}\label{E:levitransform}
  \xi h_{\bar a b} = \gm_a^B g_{\bar b B};
\end{equation}
\begin{equation}\label{E:xitransform}
 L_{\bar a} \xi + \xi h_{\bar a} = g_{\bar a B} \eta^B + g_{\bar a
 } \xi.
 \end{equation}
This follows comparing the coefficients of $D_{\bar a} \phi \tau$
and $\phi D_{\bar a } \tau$. We also need the following
equalities:
\begin{equation}\label{E:barredderivativeofgamma}
  L_{\bar a} \gm_{b}^B = - \eta^B h_{\bar a b},
\end{equation}
\begin{equation}\label{E:barredderivativeofeta}
  L_{\bar a} \eta^B = - \eta^B h_{\bar a},
\end{equation}
\begin{equation}\label{E:Tderivativeofgamma}
  T\gm_a^A = L_a \eta^A + \eta^A h_a,
\end{equation}
which follow as in \cite{E5}. We observe that the identity
\eqref{E:levitransform} implies that if $f$ is transversal at $p$
(i.e.\ $\xi(p)\neq 0$) and $M$ is Levi nondegenerate at $p$ (i.e.
the matrix $(h_{\bar a b})(p)$ is invertible), then the rank of
the matrix $(\gamma^B_a)(p)$ must be maximal, i.e. $=n$. Thus, if
$f$ is transversal and $M$ is Levi nondegenerate, then $f$ is in
fact a local embedding.



\section{Jet reflection identities for nondegenerate mappings }
In this section we are going to establish the jet reflection
identities necessary to use the machinery developed in \cite{E5}.
Our first goal is the reflection identities for nondegenerate
mappings. Fix an integer $k_0$. We denote by $\indexset$ the set
of all strings $\stringA = \bar a_1 \dots \bar a_k$ of elements of
$\{\bar 1,\dots, \bar n\}$ with $k\leq k_0$, and we write
$\indexsettwo$ for the set of all $\hat n$-tuples of elements of
$\indexset$. For $\stringA = \bar a_1 \dots \bar a_k$ we let
$\ell(\stringA) = k$ denote the length of $\stringA$.


\begin{thm}\label{T:reflectionnondegenerate} Let $k_0\in \N$. There exist
functions
\begin{equation*}
\Da_{\dblind}\left( \overline{ L^J \gm_c^D }\right) , \quad
r_{a,\dblind}^{B} \left( \overline{ L^J \gm_c^D }, \overline{L^I
\xi}
  \right), \quad r^{B}_{\dblind}\left( \overline{ L^J \gm_c^D }, \overline{L^I
\xi}
  \right),
\end{equation*}
where  $|J|\leq k_0 - 1, |I| \leq k_0$, for $\dblind =
(\mathbf{A}_0,\dots,\mathbf{A}_n) \in\indexsettwo^{n+1}$, $a\in \{
1,\dots ,n\}$ and $B\in \{1,\dots,\hat n\}$ with the following
property: If $f:M\to \hat M$ is $k$-nondegenerate at $0$ with
$k\leq k_0$, then there exists $\dblind^0$ with $\Da_{\dblind^0}
\left( \overline{L^J \gm_c^D} (0) \right) \neq 0$, and for any
$\dblind \in\indexsettwo^{n+1}$ with $\Da_{\dblind} \left(
\overline{L^J \gm_c^D} (0) \right) \neq 0$ the following
equalities hold:
\begin{equation}\label{E:reflectionnondegenerate1}
  \gm_a^B = r_{a,\dblind}^B \left( \overline{ L^J \gm_c^D }, \overline{L^I
\xi}
  \right),
\end{equation}
\begin{equation}\label{E:reflectionnondegenerate2}
  \eta^B = r^B_\dblind \left( \overline{ L^J \gm_c^D }, \overline{L^I \xi}
  \right).
\end{equation}
The $\Da_\dblind$ are polynomials in $ \overline{L^J \gm_c^D}$
with complex coefficients; the functions $r_{a,\dblind}^B$ and
$r_\dblind^B$ are rational in $ \overline{L^J \gm_c^D}$,
polynomial in $\overline{L^I\xi}$, with coefficients smooth on
$M\times\hat M$. Furthermore, $\Da_\dblind$, $r_{a,\dblind}^B$,
and $r_\dblind^B$ only depend on $M$, $\hat M$, and $k_0$ (and
not on the mapping $f$).
\end{thm}


In order to define the functions $\Da_\dblind$ of the Theorem, we
need the following computational Lemma.
\begin{lem}\label{L:complemma} Let $f:M\to \hat M$ be a CR mapping.
Then for each $k\in \N$, all $a \in \{1,\dots, n\}$, and all
$\stringA = \bar a_1 \dots \bar a_k$ the following identities
hold:
\begin{equation}\label{E:complemma1}
  g_{\stringA B}\eta^B =
  \sum_{\ell(\stringA^\p) < \ell(\stringA) } g_{\stringA^\p B} \eta^B \,
v_{\stringA}^{\stringA^\p}
    \left( \overline{L^J \gm_c^D}\right) + w_{\stringA} \left( \overline{L^J
\gm_c^D},\overline{L^I \xi} \right);
\end{equation}
\begin{multline}\label{E:complemma2}
     g_{\stringA B} \gm_a^B =
  \sum_{\ell(\stringA^\p) < \ell(\stringA)}
  g_{\stringA^\p B} \gm_a^B \, p_{a \stringA}^{\stringA^\p}
    \left( \overline{L^J \gm_c^D}\right) + \\
  \sum_{\ell(\stringA^\p) < \ell(\stringA) } g_{\stringA^\p} \eta^B \, q_{a
\stringA}^{\stringA^\p}
    \left( \overline{L^J \gm_c^D}\right) + r_{a \stringA} \left( \overline{L^J
\gm_c^D},\overline{L^I \xi} \right);
\end{multline}
where $|J|\leq k-1$, $|I|\leq k$. The functions $v$, $w$, $p$,
$q$, and $r$ are rational in $\overline{L^I \gm_c^D}$, the
functions $w$ and $r$ are polynomial in $\overline{L^I \xi}$, with
coefficients smooth on $M\times\hat M$. Furthermore, $v$, $w$,
$p$, $q$, and $r$ only depend on $M$ and $\hat M$.
\end{lem}
\begin{proof} We start with \eqref{E:complemma1}. For $k=1$, by
\eqref{E:xitransform},
\[g_{\bar a B} \eta^B = L_{\bar a} \xi + \xi h_{\bar a} - \xi
g_{\bar a},
\]
and recalling \eqref{E:transformationlaw2}, $ g_{\bar a} =
\overline{\gm_a^A} h_{\stringA} $, we have that
\[ g_{\bar a B} \eta^B =
L_{\bar a} \xi + \xi h_{\bar a} - \xi
\overline{\gm_a^A}h_{\stringA}= w_{\bar a} \left(
\overline{\gm_c^D}, \overline{L^I \xi} \right),
\]
which is \eqref{E:complemma1} for $k = 1$. To get
\eqref{E:complemma1} for $k>1$, we use \eqref{E:complemma1} for
$k-1$ to express $g_{\bar a_1 \dots \bar a_{k-1} B} \eta^B$ and
apply $L_{\bar a_k}$, using Lemma~\ref{L:derivativrule} and
\eqref{E:barredderivativeofeta}.


The proof of \eqref{E:complemma2} is similar: We start with
\eqref{E:levitransform} for the case $k=1$, and for $k>1$, we use
induction, Lemma~\ref{L:derivativrule},
\eqref{E:barredderivativeofgamma}, and
\eqref{E:barredderivativeofeta}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T:reflectionnondegenerate}]
Let $\fatA = (\stringA_1,\dots, \stringA_{\hat n})$. We rewrite
the $\hat n$ equations obtained from  \eqref{E:complemma1} by
replacing  $\stringA$ with $\stringA_1,\dots, \stringA_{\hat n}$
in the following form:
\begin{equation}\label{E:systemnondegenerateforeta}
 M_{\fatA} \eta =
w_{\fatA},
\end{equation}
where
\[\eta = \begin{pmatrix}
  \eta^1 \\
  \vdots \\
  \eta^{\hat n}
\end{pmatrix},\quad
 w = \begin{pmatrix}
  w_{\stringA_1} \\
  \vdots \\
  w_{\stringA_{\hat n}}
\end{pmatrix},
\]
and the $(i,B)$th entry of $M_{\fatA}$ is
\[g_{\stringA_i B} -
  \sum_{\ell(\stringA^\p) < \ell(\stringA_i)}
  g_{\stringA^\p B} \, v_{\stringA_i}^{\stringA^\p}.
\]
Similarly, for each $a\in\{1,\dots,n\}$ we rewrite
\eqref{E:complemma2} as
\begin{equation}\label{E:systemnondegenerateforgamma}
N_{a,\fatA} \gm_a = K_{a,\fatA} \eta + r_\fatA.
\end{equation}
By Lemma~\ref{L:gisequalrational} and Lemma~\ref{L:complemma} the
matrices $M_\fatA$ and $N_{a,\fatA}$ are polynomial in
$\overline{L^J \gm_c^D}$, with coefficients smooth on $M\times
\hat M$ and they only depend on $M$ and $\hat M$. We define
$\Da_{0,\fatA} = \det M_\fatA$ and $\Da_{j,\fatA} = \det
N_{j,\fatA}$, $j=1,\dots, n$, where we evaluate the coefficients
of $\overline{L^J \gm_c^D}$ at $(0,0)$, and for $\dblind =
(\fatA_0, \dots ,\fatA_n)$, we set $\Da_{\dblind}=\prod_{j=0}^n
\Da_{j,\fatA_j}$. By Cramer's rule, if $\Da_{\dblind}
\left(\overline{L^J \gm_c^D}(0) \right) \neq 0$, in a
neighbourhood of $0$ we can solve the equations
\eqref{E:systemnondegenerateforeta} with $\fatA = \fatA_0$ and
\eqref{E:systemnondegenerateforgamma} with $(a, \fatA) =
(a_1,\fatA_1),\dots,(a_n,\fatA_n)$ for $\eta^B$ and $\gm_a^B$ in
the required form.


We finish the proof by showing that if $f$ is $k$-nondegenerate
at $0$  with $k\leq k_0$, then there exists $\dblind^0$ with
$\Da_{\dblind^0} \left(\overline{L^J \gm_c^D}(0) \right) \neq 0$.
First recall that by Lemma~\ref{L:degeneracyincoords}, $f$ is
$k$-nondegenerate at $0$ if and only if
\begin{equation}\label{E:nondegenerateifandonlyif}
\spanc \{ (g_{\stringA B}(0))_B  \colon \ell(\stringA)\leq k \} =
\Cnh.
\end{equation}
Consider a system of vectors of the form
\begin{equation}\label{E:definitionofthevs}
  v_{\stringA} =  \left(g_{\stringA B} (0) - \sum_{\ell(\stringA^\p) <
\ell(\stringA )}
c_{\stringA B}^{\stringA^\p} g_{\stringA^\p B} (0)\right)_B, \quad
\ell(\stringA) \leq k.
\end{equation}
We claim that if $f$ is $k$-nondegenerate, then $\spanc \{
v_{\stringA} \colon \ell(\stringA)\leq k \} = \Cnh$, which implies
that $\Da_{\dblind^0} \left(\overline{L^J \gm_c^D}(0) \right) \neq
0$ for some $\dblind^0$. To prove the last claim, we let $V_l =
\spanc \{(g_{\stringA B})_B (0) \colon \ell(\stringA)\leq l \}$
and $W_l = \spanc \{ v_{\stringA} \colon \ell(\stringA)\leq l \}$.
By \eqref{E:definitionofthevs}, $V_1 = W_1$. But then, again by
\eqref{E:definitionofthevs}, $V_2 = W_2$. By induction, we see
that $W_k = V_k = \Cnh$.
\end{proof}


Theorem \ref{T:reflectionnondegenerate} is completely analogous to
Theorem 2.4 in \cite{E5}. The proof of Theorem
\ref{T:nondegenerate} can now easily be completed using the
arguments in \cite{E5} and a result from \cite{L2}.


\begin{proof}[Proof of Theorem~\ref{T:nondegenerate}]
By repeating the arguments in \cite{E5}, based on the identities
in Theorem~\ref{T:reflectionnondegenerate} instead of those in
Theorem 2.4 in \cite{E5}, we obtain a complete system of some
order $l_0$ (the exact value of which is not relevant for this
argument) for $f$ and $g$. That is, there exists a neighbourhood
$U$ of $p_0$ and a smooth function $\phi:U\times J^{l_0}(U,\hat M)
\to J^{l_0+1}(U,\hat M)$ such that
\[j^{l_0 + 1}_{x} f = \phi (x,j^{l_0}_x f) \text{ and }
 j^{l_0 + 1}_{x} g = \phi (x,j^{l_0}_x g), \quad x\in U.\] By
Theorem 2 in \cite{L2}, $j^l_{p_0} f = j^l_{p_0} g$ for all $l$
and, in particular then, $j^{l_0}_{p_0} f = j^{l_0}_{p_0} g$. The
conclusion of Theorem~\ref{T:nondegenerate} follows from the
uniqueness of solutions of complete systems (see e.g. \cite{BCG}).
\end{proof}
\section{Jet reflection identities for Levi-nondegenerate hypersurfaces}
In this section, we consider the case $\hat N = N + 1$, where $M$
is Levi-nondegenerate at $0$, and $\hat M$ is Levi-nondegenerate
at $\hat 0$. We are also assuming that $f$ is transversal, which
in our notation is equivalent to $\xi (0) \neq 0$.


For the proof of Theorem~\ref{T:codimension1} we first note that
by Lemma 20 in \cite{L2}, the degeneracy of $f$ is either $0$ or
$1$. In the case where the degeneracy of $f$ is $0$,
Theorem~\ref{T:codimension1} follows from
Theorem~\ref{T:nondegenerate}, so that we only have to deal with
the case where $f$ is constantly $1$-degenerate. Using
\eqref{E:levitransform} it is easy to see that with our
assumptions, the vectors  $g_{a B} (0)$, $ a=1,\dots, n$ are
linearly independent, as are the vectors $ \gm_c^D (0)$,  $
c=1,\dots, n$. We conclude that $f$ is constantly
$(1,1)$-degenerate. Our goal in this section is to derive the
following jet reflection identity for such mappings. The unique
determination theorem will be proved in
section~\ref{S:completesystem} by producing a complete system
using this jet reflection identity.
\begin{thm}\label{T:reflectionlevinondegenerate}
Let $M\subset\CN$ and $\hat M\subset\CNplus$ be hypersurfaces
which are Levi-nondegenerate at the points $p_0$ and $\hat p_0$,
$f\colon M\to \hat M$ a smooth CR map which is constantly
$(1,1)$-degenerate and transversal at $p_0$. Then for any $I$ with
$|I| = k$ there exist functions $r^{I,A}_b$ and $s^{I,A}$ for
$a,b\in \{1,\dots,n\}$ and $A\in \{1,\dots,n+1\}$ such that
\begin{align}\label{E:reflectionlevinondegenerategamma}
  L^I \gm_b^A &= r_{b}^{I,A} \left( L^M \gm_c^D, L^M \eta^D,
  \overline{\gm_c^D}, \overline{\eta^D} \right),\\
\label{E:reflectionlevinondegenerateeta}
  L^I \eta^A &= s^{I,A} \left( L^M  \gm_c^D,L^M \eta^D,
  \overline{L_d\gm_c^D}, \overline{L_d\eta^D},\overline{T\gm_c^D},
  \overline{T\eta^D} \right)
\end{align}
Here $|M| \leq k-1$. The $r^{I,A}_b$ and $s^{I,A}$ are rational in
their arguments with coefficients which are smooth on $M\times
\hat M$, and they only depend on $M$ and $\hat M$.
\end{thm}


\begin{proof} Since $\hat M$ is Levi-nondegenerate, we can choose the basis
of CR vector fields tangent to $\hat M$ such that
\begin{equation}\label{E:conventionlevinondegenerate}\hat h_{\bar A B} (0)
= \eps_{\bar A B} = \begin{cases} \pm i, & A = B \\
0, & A\neq B \end{cases}.\end{equation}


Since $f$ is constantly $(1,1)$-degenerate, for any ${\bar a}$,
$\bar b$ the determinant of the matrix
\begin{equation}\label{E:bigmatrix}
\begin{pmatrix}
  g_{\bar 1 \, 1} & \dots & g_{\bar 1 \, n+1} \\
  \vdots &  & \vdots \\
  g_{\bar n \, 1} & \dots & g_{\bar n \, n+1} \\
  L_{\bar a} g_{\bar b \, 1} & \dots  & L_{\bar a} g_{\bar b \, n+1}
\end{pmatrix}
\end{equation}
vanishes. Let us write $(-1)^{B+1} \Da^B$ for the determinant of
the matrix obtained from the matrix \eqref{E:bigmatrix} by
dropping the last row and the $B$th column. We fix $a$ and $b$
for the moment. By developing the determinant of
\eqref{E:bigmatrix} along the last row,
\begin{equation}\label{E:detdeveloped}
  \Da^B L_{\bar a} g_{\bar b B} = 0.
\end{equation}
We compute
\begin{equation}\label{E:Lbaragisequal}
  L_{\bar a} g_{\bar b B} = L_{\bar a}\left( \overline{\gm_b^A}
  {\hat  h}_{\bar A B}\right) =
   \overline{L_{ a}\gm_b^A} {\hat  h}_{\bar A B} +
     \overline{\gm_b^A \gm_a^D} {\hat L}_{\bar D}
     {\hat h}_{\bar A B}.
\end{equation}
Plugging \eqref{E:Lbaragisequal} into \eqref{E:detdeveloped} and
using Lemma~\ref{L:gisequalrational} we conclude that
\begin{equation}\label{E:equationforbarlgamma}
 \Da^B \overline{L_{ a}\gm_b^A} {\hat  h}_{\bar A B} = r\left(
 \overline{ \gm_c^D } \right),
\end{equation}
where $r$ is rational in $\overline{\gm_c^D}$ with coefficients
smooth on $M\times \hat M$ and $r$ is determined only by $M$ and
$\hat M$. We take the complex conjugate of this equation to obtain
\begin{equation}\label{E:equationforlgamma}
 \Da^{\bar B} {L_{ a}\gm_b^A} {\hat  h}_{ A \bar B} = {\bar r}\left(
  \gm_c^D  \right),
\end{equation}


Consider now the equations \eqref{E:levitransform} which we
rewrite as
\[\xi h_{\bar d b} = \gm_b^A g_{\bar d A}.
\]
We apply $L_a$ to this equation and conclude, using also equations
\eqref{E:levitransform} and \eqref{E:xitransform},
\begin{equation}\label{E:lalevitransform}
  \left( L_a \gm_b^A \right) g_{\bar d A} = \td p_{\bar d} \left( \gm_c^D,
  \overline{\gm_c^D}, \xi, L_a \xi \right) = p_{\bar d} \left(
  \gm_c^D,\eta^D
  \overline{\gm_c^D}, \overline{\eta^D} \right),
\end{equation}
where $p_{\bar d}$ is rational in its arguments with coefficients
smooth on $M\times \hat M$ and $p_{\bar d}$ is determined only by
$M$ and $\hat M$. We claim that the matrix
\begin{equation}\label{E:matrixforsystem}
  \begin{pmatrix}
    g_{\bar 1\, 1}(0)  & \dots  & g_{\bar 1 \, n+1}(0) \\
    \vdots &  & \vdots \\
    g_{\bar n \, 1}(0) & \dots & g_{\bar n \, n+1} (0) \\
    \Da^{\bar 1}(0) \eps_{\bar 1 1} & \dots & \Da^{\overline{n+1}} (0)
  \eps_{\overline{n+1}n+1}
  \end{pmatrix}
\end{equation}
is invertible, so that the system of equations
\begin{align}\label{E:systemforlevinondegenerate}
  X^A g_{\bar d A} &= p_{\bar d}, \quad d\in\{1,\dots ,n\},\\
  X^A \Da^{\bar B} {\hat h}_{ A \bar B} &= \bar
r,\label{E:systemforlevinondegenerate2}
\end{align}
can be solved uniquely for the $X^A$ (in a neighbourhood of $0$).
Developing the determinant $\Da$ for the matrix
\eqref{E:matrixforsystem} along the last row, we conclude that
\begin{equation}\label{E:deltaisequal}
  \Da = \eps_{\bar A A} \Da^{\bar A} \Da^A.
\end{equation}
We are now going to show that this last expression is not equal
to $0$. Consider the matrices
\begin{equation}\label{E:matricesGandGm}
  G= \begin{pmatrix}
    g_{\bar 1\, 1}(0)  & \dots  & g_{\bar 1 \, n+1}(0) \\
    \vdots &  & \vdots \\
    g_{\bar n \, 1}(0) & \dots & g_{\bar n \, n+1} (0)
  \end{pmatrix},
  \quad
  \Gm = \begin{pmatrix}
    \gm_{\bar 1}^{ 1}(0)  & \dots  & \gm_{\bar n}^{1}(0) \\
    \vdots &  & \vdots \\
    \gm_{\bar 1}^{ n+1}(0) & \dots & \gm_{\bar n}^{ n+1} (0),
  \end{pmatrix}
\end{equation}
and the $n\times n$ matrix
\begin{equation}\label{E:matrixH}
  H=\begin{pmatrix}
    h_{\bar 1\, 1}(0)  & \dots  & h_{\bar 1 \, n}(0) \\
    \vdots &  & \vdots \\
    h_{\bar n \, 1}(0) & \dots & h_{\bar n \, n} (0) \\
  \end{pmatrix}.
\end{equation}
In matrix notation, \eqref{E:levitransform} then reads as $\xi H
= G \Gm$. Let us write $G_A$ for the matrix $G$ with the $A$th
column dropped, and $\Gm^A$ for the matrix $\Gm$ with the $A$th
row dropped. The Cauchy-Binet Theorem from linear algebra states
that $\det G \Gm^t = \det G_A\det \Gm^A$. By our convention
\eqref{E:conventionlevinondegenerate} and
\eqref{E:transformationlaw2} it follows that
\[ g_{\bar a B} (0) = \overline{\gm_a^B}(0) \eps_{\bar B B}.
\]
Writing $\eps = \prod_B \eps_{\bar B B}$ we obtain from the
equation above
\[ \Da^A = \frac{\eps}{\eps_{\bar A A}} \det \overline{\Gm^A}.
\]
Recalling that $\Da^A = \det G_A$, we see that the Cauchy-Binet
Theorem implies that
\begin{align*}\label{E:deltaisequal2}
  \Da &= \eps_{\bar A A} \Da^{\bar A} \Da^A \\
  &= \eps \det {\overline{G_A}} \det \overline{\Gm^A} \\
  &= \eps \xi(0)^n \det H \neq 0,
\end{align*}
and our last claim is proved. Using that the matrix
\eqref{E:matrixforsystem} is invertible to solve
\eqref{E:equationforlgamma} and \eqref{E:lalevitransform} for $
L_a \gm^A_b$, we obtain (by repeatedly differentiating the
resulting equations and using the identities
\eqref{E:barredderivativeofgamma},
\eqref{E:barredderivativeofeta}, and \eqref{E:Tderivativeofgamma})
the first part of Theorem \ref{T:reflectionlevinondegenerate}.


We also need reflection identities
\eqref{E:reflectionlevinondegenerateeta} for $L_a \eta^B$. We will
deduce those from the equation
\eqref{E:reflectionlevinondegenerategamma} we just proved. We
first compute
\[ L_{\bar c} L_b \gm_e^B = L_b L_{\bar c} \gm_e^B - h_{\bar c b}
T\gm_e^B,
\]
which follows from \eqref{E:habasLeviform} and the fact that
$R^b_{\bar a c}=0$ (cf. \cite{E5}, Lemma 1.33). By
\eqref{E:barredderivativeofgamma} and
\eqref{E:Tderivativeofgamma}, we conclude that
\begin{equation}\label{E:new} L_{\bar c} L_b \gm_e^B = - L_b
\left( \eta^B h_{\bar c e} \right) - h_{\bar c b}\left( L_e \eta^B
+ \eta^B h_e \right)= -( h_{\bar c e} L_b \eta^B+h_{\bar c b} L_e
\eta^B) + r(\eta^B) ,\end{equation} where $r(\eta^B)$ is a
(linear) polynomial in $\eta^B$ with coefficients which are smooth
on $M$ (and only depend on $M$). The identity
\eqref{E:reflectionlevinondegenerateeta} with $k=1$ now follows by
solving for $L_b\eta^B$ in \eqref{E:new} (e.g.\ set $b=c=e$ and
recall that $h_{\bar b b}\neq 0$) and then applying $L_{\bar c}$
to \eqref{E:reflectionlevinondegenerategamma} with $k=1$, and
finally using \eqref{E:barredderivativeofgamma} and
\eqref{E:barredderivativeofeta} to substitute for $L_{\bar
c}\gamma^B_e$ and $L_{\bar c}\eta^B$. The general case follows by
differentiating. The details are left to the reader.
\end{proof}


\section{Strictly pseudoconvex targets}
Our goal in this section is to derive the following reflection
identity; in section~\ref{S:completesystem} we will use it to
derive a complete system for the mapping $f$.


\begin{thm}\label{T:reflectionstrpscx} Let $M\subset\CN$, $\hat M \subset\CNh$
be smooth
hypersurfaces, $p_0 \in M$, $\hat p_0 \in \hat M$, with $\hat M$
strictly pseudoconvex at $\hat p_0$. Also assume that $f$ is a
smooth CR mapping from $M$ to $\hat M$ with $f(p_0) = \hat p_0$
which is transversal and constantly $(k_0,s)$-degenerate at $p_0$.
Then for any $I$ with $|I| = k$, $a\in\{1,\dots,n\}$,
$B\in\{1,\dots,\hat n\}$ there exist functions $r_a^{I,B}$ and
$s^{I,B}$ which are rational in their arguments such that
\begin{align}\label{E:reflectionpseudoconvexgamma}
 L^I \gm_a^B &= r_a^{I,B} \left( L^M T^l \gm_c^D,L^M T^l \eta^D,
 \overline{ L^J T^m \gm_c^D}, \overline{ L^J T^m \eta^D}
 \right),\\ \label{E:reflectionpseudoconvexeta}
 L^I \eta^B &= s^{I,B} \left( L^M T^l \gm_c^D, L^M T^l \eta^D,
 \overline{ L^J T^m \gm_c^D}, \overline{ L^J T^m \eta^D}
 \right).
\end{align}
Here $l \in \{ 0,1 \}$, $|M| + l \leq \max (k_0,k-1)$, $|J| + m
\leq k_0+1$, $m\leq \min(k,k_0)$. Furthermore, $r_a^{I,B}$ and
$s^{I,B}$ only depend on $M$, $\hat M$, and $j^{k_0}_{p_0} f$.
\end{thm}


\begin{proof} Assuming that $\hat M$ is strictly pseudoconvex, we choose a
local basis of CR vector fields tangent to $\hat M$ such that
$\hat h_{\bar A B} (0) = i \da_A^B$. Since $f$ is transversal, it
follows that $M$ is strictly pseudoconvex at $0$, and we likewise
choose a local basis of CR vector tangent to $M$ such that
$h_{\bar a b} (0) = i \da_a^b$.
 As in
Theorem~\ref{T:strpcx}, we assume that $f$ is constantly
$(k_0,s)$-degenerate at $p_0$. We let
\[ V_k = \spanc \{ g_{\stringA B} (0) \colon \ell (\stringA) \leq
k \}
\]
and claim that we can assume that $V_{k_0} \subset \Cnh$ is just
the subspace spanned by the first $s$ unit vectors. To see this,
recall that
\[g_{\bar a_1\dots \bar a_k B} = \br{D_{\bar a_k} D_{\bar a_{k-1}}
\dots D_{\bar a_1} \tau , \hat L_B}.
\]
Hence, under a change of basis $\hat L_B^\p = u_B^C \hat L_C$,
the $g$ transform as $g_{\stringA B}^\p = u_B^D g_{\stringA D}$.
Also, since
\[ \hat h_{\bar A B} = \br{ d \hat \gth, \hat L_{\bar A} \wedge \hat L_B},\]
the $\hat h_{\bar A B}$ transform as $\hat h_{\bar A B}^\p =
\overline{ u_A^C } u_B^D \hat h_{\bar C D}$, and any $u_A^B$
which is unitary at $0$ will respect the normalization condition
$\hat h_{\bar A B} (0) = i \da_A^B$. So, our claim is proved.


Since $f$ is constantly $(k_0,s)$-degenerate, we can choose
$\stringA_{1,a},\dots,\stringA_{s,a}$ for $a\in\{1,\dots,n\}$ and
$\stringA_1,\dots,\stringA_s$, whose length does not exceed $k_0$,
with such that for any $a\in \{1,\dots,n\}$ the vectors
\[v_{a,j} = \left(g_{\stringA_{j,a}
B} -
  \sum_{\ell(\stringA^\p) < \ell(\stringA_{j,a})}
  g_{\stringA^\p B} \, v_{ \stringA_j}^{\stringA^\p}
    \left( \overline{L^J \gm_c^D} \right)\right),
    \quad j=1,\dots,s,
    \]
as well as
\[v_{j} = \left(g_{\stringA_j
B} -
  \sum_{\ell(\stringA^\p) < \ell(\stringA_j)}
  g_{\stringA^\p B} \, p_{a \stringA_j}^{\stringA^\p}
    \left( \overline{L^J \gm_c^D} \right)\right),
    \quad j=1,\dots,s,
    \]
 are linearly independent at $0$, and hence each of these $n+1$
 families of vectors span
$V_{k_0}$. For any  $t\in\{1,\dots,n\}$, $C\in\{1,\dots,\hat n\}$,
and $I=(I_1,\dots,I_k)$, the determinants of
\begin{equation}\label{E:detvanishesstrpscx}
  \begin{pmatrix}
    v_{a,1}^1 & \dots & v_{a,1}^s & v_{a,1}^C\\
    \vdots & & \vdots &\vdots  \\
    v_{a,s}^1 & \dots & v_{a, s}^s & v_{a,s}^C \\
    L^{\bar I} g_{\bar a 1} & \dots & L^{\bar I} g_{\bar a s} &L^{\bar I}
g_{\bar a C}
  \end{pmatrix}, \quad a=1,\dots,n,
\end{equation}
and
\begin{equation}\label{E:det2vanishesstrpscx}
  \begin{pmatrix}
    v_{1}^1 & \dots & v_{1}^s & v_{1}^C\\
    \vdots & & \vdots &\vdots  \\
    v_{s}^1 & \dots & v_{ s}^s & v_{s}^C \\
    L^{\bar I} g_{\bar t 1} & \dots & L^{\bar I} g_{\bar t s} &L^{\bar I}
g_{\bar t C}
  \end{pmatrix}
\end{equation}
vanish on $M$. Since
\[ L^{\bar I} g_{\bar t B} = \overline{ L^I\left( \gm_t^A \hat h_{A \bar
B}\right)} = \overline{ L^I \gm_t^A } \hat h_{\bar A B} +
q^I_{t,B} \left( \overline{L^J \gm_c^D}\right),
\]
where $|J|<k$, expanding \eqref{E:detvanishesstrpscx} and
\eqref{E:det2vanishesstrpscx} along the last row we obtain
\begin{equation}\label{E:detstrpscxexpanded}
  \Da^{\bar b}_{\bar a, C_0} \left( L^I \gm_a^A \right) \hat h_{A\bar b}
  + \bar \Da_a \left(  L^I \gm_a^A \right) \hat h_{A \bar C_0}=
  r^I_{a,C_0} \left({L^J \gm_a^B}\right), \quad
  a=1,\dots,n,
\end{equation}
\begin{equation}\label{E:det2strpscxexpanded}
  \Da^{\bar b}_{\bar C_0} \left(L^I \gm_t^A\right) \hat h_{A\bar b}
  + \bar \Da  \left(L^I \gm_t^A\right) \hat h_{A \bar C_0}=
  r^I_{C_0} \left({L^J \gm_a^B}\right), \quad
\end{equation}
where again $|J|<k$, $r^I_{a,C_0}$ and $r^I_{C_0}$ are polynomial
in their arguments with smooth coefficients, and $\Da^{\bar
b}_{\bar a, C_0}$, $\bar \Da_a$, $\Da^{\bar b}_{\bar C_0}$, $\bar
\Da_a$ denotes the various cofactors produced by the row
expansion. These are rational in $L^J \gm_a^B$, where $|J|\leq
k_0-1$, and polynomial in $L^K \xi$, where $|K|\leq k_0$.


We need the following computational fact, which the reader can
verify easily.


\begin{lem}\label{L:interchanging1}
Given $P = p_1 \dots p_k$ with $1\leq p_j\leq n$, the following
holds for any $a$.
\begin{equation}\label{E:interchanging1}
  L_{\bar a} L^P = L^P L_{\bar a} - \sum_{j=1}^k h_{\bar a p_j}
  L^{P_j} T + \sum_{|K| \leq k-2} C_K L^K T,
\end{equation}
where $P_j$ is obtained from $P$ by removing the $j$th entry.
\end{lem}


Now given any $I = i_1 \dots i_k$ with $i_1\leq i_2 \leq \dots
\leq i_k$, we first apply the operator $L^I$ to the equations
\eqref{E:complemma1} and \eqref{E:complemma2} to obtain
\begin{multline} \label{E:firstforetas} g_{\stringA B}\left( L^I \eta^B
\right) -
  \sum_{\ell(\stringA^\p) < \ell(\stringA) } g_{\stringA^\p B} \left(L^I
\eta^B\right) \, v_{\stringA}^{\stringA^\p}
    \left( \overline{L^J \gm_c^D}\right)\\ = \td w_{\stringA}
    \left(L^M \gm_c^D, \overline{L^J T^m \gm_c^D},\overline{L^J T^m \eta^D}
\right)
\end{multline}
and
\begin{multline}\label{E:thenforgammas}
     g_{\stringA B} \left( L^I \gm_a^B \right) -
  \sum_{\ell(\stringA^\p) < \ell(\stringA)}
  g_{\stringA^\p B} \left( L^I \gm_a^B \right) \, p_{a \stringA}^{\stringA^\p}
    \left( \overline{L^J \gm_c^D}\right) - \\
  \sum_{\ell(\stringA^\p) < \ell(\stringA) } g_{\stringA^\p B} \left( L^I
\eta^B \right) \, q_{a \stringA}^{\stringA^\p}
    \left( \overline{L^J \gm_c^D}\right) = \td r_{a \stringA} \left(L^M
\gm_c^D, \overline{L^J T^m \gm_c^D},
    \overline{L^J T^m \eta^D}\right),
\end{multline}
where $|M|<k$,  $|K|\leq k_0 -1$, $|J|\leq k_0$, $m\leq k$, and
$|J| + m \leq k_0$. We have also simplified matters a bit by
applying \eqref{E:levitransform} in order to get rid of the terms
involving $\xi$.


Let us recall \eqref{E:det2strpscxexpanded}:
\[ \Da^{\bar b}_{\bar C_0} \left(L^{I} \gm_t^A\right) \hat h_{A\bar b}
  + \bar \Da  \left(L^{I} \gm_t^A\right) \hat h_{A \bar C_0}=
  r^I_{C_0} \left({L^M \gm_a^B}\right),\]
where  $|M| \leq k-1$. Applying $L_{\bar t}$ to this equation, and
applying Lemma \ref{L:interchanging1} and
\eqref{E:Tderivativeofgamma}, we obtain
\begin{multline}\label{E:missingequationforeta}
\left(L^{I} \gm_t^A\right) L_{\bar t} \left( \Da^{\bar b}_{\bar
C_0}  \hat h_{A\bar b}\right)
  + \left(L^{I} \gm_t^A\right) L_{\bar t} \left( \bar \Da   \hat h_{A \bar
C_0}
  \right)
  - \Da^{\bar b}_{\bar C_0} \left(L^{I} \eta^A\right) h_{\bar t t} \hat
h_{A\bar
  b}\\
  - \bar \Da  \left(L^{I} \eta^A\right) h_{\bar t t} \hat h_{A \bar
  C_0}
  - \sum_{j=1}^k h_{\bar t i_j} \left( L^{I_j} L_t \eta^A \right) \Da^{\bar
b}_{\bar C_0}\hat h_{A\bar b}
  - \sum_{j=1}^k h_{\bar t i_j} \left( L^{I_j} L_t \eta^A \right) \bar \Da
\hat h_{A\bar C_0}
  \\
  =
  \td r^I_{C_0} \left(L^M \eta^B, {L^M \gm_a^B}, L^J T \eta^B, {L^J T
\gm_a^B}\right).
\end{multline}
In this equation, $|M|<k$, and $|J|<k-1$.


We consider the system of linear equations in the $L^I \gm_a^B$
and $L^I \eta^B$ where $I$ ranges over all $I = i_1 \dots i_k$
with $i_1 \leq i_2 \leq \dots \leq i_k$, $a\in \{ 1,\dots , n\}$,
$B\in \{1\dots,\hat n \}$ which is obtained in the following way.
In \eqref{E:firstforetas} we set $\stringA = \stringA_1 ,\dots ,
\stringA_s$. For each $I$, this gives $s$ linear equations in $L^I
\eta^B$. We next consider the $\hat n - s$ equations obtained from
\eqref{E:missingequationforeta} by replacing $t$ by $i_k$ and
letting $C_0 = s+1,s+2,\dots,\hat n$. For the equations in $L^I
\gm_a^B$ ($a\in\{1,\dots ,n\}$  fixed) we replace $\stringA$ in
\eqref{E:thenforgammas} by $\stringA_{a,1}\dots \stringA_{a,s}$,
which gives us (for each $I$ and each $a$) $s$ linear equations in
$L^I \gm_a^B$. For each $I$ and each $a$ we also consider the
equations obtained from \eqref{E:det2strpscxexpanded} by replacing
$t$ by $a$, and letting $C_0 = s+1,\dots ,\hat n$.


With the simplifications above, it is easy to see that this linear
system can be solved for $L^I \gm_a^B$ and $L^I \eta^B$. We can
now lift the restriction that $I$ is increasing by noting that
rearranging the order of the $i_j$s only produces error terms of
order less than $k$. This finishes the proof.
\end{proof}


\section{Constructing the complete system; proof of Theorem
\ref{T:complsys}}\label{S:completesystem} We are now going to
construct the complete system for the jets of $f$. Since Theorems
\ref{T:reflectionlevinondegenerate} and \ref{T:reflectionstrpscx}
are a bit different than the reflection identities in \cite{E5},
we will go into some detail. For the rest of this section we
assume that we are either in the situation of Theorem
\ref{T:reflectionlevinondegenerate} or in the situation of Theorem
\ref{T:reflectionstrpscx}. First, let us recall the following fact
from \cite{E5} (the second part of Proposition 3.18):


For any multi-index $J$, integer $k\geq 1$, and index
$c\in\{1,\ldots, n\}$ there exist smooth functions  $b^{e_1\ldots
e_m}_s$  such that
\begin{equation}\label{E:maincommutatoridentity}
\sum_{m=1}^{|J|+k}\sum_{s=0}^{k} b_s^{e_1\ldots e_m}
\underbrace{[\ldots [L_{e_1}\ldots L_{e_m},L_{\bar c}],L_{\bar
c}]\ldots,L_{\bar c}]}_{\text{\rm length $s$ }}=(h_{\bar c
1})^{p}L^{J}T^k.
\end{equation}
Here, $p=k+|J|-|J|_1+1$, $|J|_1$ denotes the number of occurences
of the index $1$ in the multi-index $J$, and the length of the
commutator $[\ldots [X,Y_{1}],Y_{2}]\ldots,Y_{s}]$ is $s$. Since
in any case $M$ is Levi-nondegenerate at $0$, we can use this
identity in order to find a formula for $L^J T^k \gm_a^B$ and $L^J
T^k \eta^B$. We first apply $L_{\bar c}$ (at most $k$ times) to
$\gm_a^B$ and $\eta_a^B$, and using equations
\eqref{E:barredderivativeofgamma} and
\eqref{E:barredderivativeofeta} we see that the result is
polynomial in $\gm_c^D$, $\eta_c^D$, and $\overline{L_{\bar c}^j
\gm_c^D}$, where $j\leq k-1$. Next, we apply $L_{e_1}\dots
L_{e_m}$ to the result, and we apply our reflection identities
(either \eqref{E:reflectionlevinondegenerategamma} and
\eqref{E:reflectionlevinondegenerateeta} or
\eqref{E:reflectionpseudoconvexgamma} and
\eqref{E:reflectionpseudoconvexeta}) to see that the result is
rational in
\[L^M T^l \gm_c^D, L^M T^l \eta^D,
 \overline{ L^K T^m \gm_c^D}, \overline{ L^K T^m \eta^D},
\]
where $|M|+l \leq \max(k_0,|J| + k -1)$,  $|K| + m  \leq k_0 + 1$,
and $m\leq |J|$. Another application of $L_{\bar c}$ (at most $k$
times) then gives us that
\begin{align}\label{E:LTgammaisequal1}
L^J T^k \gm_a^B &= u(L^M T^l \gm_c^D, L^M T^l \eta^D,
 \overline{ L^K T^m \gm_c^D}, \overline{ L^K T^m \eta^D}), \\
 \label{E:LTetaisequal1}
 L^J T^k \eta^B &= v(L^M T^l \gm_c^D, L^M T^l \eta^D,
 \overline{ L^K T^m \gm_c^D}, \overline{ L^K T^m \eta^D}),
\end{align}
where $|M|+l \leq \max(k_0,|J| + k -1)$,  $|K| + m  \leq k + k_0 +
1$, and $m\leq \min(|J| + k, k_0)$. We are abusing notation by
denoting by $u$, $v$ and $w$ functions which are rational in their
arguments with coefficients smooth on $M\times\hat M$ and which
only depend on $M$ and $\hat M$ (and are allowed to change). Now
in these equations for $|J| + k = 2 k_0 + 2$ we can substitute for
the conjugated terms, yielding the same equations, but with the
bounds $|M|+l \leq 2k_0 + 1 $, $|K| + m \leq 2k_0 + 1$. Another
use of equations \eqref{E:barredderivativeofgamma} and
\eqref{E:barredderivativeofeta} lets us conclude that for all $J$,
$k$, $R$ with $|J| + k + |R| = 2k_0 + 2$,
\begin{align}\label{E:LTLbargammaisequal}
L^J T^k L^{\bar R}\gm_a^B &= u(L^M T^l \gm_c^D, L^M T^l \eta^D,
 \overline{ L^K T^m \gm_c^D}, \overline{ L^K T^m \eta^D}), \\
 \label{E:LTLbaretaisequal1}
 L^J T^k L^{\bar R}\eta^B &= v(L^M T^l \gm_c^D, L^M T^l \eta^D,
 \overline{ L^K T^m \gm_c^D}, \overline{ L^K T^m \eta^D}),
\end{align}
with $|M|+l \leq 2k_0 + 1 $, $|K| + m  \leq 2k_0 + 1$. These two
equations together with \eqref{E:levitransform} let us also see
that for all $J$, $k$, $R$ with $|J| + k + |R| = 2k_0 + 2$,
\begin{equation}\label{E:LTLxiisequal}
  L^J T^k L^{\bar R}\xi = w(L^M T^l \gm_c^D, L^M T^l \eta^D,
 \overline{ L^K T^m \gm_c^D}, \overline{ L^K T^m \eta^D}),
\end{equation}
again with $|M|+l \leq 2k_0 + 1 $, $|K| + m  \leq 2k_0 + 1$.
Equations \eqref{E:LTLbargammaisequal},
\eqref{E:LTLbaretaisequal1}, and \eqref{E:LTLxiisequal}, written
in some local coordinate systems, constitute the complete system
of differential equations described in Theorem \ref{T:complsys}.
Also note that the same complete system works for mappings whose
jets are close to the $2k_0 + 2$ jet of $f$. This finishes the
proof of Theorem \ref{T:complsys}.


\bibliographystyle{plain}
\bibliography{050102}
\end{document}


EL_finite_submission.bbl
\begin{thebibliography}{10}

\bibitem{BERbull}
M.~S. Baouendi, P.~Ebenfelt, and L. ~P.~Rothschild.
\newblock Local geometric properties of real submanifolds in complex space.
\newblock {\em Bull. Amer. Math. Soc. (N.S.)}, 37(3):309--336 (electronic),
  2000.

\bibitem{BERbook}
M.~S. Baouendi, P.~Ebenfelt, and L.~P. Rothschild.
\newblock {\em Real submanifolds in complex space and their mappings}.
\newblock Princeton University Press, Princeton, NJ, 1999.

\bibitem{BHR1}
M.~S. Baouendi, X.~Huang, and L.~P.~Rothschild.
\newblock Regularity of {C}{R} mappings between algebraic hypersurfaces.
\newblock {\em Invent. Math.}, 125(1):13--36, 1996.

\bibitem{BCG}
R.~L. Bryant, S.~S. Chern, R.~B. Gardner, H.~L. Goldschmidt, and P.~A.
  Griffiths.
\newblock {\em Exterior Differential Systems}, volume~18 of {\em Mathematical
  Sciences Research Institute Publications}.
\newblock Springer-Verlag, New York, 1991.

\bibitem{E5}
P.~Ebenfelt.
\newblock Finite jet determination of biholomorphic mappings at the boundary.
\newblock To appear, {\em Asian J. Math.}

\bibitem{FO1}
F.~Forstneri{\v{c}}.
\newblock Extending proper holomorphic mappings of positive codimension.
\newblock {\em Invent. Math.}, 95(1):31--61, 1989.

\bibitem{GH}
P.~Griffiths and J.~Harris.
\newblock {\em Principles of algebraic geometry}.
\newblock John Wiley \& Sons Inc., New York, 1994.
\newblock Reprint of the 1978 original.

\bibitem{L2}
B.~Lamel.
\newblock Holomorphic maps of real submanifolds in complex spaces of different
  dimensions.
\newblock {\em Pacific J. Math.}, 201(2):357--387, 2001.

\bibitem{L1}
B.~Lamel.
\newblock A reflection principle for real-analytic submanifolds of complex
  spaces.
\newblock {\em J. Geom. Anal.}, 11(4):625--631, 2001.

\bibitem{KZ1}
D.~Zaitsev and S.-Y. Kim.
\newblock The equivalence and the embedding problems for cr-structures of any
  codimension.
\newblock Preprint, math.CV/0108093, 2001.

\end{thebibliography}
