# Local orthogonal maps and rigidity of holomorphic mappings between real
hyperquadrics

Yun Gao111School of Mathematical Sciences, Shanghai Jiaotong University,
Shanghai, People’s Republic of China. Email<EMAIL_ADDRESS>Sui-Chung
Ng222School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East
China Normal University, Shanghai, People’s Republic of China. Email:
<EMAIL_ADDRESS>
###### Abstract

We introduced a new coordinate-free approach to study the Cauchy-Riemann (CR)
maps between the real hyperquadrics in the complex projective space. The
central theme is based on a notion of orthogonality on the projective space
induced by the Hermitian structure defining the hyperquadrics. There are
various kinds of special linear subspaces associated to this orthogonality
which are well respected by the relevant CR maps and this is where the
rigidities come from. Our method allows us to generalize a number of well-
known rigidity theorems for the CR mappings between real hyperquadrics with
much simpler arguments.

##  1 Introduction

The real hyperquadrics in the complex projective space plays a special role in
the field of CR (Cauchy-Riemann) Geometry. On the one hand, they are the among
simplest CR-manifolds (e.g. the boundaries of the unit balls) which can serve
as model spaces on which one can formulate or verify various statements or
theories. On the other hand, to CR geometry they play a role like what the
Euclidean spaces to Riemannian Geometry and what the projective spaces to
Algebraic Geometry. For instance, there is the well-known problem about which
CR manifolds can be embedded into these hyperquadrics. We refer the reader to
[BEH, Da, Fo, HJ, Za1] (and the references therein) for the related works in
this area.

The traditional approach to the study is based on Chern-Moser’s normal form
theory, in which the central theme is that one can choose good coordinates
such that the CR manifolds and the relevant holomorphic maps take certain
normal forms. This is a powerful method which has been used to solve many
problems but it usually requires formidable calculation. Nevertheless, we
observed that when the CR manifolds being concerned are real hyperquadrics,
there are a certain type of orthogonality and a number of related objects such
as null spaces and orthogonal complements which interact well with the CR
maps. Our approach to the study of real hyperquadrics is to work on these
geometric objects directly. While our method might not be able to produce as
much fine detail of the relevant maps as the traditional normal form theory,
it has the advantages of being coordinate free and geometrically more
transparent. It is especially suited for obtaining rigidities and studying
general behaviors. In addition, the arguments involved are much easier.

Let $r,s,t\in\mathbb{N}$ and denote by $\mathbb{C}^{r,s,t}$ be the Euclidean
space equipped with the standard (possibly degenerate) Hermitian bilinear form
whose eigenvalues are $+1$, $-1$ and $0$ with multiplicities $r$, $s$ and $t$
respectively. Consider its projectivization
$\mathbb{P}^{r,s,t}:=\mathbb{P}\mathbb{C}^{r,s,t}$. The notions of positive,
negative and null points are well defined on $\mathbb{P}^{r,s,t}$. Among
these, the set of positive points, denoted by
$\mathbb{B}^{r,s,t}\subset\mathbb{P}^{r,s,t}$, is called a generalized ball
and its boundary $\partial\mathbb{B}^{r,s,t}$, which consists of the set of
null points, is a CR hypersurface in $\mathbb{P}^{r,s,t}$ defined by a real
quadratic equation. The name “generalized balls” comes from the fact that
$\mathbb{B}^{1,s,0}$ is just the ordinary $s$-dimensional complex unit ball
embedded in $\mathbb{P}^{s}$.

Suppose $U\subset\mathbb{P}^{r,s,t}$ is a connected open set,
$U\cap\partial\mathbb{B}^{r,s,t}\neq\varnothing$ and
$f:U\rightarrow\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ is a holomorphic
map such that
$f(U\cap\partial\mathbb{B}^{r,s,t})\subset\partial\mathbb{B}^{r^{\prime},s^{\prime},t^{\prime}}$.
A classical problem in CR geometry is under what conditions $f$ is rigid in
the sense that it comes from some “standard” map between the projective
spaces, e.g. from a linear map. Our starting point is the observation that $f$
preserves the orthogonality induced by the Hermitian bilinear form on
$\mathbb{C}^{r,s,t}$ which remains defined on $\mathbb{P}^{r,s,t}$
(Proposition 2.5). This leads to the definition of local orthogonal maps (see
Definition 2.3 and the remark there regarding the relation to CR maps between
hyperquadrics). In what follows, we call $f$ null if
$f(U)\subset\partial\mathbb{B}^{r^{\prime},s^{\prime},t^{\prime}}$. This is a
kind of triviality in the current setting analogous to constant maps. We call
$f$ quasi-linear (resp. quasi-standard) if it is in some sense a “direct sum”
of two parts, of which one comes from a linear map (resp. linear isometry)
from $\mathbb{C}^{r,s,t}$ to $\mathbb{C}^{r^{\prime},s^{\prime},t^{\prime}}$
and the other one is null. The precise definitions are given in Definitions
3.1 and 3.2 and for the moment we use a simple example of quasi-standard maps
for illustration:

Example. We split the coordinates of $\mathbb{C}^{r,s}$ as $(z^{+},z^{-})$
corresponding to the positive and negative eigenvalues of the standard
Hermitian form. Let $\phi$, $\psi$ and $\chi$ be homogeneous polynomials
function on $\mathbb{C}^{r,s}$ such that $\deg(\psi)=\deg(\chi)=\deg(\phi)+1$.
Then a rational map from $\mathbb{P}^{r,s}$ to $\mathbb{P}^{r+1,s+1,1}$ of the
form $[z^{+},z^{-}]\mapsto[\phi z^{+},\psi,\phi z^{-},\psi,\chi]$ is a quasi-
standard orthogonal map, in which $[\phi z^{+},0,\phi
z^{-},0,0]\sim[z^{+},0,z^{-},0,0]$ is the standard (i.e. isometry) part and
$[0,\psi,0,\psi,\chi]$ is the null part.

We are now ready to summarize our main results for local orthogonal maps:

######  Theorem 1.1.

Let $U\subset\mathbb{P}^{r,s,t}$ be a connected open set such that
$U\cap\partial\mathbb{B}^{r,s,t}\neq\varnothing$ and
$f:U\rightarrow\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ be a local
orthogonal map. Then $f$ is either null or quasi-linear if one of the
conditions below is satisfied:

$\begin{matrix}[ccccl](i)&r,s\geq
2&\textrm{and}&\min\\{r^{\prime},s^{\prime}\\}\leq\min\\{r,s\\};&{\rm(Theorem~{}\ref{same})}\\\
(ii)&t=0&\textrm{and}&\min\\{r^{\prime},s^{\prime}\\}\leq
2\min\\{r,s\\}-2;&{\rm(Theorem~{}\ref{main})}\\\
(iii)&t=0&\textrm{and}&r^{\prime}+s^{\prime}\leq
2\dim(\mathbb{P}^{r,s})-1.&{\rm(Theorem~{}\ref{double dim thm})}\end{matrix}$

In addition, $f$ is quasi-standard if it maps a positive point to a positive
point under any of the conditions above, or

$\begin{matrix}[ccccl](iv)&r,s\geq
2,\,\,r=r^{\prime}&\textrm{and}&f(U\cap\mathbb{B}^{r,s,t})\subset\mathbb{B}^{r^{\prime},s^{\prime},t^{\prime}}.&{(\rm
Theorem~{}\ref{same2})}\end{matrix}$

From Theorem 1.1, we can deduce and generalize a number of well-known rigidity
theorems for the holomorphic maps between real hyperquadrics, including those
of Baouendi-Huang [BH] (from $(i)$ and $(iv)$); Baouendi-Ebenfelt-Huang [BEH]
(from $(ii)$); Faran [Fa1] (from $(iii)$); and Xiao-Yuan [XY] (from $(iii)$).
The detail of the relation of these results to ours is in Section 4.

Here we briefly explain the basic philosophy behind our method, which is
essentially the interplay between orthogonality and linear subspaces. First of
all, orthogonality can be used to obtain important information about the
images of the linear subspaces. For one thing, the maximal null spaces are
easily seen to be respected by an orthogonal map $f$. Furthermore, since
maximal null spaces constitute in some sense “a distinguished subset” in the
Grassmannian, we can then bound the dimension of the linear spans of the
images of all subspaces of the same dimension. For another thing, if we take a
pair of orthogonal subspaces, then the linear spans of their images under an
orthogonal map $f$ remain orthogonal. However, the sum of the dimensions of
two orthogonal subspaces is bounded for a given target space and so this will
again control the linear spans of the images of linear subspaces. Once such
dimension estimates are obtained, by making appropriate hypotheses on the
parameters defining the Hermitian forms and with the help of a technical
proposition relating the images of the subspaces in different dimensions
(Proposition 3.5), we can deduce the desired rigidity.

##  2 Definitions and basic properties

Let $r,s,t\in\mathbb{N}$ and $n:=r+s+t>0$. Define the (possibly degenerate)
indefinite inner product of signature $(r;s;t)$ on $\mathbb{C}^{n}$:

$\langle
z,w\rangle_{r,s,t}=z_{1}\bar{w}_{1}+\cdots+z_{r}\bar{w}_{r}-z_{r+1}\bar{w}_{r+1}-\cdots-
z_{r+s}\bar{w}_{r+s},$

where $z=(z_{1},\ldots,z_{n})$ and $w=(w_{1},\ldots,w_{n})$. We also define
the indefinite norm $\|z\|^{2}_{r,s,t}=\langle z,z\rangle_{r,s,t}$. Then, for
any $z\in\mathbb{C}^{r,s,t}$, we call it a positive point if
$\|z\|^{2}_{r,s,t}>0$; a negative point if $\|z\|^{2}_{r,s,t}<0$ and a null
point if $\|z\|^{2}_{r,s,t}=0$. If $\langle z,w\rangle_{r,s,t}=0$, we say that
$z$ is orthogonal to $w$ and write $z\perp w$. In addition, the orthogonal
complement of $z$ is defined as

$z^{\perp}=\\{w\in\mathbb{C}^{r,s,t}\mid\langle z,w\rangle_{r,s,t}=0\\}.$

We denote by $\mathbb{C}^{r,s,t}$ the $\mathbb{C}^{n}$ with the Hermitian
inner product defined above and by
$\mathbb{P}^{r,s,t}:=\mathbb{P}\mathbb{C}^{r,s,t}$ its projectivization. We
write $\mathbb{C}^{r,s}$ and $\mathbb{P}^{r,s}$ instead of
$\mathbb{C}^{r,s,0}$ and $\mathbb{P}^{r,s,0}$. A biholomorphism on
$\mathbb{P}^{r,s,t}$ induced by a linear isometry of $\mathbb{C}^{r,s,t}$ is
said to be an automorphism of $\mathbb{P}^{r,s,t}$.

Although the norm $\|\cdot\|^{2}_{r,s,t}$ of course does not descend to
$\mathbb{P}^{r,s,t}$, the positivity, negativity or nullity of a line
(1-dimensional subspace) remains well defined and thus we can talk about
positive points, negative points and null points on $\mathbb{P}^{r,s,t}$.
Furthermore, the orthogonality of two points on $\mathbb{P}^{r,s,t}$ and hence
the notion of orthogonal complement also make sense on $\mathbb{P}^{r,s,t}$.

More generally, let $V$ be a complex vector space equipped with a Hermitian
inner product (possibly degenerate or indefinite) $H_{V}$ of signature
$(r;s;t)$, where $\dim(V)=r+s+t$. Let $\mathbb{P}V$ be its projectivization.
The notion of positivity, negativity, nullity and orthogonality can be
similarly defined on $\mathbb{P}V$. In addition, any linear isometry
$F:\mathbb{C}^{r,s,t}\rightarrow V$ induces a biholomorphic map
$\tilde{F}:\mathbb{P}^{r,s,t}\rightarrow\mathbb{P}V$ preserving all these
notions. Sufficient for our purpose, we can simply identify any such
projective space $\mathbb{P}V$ with $\mathbb{P}^{r,s,t}$ through any such
biholomorphism and we write $\mathbb{P}V\cong\mathbb{P}^{r,s,t}$ for such
identification.

Now let $H$ be a complex linear subspace in $\mathbb{C}^{r,s,t}$ and the
restriction of $\langle\cdot,\cdot\rangle_{r,s,t}$ on $H$ has the signature
$(a;b;c)$. Obviously, we have $0\leq a\leq r$, $0\leq b\leq s$, $0\leq
c\leq\min\\{r-a,s-b\\}+t$ and $a+b+c=\dim(H)$. Then
$\mathbb{P}H\cong\mathbb{P}^{a,b,c}$. We call $\mathbb{P}H$ an
$(a,b,c)$-subspace of $\mathbb{P}^{r,s,t}$. Usually, we denote an
$(a,b,c)$-subspace by $H^{a,b,c}$.

If $a=b=0$, $\mathbb{P}H$ is called a null space. Similarly, it is called a
positive space (resp. negative space) if $b=c=0$ (resp. $a=c=0$). We will also
use the terms null $k$-plane, positive $k$-plane and negative $k$-plane when
the $\dim(\mathbb{P}H)=k$. Obviously, the maximum dimension of the null spaces
in $\mathbb{P}^{r,s,t}$ is $\min\\{r,s\\}+t-1$. The null spaces with the
maximal dimension are called the maximal null spaces.

We now recall the definition of type-I irreducible bounded symmetric domain
and with that we can give some useful parametrizations for the positive,
negative and null spaces in $\mathbb{P}^{r,s,t}$.

######  Definition 2.1.

Let $M_{r,s}$ be the set of $r\times s$ complex matrices. The type-I
irreducible bounded symmetric domain $\Omega_{r,s}$ is the domain in
$M_{r,s}\cong\mathbb{C}^{rs}$ defined by $\Omega_{r,s}=\\{A\in
M_{r,s}:I-AA^{H}>0\\}$, where $A^{H}$ denotes the Hermitian transpose of $A$.

In what follows, for a point $[z]=[z_{1},\ldots,z_{n}]\in\mathbb{P}^{r,s,t}$,
we split the homogeneous coordinates as $[z]=[z^{+},z^{-},z^{0}]$, where
$z^{+}=[z_{1},\cdots,z_{r}]$, $z^{-}=[z_{r+1},\cdots,z_{r+s}]$,
$z^{0}=[z_{r+s+1},\cdots,z_{n}]$. If $t=0$, $[z]$ is split as
$[z]=[z^{+},z^{-}]$.

Let $A\in M_{r,s}$ and $B\in M_{r,t}$. Consider the $(r-1)$-plane defined by

$H_{A,B}=\\{[z^{+},z^{-},z^{0}]\in\mathbb{P}^{r,s,t}\mid
z^{-}=z^{+}A\,\,\textrm{and}\,\,z^{0}=z^{+}B\\}\cong\mathbb{P}^{r-1}\subset\mathbb{P}^{r,s,t}.$

Using the definition of $H_{A,B}$, we see that $M_{r,s}\times M_{r,t}$ can be
identified naturally as an open subset of the Grassmannian
$\mathbb{G}(r-1,\mathbb{P}^{r,s,t})$ which is the set of $(r-1)$-planes in
$\mathbb{P}^{r,s,t}$. (Note that the dimensions of $M_{r,s}\times M_{r,t}$ and
$\mathbb{G}(r-1,\mathbb{P}^{r,s,t})$ are the same.)

The $(r-1)$-plane $H_{A,B}$ is a positive subspace if and only if
$\|z^{+}\|^{2}>\|z^{-}\|^{2}=\|z^{+}A\|^{2}$ for all
$z^{+}\in\mathbb{P}^{r-1}$, which is in turn equivalent to $A\in\Omega_{r,s}$.
Similarly, when $r\leq s$, we see that $H_{A,B}$ is a null $(r-1)$-plane if
and only if $AA^{H}=I$. We know that latter equation defines precisely the
Shilov boundary of $\Omega_{r,s}$ in $M_{r,s}\cong\mathbb{C}^{rs}$, denoted by
$S(\Omega_{r,s})$. To summarize, we have the following (cf. [NZ]):

######  Proposition 2.2.

There is an open embedding of $M_{r,s}\times M_{r,t}$ into
$\mathbb{G}(r-1,\mathbb{P}^{r,s,t})$ such that its restriction to
$\Omega_{r,s}\times M_{r,t}$ gives a parametrization of all positive
$(r-1)$-planes in $\mathbb{P}^{r,s,t}$. In addition, when $r\leq s$, the
restriction to $S(\Omega_{r,s})\times M_{r,t}$ also gives a parametrization of
the null $(r-1)$-planes in $\mathbb{P}^{r,s,t}$.

Many analyses on the mapping problems between CR manifolds begin with the fact
that the associated Segre varieties are well respected by CR maps. The study
of Segre varieties has a very important role in many problems like reflection
principle and algebracity [Za2]. For real hyperquadrics on complex projective
space, we note that their Segre varieties are just the orthogonal complements
(to points) with respect to the orthogonality described previously. Motivated
from this, we thus give the following definition.

######  Definition 2.3.

Let $U\subset\mathbb{P}^{r,s,t}$ be a connected open set containing a null
point. We call a holomorphic map
$F:U\rightarrow\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ orthogonal if
$F(p)\perp F(q)$ for any $p,q\in U$ such that $p\perp q$; sign-preserving if
$F$ maps positive points to positive points and negative points to negative
points. We also simply call $F$ a local orthogonal map or local sign-
preserving map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ in such cases.

Remark. For the sake of convenience, in our definition we require the domain
of definition $U$ of a local orthogonal map $f$ to intersect
$\partial\mathbb{B}^{r,s,t}$ so that the orthogonality or sign-preserving
condition will not become vacuous. (Otherwise, it could happen that
$p^{\perp}\cap U=\varnothing$ for any point $p\in U$.) With this requirement,
Proposition 2.4 and Proposition 2.5 below will imply that orthogonal maps and
holomorphic maps between real hyperquadrics are the same set of mappings.
However, we remark that in principle one can define orthogonal maps more
generally without requiring $U\cap\partial\mathbb{B}^{r,s,t}\neq\varnothing$
and some of our results (e.g. Theorem 3.11) will still be valid. In this
sense, a local orthogonal map is a slightly more general object.

Convention. In what follows, if we say that $F$ maps lines to lines (resp.
$k$-planes to $k^{\prime}$-planes), we mean $F$ maps the intersection of any
line (resp. $k$-plane) and $U$ into a line (resp. $k^{\prime}$-plane).

Local orthogonal map preserves null spaces, as demonstrated below.

######  Proposition 2.4.

Let $F$ be a local orthogonal map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$. Then $F$ maps null spaces to
null spaces (in particular, null points to null points).

###### Proof.

For any two points $\alpha,\,\beta$ in a null space in $\mathbb{P}^{r,s,t}$,
we have
$\langle\alpha,\,\alpha\rangle_{r,s,t}=\langle\beta,\,\beta\rangle_{r,s,t}=\langle\alpha,\,\beta\rangle_{r,s,t}=0$.
Since $F$ is a local orthogonal map, we get

$\langle
F(\alpha),\,F(\alpha)\rangle_{r^{\prime},s^{\prime},t^{\prime}}=\langle
F(\beta),\,F(\beta)\rangle_{r^{\prime},s^{\prime},t^{\prime}}=\langle
F(\alpha),\,F(\beta)\rangle_{r^{\prime},s^{\prime},t^{\prime}}=0.$

So the linear span of the image of a null space is a null space. ∎

There are two kinds of null points in $\mathbb{P}^{r,s,t}$ when $r,s,t>0$. The
first kind of null points is the null points $\alpha$ satisfying
$\langle\alpha,\beta\rangle_{r,s,t}=0$, for any $\beta\in\mathbb{P}^{r,s,t}$.
We call these null points special null points and call other null points
ordinary null points. It is easy to see from the defining equations for
special and ordinary null points that a general null point is ordinary and for
$r,s>0$, whenever an open set contains a null point, it must contain an
ordinary null point.

Both orthogonal maps and sign-preserving maps map null points to null points.
Conversely, the following proposition in particular implies that a sign-
preserving map is locally an orthogonal map.

######  Proposition 2.5.

Let $r,s>0$ and
$F:U\subset\mathbb{P}^{r,s,t}\rightarrow\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$
be a holomorphic map, where $U$ is an open set containing a null point. If $F$
maps null points to null points, then there exists an open set $V\subset U$,
such that $F:V\rightarrow\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ is an
orthogonal map.

###### Proof.

Let $n=r+s+t$, $n^{\prime}=r^{\prime}+s^{\prime}+t^{\prime}$ and write the
homogeneous coordinates of $\mathbb{P}^{r,s,t}$ and
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ as $[z_{1},\ldots,z_{n}]$ and
$[w_{1},\ldots,w_{n^{\prime}}]$ respectively. Since $r,s>0$, we know that $U$
must contain an ordinary null point and hence by shrinking $U$ if necessary,
we may assume without loss of generality that $U$ is contained in the open set
$U_{1}\cong\mathbb{C}^{n-1}\subset\mathbb{P}^{r,s,t}$ defined by $z_{1}\neq 0$
and $F(U)$ is contained in the open set of
$U^{\prime}_{1}\cong\mathbb{C}^{n^{\prime}-1}\subset\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$
defined by $w_{1}\neq 0$. We write the standard inhomogeneous coordinates in
$U_{1}$ as $(\zeta_{2},\ldots,\zeta_{n})$, where $\zeta_{j}=z_{j}/z_{1}$ and
similarly write $(\eta_{2},\ldots,\eta_{n^{\prime}})$ for $U^{\prime}_{1}$,
where $\eta_{\ell}=w_{\ell}/w_{1}$. In terms of these coordinates, the null
points in $U_{1}$ and $U^{\prime}_{1}$ are respectively given by the equations

$1+\sum^{r}_{j=2}|\zeta_{j}|^{2}-\sum^{r+s}_{j=r+1}|\zeta_{j}|^{2}=0\,\,\,\,\textrm{and}\,\,\,\,1+\sum^{r^{\prime}}_{\ell=2}|\eta_{\ell}|^{2}-\sum^{r^{\prime}+s^{\prime}}_{\ell=r^{\prime}+1}|\eta_{\ell}|^{2}=0.$

(Strictly speaking, these equations are for the cases where $r,r^{\prime}\geq
2$ but other cases can be handled in a similar fashion.)

Using the inhomogeneous coordinates above, write
$F=(F_{2},\cdots,F_{n^{\prime}})$. If $F$ maps null points to null points,
then there exist a connected open set $V\subset U$ containing a null point,
$k\in\mathbb{N}^{+}$ and a real analytic function $\rho$ on $V$ such that

$1+\sum^{r^{\prime}}_{\ell=2}|F_{\ell}|^{2}-\sum^{r^{\prime}+s^{\prime}}_{\ell=r^{\prime}+1}|F_{\ell}|^{2}=\left(1+\sum^{r}_{j=2}|\zeta_{j}|^{2}-\sum^{r+s}_{j=r+1}|\zeta_{j}|^{2}\right)^{k}\rho$
(1)

holds on $V$. Hence, by shrinking $V$ if necessary, we can polarize the
equation, i.e. for any $\zeta,\xi\in V$, we have

$1+\sum^{r^{\prime}}_{\ell=2}F_{\ell}(\zeta)\overline{F_{\ell}(\xi)}-\sum^{r^{\prime}+s^{\prime}}_{\ell=r^{\prime}+1}F_{\ell}(\zeta)\overline{F_{\ell}(\xi)}=\left(1+\sum^{r}_{j=2}\zeta_{j}\overline{\xi_{j}}-\sum^{r+s}_{j=r+1}\zeta_{j}\overline{\xi_{j}}\right)^{k}\rho(\zeta,\bar{\xi}),$

where $\zeta=(\zeta_{2},\ldots,\zeta_{n})$, $\xi=(\xi_{2},\ldots,\xi_{n})$.

Now, $\zeta$ as a point in $\mathbb{P}^{r,s,t}$ has homogeneous coordinates
$[1,\zeta_{2},\ldots,\zeta_{n}]$ and thus $\zeta^{\perp}\cap V$ consists
precisely the points $\xi=[1,\xi_{2},\ldots,\xi_{n}]\in V$ satisfying
$\displaystyle
1+\sum^{r}_{j=2}\zeta_{j}\overline{\xi_{j}}-\sum^{r+s}_{j=r+1}\zeta_{j}\overline{\xi_{j}}=0$.
With a similar consideration for $F(\zeta)^{\perp}$, we conclude from the
above equation that $F(\zeta^{\perp})\subset(F(\zeta))^{\perp}$ for $\zeta\in
V$. ∎

It has been known that for any local proper holomorphic map $f$ between two
generalized balls (excluding the unit balls), there exist
$k,k^{\prime}\in\mathbb{N}^{+}$ such that $f$ maps $k$-planes to
$k^{\prime}$-planes ([Ng1] Proposition 4.1 therein). The following proposition
is a generalization of this result to local orthogonal maps, which is very
crucial in our study.

######  Proposition 2.6.

If $F$ is a local orthogonal map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$, then $F$ maps
$(\min\\{r,s\\}-1)$-planes to
$(\min\\{r^{\prime},s^{\prime}\\}+t^{\prime}-1)$-planes.

###### Proof.

By symmetry, it suffices to prove the case for $r\leq s$ and $r^{\prime}\leq
s^{\prime}$. The case $r=1$ is trivial and so we let $r\geq 2$. Recall from
Proposition 2.2 that the null $(r-1)$-planes in $\mathbb{P}^{r,s,t}$ can be
parametrized by $S(\Omega_{r,s})\times
M_{r,t}\subset\mathbb{G}(r-1,\mathbb{P}^{r,s,t})$, where $S(\Omega_{r,s})$ is
the Shilov boundary of the type-I bounded symmetric domain $\Omega_{r,s}$. Now
from Proposition 2.4, the map $F$ maps null spaces to null spaces and thus the
image of every null $(r-1)$-plane in $\mathbb{P}^{r,s,t}$ is contained in a
maximal null space, which is an $(r^{\prime}+t^{\prime}-1)$-plane in
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$.

Note that for any point $p\in\mathbb{G}(r-1,\mathbb{P}^{r,s,t})$, there exists
a set of local holomorphic functions in a neighborhood $\mathcal{U}\ni p$ such
that they vanish at a point $q\in\mathcal{U}$ precisely when the image under
$F$ of the $(r-1)$-plane given by $q$ is contained in an
$(r^{\prime}+t^{\prime}-1)$-plane. (For instance, one can consider a set of
determinants given by the Taylor coefficients of $F$. For more detail, see
[NZ], Proof of Theorem 1.1 therein.) From the previous paragraph, we know that
when choose $p\in S(\Omega_{r,s})\times M_{r,t}$, these functions vanish at
$\mathcal{U}\cap(S(\Omega_{r,s})\times M_{r,t})$. The Shilov boundary is the
distinguished boundary of $\Omega_{r,s}$ and from this we deduce that these
functions actually vanish identically on the open set $\mathcal{U}$ (see [Ng2]
or [NZ]). Thus, $F$ maps $(r-1)$-planes to $(r^{\prime}+t^{\prime}-1)$-planes.
∎

##  3 Rigidity of local orthogonal maps

In order to simplify the presentation, we will give a couple of definitions.
In what follows, when we say that a local holomorphic map $F$ between
projective spaces is linear, we mean $F$ extends to a linear rational map.

######  Definition 3.1.

Let $F$ be a local holomorphic map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$. We call $F$ standard if it is
linear and comes from a linear isometry from $\mathbb{C}^{r,s,t}$ into
$\mathbb{C}^{r^{\prime},s^{\prime},t^{\prime}}$. We call $F$ null if its image
is contained in a null space in
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$.

For any non-trivial orthogonal direct sum decomposition
$\mathbb{C}^{r,s,t}=A\oplus B$, there are two canonical projections
$\pi_{A}:\mathbb{P}^{r,s,t}\dasharrow\mathbb{P}A$ and
$\pi_{B}:\mathbb{P}^{r,s,t}\dasharrow\mathbb{P}B$ (as rational maps). Using
these, we make the following definition.

######  Definition 3.2.

Let $F$ be a local holomorphic map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$. We call $F$ quasi-standard
(resp. quasi-linear) if either $F$ is standard (resp. linear) or there exists
a non-trivial orthogonal decomposition
$\mathbb{C}^{r^{\prime},s^{\prime},t^{\prime}}=A\oplus B$ for some subspaces
$A,B$ such that $\pi_{A}\circ F$ is standard (resp. linear) and $\pi_{B}\circ
F$ is null.

Since the projection from $\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ to
$\mathbb{P}^{r^{\prime},s^{\prime}}$ is also an orthogonal map (see the lemma
below), therefore a local orthogonal map $F$ from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ naturally gives rise to a
local orthogonal map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime}}$ (unless the image lies entirely in the
space of special null vectors, which is the set of indeterminacy for the
projection). The same thing happens for local sign-preserving maps. This
allows us to reduce the problem to the case for $t^{\prime}=0$ in many
situations. Moreover, the orthogonality of mappings also interacts nicely with
orthogonal decompositions. The following two propositions should be evident to
the reader and we omit their proofs.

######  Proposition 3.3.

The projection $\pi:\mathbb{P}^{r,s,t}\dashrightarrow\mathbb{P}^{r,s}$ is
standard. In particular, it is sign-preserving map and orthogonal.

######  Proposition 3.4.

Let $F$ be a local holomorphic map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ and
$\mathbb{C}^{r^{\prime},s^{\prime},t^{\prime}}=A\oplus B$ be an orthogonal
decomposition such that the image of $F$ does not lie entirely in
$\mathbb{P}A$ or $\mathbb{P}B$. Let
$\pi_{A}:\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}\dasharrow\mathbb{P}A$
and
$\pi_{B}:\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}\dasharrow\mathbb{P}B$
be the canonical projections. If $F$ and $\pi_{A}\circ F$ are orthogonal, then
so is $\pi_{B}\circ F$.

We have already seen that how orthogonal maps respect various linear subspaces
associated to the Hermitian structure defining the real hyperquadrics. The
following proposition serves as a bridge connecting that property of
orthogonal maps to their rigidity.

######  Proposition 3.5.

Let $g:U\subset\mathbb{P}^{m}\rightarrow\mathbb{P}^{m^{\prime}}$ be a local
holomorphic map which maps $\ell$-planes to $\ell^{\prime}$-planes.

$(i)$ If $\ell^{\prime}\leq\ell$, then either the image of $g$ is contained in
an $\ell^{\prime}$-plane or $g$ extends to a linear rational map. In addition,
the image of $g$ must be contained in an $\ell^{\prime}$-plane if
$\ell^{\prime}\leq\ell-1$;

$(ii)$ If $\ell\leq\ell^{\prime}\leq 2\ell-1$, then $g$ maps $(\ell+k)$-planes
to $(\ell^{\prime}+k)$-planes for $k\geq 0$.

###### Proof.

For $(i)$, if $\ell^{\prime}\leq\ell$ and the image of $g$ is not contained in
an $\ell^{\prime}$-plane, by considering a general pair of $\ell$-planes such
that their intersection is an $(\ell-1)$-plane, we see $g$ maps
$(\ell-1)$-planes to $(\ell^{\prime}-1)$-planes. Inductively, we deduce that
$g$ maps lines to lines and hence $g$ is linear. For a proof of the last fact,
see [Ng3] (Lemma 4.1 therein). If in addition, $\ell^{\prime}\leq\ell-1$ and
the image of $g$ is not contained in an $\ell^{\prime}$-plane, we conclude
again that $g$ is linear. We can then find an $(\ell^{\prime}+1)$-plane $E$
with $E\cap U\neq\varnothing$ such that $g(E\cap U)$ is an open piece of an
$(\ell^{\prime}+1)$ plane, but $\dim(E)=\ell^{\prime}+1\leq\ell$, this
contradicts the hypothesis that $g$ maps $\ell$-planes to
$\ell^{\prime}$-planes.

For $(ii)$, it follows from a hyperplane restriction theorem of holomorphic
mappings (see [GN], Theorem 1.2). For an independent proof, see also [Ga],
Lemma 5.1 therein. ∎

We are now ready to prove our rigidity theorems for local orthogonal maps.

######  Theorem 3.6.

Let $F$ be a local orthogonal map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$, where $r,s\geq 2$. If

$\min\\{r^{\prime},s^{\prime}\\}\leq\min\\{r,s\\},$

then $F$ is either null or quasi-linear (linear for $t^{\prime}=0$). If in
addition $F$ preserves the sign of any single positive or negative point, then
$F$ is quasi-standard (standard for $t^{\prime}=0$).

###### Proof.

We first consider the case $t^{\prime}=0$. From Proposition 2.6, we know that
$F$ maps $(\min\\{r,s\\}-1)$-planes into
$(\min\\{r^{\prime},s^{\prime}\\}-1)$-planes. Suppose $F$ is not linear. Since
$r,s\geq 2$ and $\min\\{r^{\prime},s^{\prime}\\}\leq\min\\{r,s\\}$, by
Proposition 3.5$(i)$, the linear span the image of $F$, denoted by $S$, is of
dimension at most $(\min\\{r^{\prime},s^{\prime}\\}-1)$.

We claim that $S$ is a null space. If on the contrary $S$ is not null, then
$S\cong\mathbb{P}^{a,b,c}$ for some $a,b,c$ and the dimension of the maximal
null space of $\mathbb{P}^{a,b,c}$ is $\min\\{a,b\\}+c-1$, which is strictly
less than $\dim(S)$. Now $F$ can be regarded as a local orthogonal map from
$\mathbb{P}^{r,s,t}$ into $\mathbb{P}^{a,b,c}$ which is not linear, and thus
by Propositions 2.6 and 3.5$(i)$ again, the image of $F$ is contained in a
linear subspace of dimension $\min\\{a,b\\}+c-1<\dim(S)$, contradicting to the
fact that $S$ is the linear span of the image of $F$. So $S$ is a null space
and hence $F$ is null. We have thus shown that $F$ is either linear or null.
If in addition $F$ preserves the sign of a positive or negative point, then
$F$ cannot be null and from Lemma 3.7 below, we see that $F$ must be standard.

Since
$\mathbb{C}^{r^{\prime},s^{\prime},t^{\prime}}=\mathbb{C}^{r^{\prime},s^{\prime}}\oplus\mathbb{C}^{0,0,t^{\prime}}$
is an orthogonal direct sum, the desired result for the general case now
follows directly from the case $t^{\prime}=0$ and Definition 3.2. ∎

######  Lemma 3.7.

Let
$G:\mathbb{C}^{r,s,t}\rightarrow\mathbb{C}^{r^{\prime},s^{\prime},t^{\prime}}$
be linear and it maps null vectors to null vectors. Then, there exists
$\lambda\in\mathbb{R}$ such that $\langle
G(u),G(v)\rangle_{r^{\prime},s^{\prime},t^{\prime}}=\lambda\langle
u,v\rangle_{r,s,t}$ for any $u,v\in\mathbb{C}^{r,s,t}$. Moreover, if $G$
preserves the sign of any single positive or negative vector, then $r\leq
r^{\prime}$, $s\leq s^{\prime}$ and $\lambda>0$.

###### Proof.

The proof is just standard linear algebra. For the detail, we refer the reader
to [NZ], Lemma 4.2 therein. ∎

We can say more when $\min\\{r^{\prime},s^{\prime}\\}<\min\\{r,s\\}$:

######  Theorem 3.8.

If $\min\\{r^{\prime},s^{\prime}\\}<\min\\{r,s\\}$, then any local orthogonal
map $F$ from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ is null.

###### Proof.

When $\min\\{r,s\\}=1$, the result is obvious. Now, suppose $\min\\{r,s\\}\geq
2$. We start from the case $t^{\prime}=0$. By Proposition 2.6, $F$ maps
$(\min\\{r,s\\}-1)$-planes to $(\min\\{r^{\prime},s^{\prime}\\}-1)$-planes.
Then Proposition 3.5$(i)$ says that the linear span of the image of $F$,
denoted by $S$ is of dimension at most $(\min\\{r^{\prime},s^{\prime}\\}-1)$.
Exactly the same argument as in Theorem 3.6 shows that $S$ is a null space and
hence $F$ is null.

Now, when $t^{\prime}>0$, we know that $\pi\circ F$ is null, where $\pi$ is
the projection from $\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ to
$\mathbb{P}^{r^{\prime},s^{\prime}}$. Hence, $F$ is also null. ∎

######  Proposition 3.9.

If $r^{\prime}<r$ or $s^{\prime}<s$, then there is no local sign-preserving
map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$.

###### Proof.

By symmetry, it suffices to prove the statement for $r^{\prime}<r$. The
statement is trivial if $r^{\prime}=0$. Suppose $1\leq r^{\prime}<r$ and
$F:U\subset\mathbb{P}^{r,s,t}\rightarrow\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$
is a local sign-preserving map.

For a null point $x\in U$, we have $x\in x^{\perp}\cap U$ and so if we take a
positive point $p_{1}\in U$ close enough to $x$, then $p_{1}^{\perp}\cap U$ is
also non-empty. Note that $p_{1}^{\perp}$ is an $(r-1,s,t)$-subspace
$H^{r-1,s,t}$ and by the sign-preserving property $(F(p_{1}))^{\perp}$ is an
$(r^{\prime}-1,s^{\prime},t^{\prime})$-subspace
$H^{r^{\prime}-1,s^{\prime},t^{\prime}}$. So the restriction of $F$ on
$U_{1}:=H^{r-1,s,t}\cap U$ is a local sign-preserving map from $H^{r-1,s,t}$
to $H^{r^{\prime}-1,s^{\prime},t^{\prime}}$. By choosing a convex $U$ with
respect to the standard coordinates, we may assume that $U_{1}$ remains convex
and connected. Now, as $r-1\geq 1$ and $s\geq 1$, we can always choose $p_{1}$
such that $U_{1}$ contains both positive and negative points and hence also
null points.

We repeat the same procedures on $U_{1}$. Inductively, after taking
$r^{\prime}$ positive points $p_{1},\ldots,p_{r^{\prime}}$, we get from the
restriction of $F$ a local sign-preserving map from $H^{r-r^{\prime},s,t}$ to
$H^{0,s^{\prime},t^{\prime}}$. This is a contradiction since
$H^{r-r^{\prime},s,t}$ contains positive points but
$H^{0,s^{\prime},t^{\prime}}$ does not. ∎

######  Theorem 3.10.

Let $F$ be a local sign-preserving map from $\mathbb{P}^{r,s,t}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$, where $r,s\geq 2$. If
$r=r^{\prime}$ or $s=s^{\prime}$ then $F$ is quasi-standard (standard for
$t^{\prime}=0$).

###### Proof.

By symmetry, it suffices to prove the theorem for $r=r^{\prime}$. We begin
with the case $t^{\prime}=0$. If $r\leq s$, then everything follows from
Theorem 3.6. Suppose now $r>s$.

By following the same procedures of taking orthogonal complements as in the
proof of Proposition 3.9, we can pick $(r-s)$ orthogonal positive points in
$U$ such that the restriction of $F$ on their orthogonal complement is a local
sign-preserving map from some $(s,s,t)$-subspace
$H^{s,s,t}\subset\mathbb{P}^{r,s,t}$ to an $(s,s^{\prime})$-subspace
$H^{s,s^{\prime}}\subset\mathbb{P}^{r^{\prime},s^{\prime}}$.

As $s\geq 2$, by Theorem 3.6, $F|_{H^{s,s,t}}$ is standard. In particular,
$F|_{H^{s,s,t}}$ maps the positive $(s-1)$-planes in $H^{s,s,t}$ to positive
$(s-1)$-planes in $\mathbb{P}^{r^{\prime},s^{\prime}}$. Note that our argument
would give the same conclusion for any other $(s,s,t)$-subspace
$\tilde{H}^{s,s,t}$ close enough to $H^{s,s,t}$ (as points in the
Grassmannian). Since the set of positive $(s-1)$-planes is open in the
Grassmannian $\mathbb{G}(s-1,\mathbb{P}^{r,s,t})$, it follows that $F$ maps
$(s-1)$-planes to $(s-1)$-planes. Thus, by Proposition 3.5$(i)$, $F$ is either
linear or the image of $F$ is contained in an $(s-1)$-plane. The latter is
impossible since we already know that $F|_{H^{s,s,t}}$ is standard. So $F$ is
linear and together with the sign-preserving hypothesis, we conclude that $F$
is standard by Lemma 3.7.

Finally, if $t^{\prime}\geq 1$, by considering again the projection from
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ to
$\mathbb{P}^{r^{\prime},s^{\prime}}$ as in Theorem 3.6, we see that $F$ is
quasi-standard. ∎

######  Theorem 3.11.

Let $F$ be a local orthogonal map from $\mathbb{P}^{r,s}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$. If

$r^{\prime}+s^{\prime}\leq 2\dim(\mathbb{P}^{r,s})-1,$

then $F$ is either null or quasi-linear. If in addition $F$ preserves the sign
of any single positive or negative point, then $F$ is quasi-standard.

###### Proof.

The statement is trivial if $\dim(\mathbb{P}^{r,s})=1$. We assume
$\dim(\mathbb{P}^{r,s})\geq 3$ for the moment and the two dimensional case
will be included later in the argument.

Suppose $F$ is not null and $r^{\prime}+s^{\prime}\leq
2\dim(\mathbb{P}^{r,s})-1$. Let $F_{1}:=\pi_{1}\circ F$, where
$\pi_{1}:\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}\dasharrow\mathbb{P}^{r^{\prime},s^{\prime}}$
is the standard projection. If $F_{1}$ is not linear, then by Proposition
3.5$(i)$, either image of $F_{1}$ is contained in a line or the linear span
$S$ of the image of a general line $L$ under $F_{1}$ is of dimension $d\geq
2$. In the latter situation, since $\dim(L^{\perp})=\dim(\mathbb{P}^{r,s})-2$
and $\dim(S^{\perp})=\dim(\mathbb{P}^{r^{\prime},s^{\prime}})-d-1$ (here
$S^{\perp}$ is the orthogonal complement of $S$ in
$\mathbb{P}^{r^{\prime},s^{\prime}}$) and any
$(\dim(\mathbb{P}^{r,s})-2)$-plane in $\mathbb{P}^{r,s}$ is the orthogonal
complement of some line, we see by orthogonality that $F_{1}$ maps
$(\dim(\mathbb{P}^{r,s})-2)$-planes to
$(\dim(\mathbb{P}^{r^{\prime},s^{\prime}})-d-1)$-planes. (This in particular
also implies that we must have $\dim(\mathbb{P}^{r^{\prime},s^{\prime}})\geq
3$.)

As $\dim(\mathbb{P}^{r,s})-2\geq 1$ and

$\dim(\mathbb{P}^{r^{\prime},s^{\prime}})-d-1\leq r^{\prime}+s^{\prime}-4\leq
2\dim(\mathbb{P}^{r,s})-5=2(\dim(\mathbb{P}^{r,s})-2)-1,$

Proposition 3.5($ii$) implies that the linear span $H_{1}$ of the image of
$F_{1}$ is of dimension at most

$\dim(\mathbb{P}^{r^{\prime},s^{\prime}})-d-1+2=\dim(\mathbb{P}^{r^{\prime},s^{\prime}})-d+1\leq\dim(\mathbb{P}^{r^{\prime},s^{\prime}})-1.$

Therefore, we have shown that if $F_{1}$ is not linear, then the image of
$F_{1}$ lies in a hyperplane of $\mathbb{P}^{r^{\prime},s^{\prime}}$.

If we write
$H_{1}\cong\mathbb{P}^{r_{1}^{\prime},s_{1}^{\prime},t_{1}^{\prime}}$, then we
have $r_{1}^{\prime}+s_{1}^{\prime}\leq r^{\prime}+s^{\prime}\leq
2\dim(\mathbb{P}^{r,s})-1$. So the hypotheses of the theorem still hold for
the local orthogonal map $F_{1}$ from $\mathbb{P}^{r,s}$ to $H_{1}$ and thus
by similarly considering the standard projection $\pi_{2}:H_{1}\dasharrow
K_{1}$ for some $(r_{1}^{\prime},s_{1}^{\prime})$-subspace $K_{1}\subset
H_{1}$ and $F_{2}:=\pi_{2}\circ F_{1}$, etc., we deduce inductively there
exists some $(r^{\prime\prime},s^{\prime\prime})$-subspace
$\Phi\subset\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$ with
$F_{\Phi}:=\pi_{\Phi}\circ F$, where
$\pi_{\Phi}:\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}\dasharrow\Phi$ is
the associated projection, such that we have two possibilities:

$(i)$ $F_{\Phi}$ is orthogonal and linear; or

$(ii)$ $F_{\Phi}$ is orthogonal and there is a line $L\subset\Phi$ containing
the image of $F_{\Phi}$.

(We note here that if $\dim(\mathbb{P}^{r,s})=2$ (which implies also
$\dim(\mathbb{P}^{r^{\prime},s^{\prime}})\leq 2$), by replacing a general line
$L$ in the argument above by a general point, we see immediately by
orthogonality that $F_{1}$ maps lines to lines and so we still end up with the
two possibilities above (with $F_{\Phi}$ being $F_{1}$).)

We are going to first show that case $(ii)$ will lead to a contradiction. If
$L\cong\mathbb{P}^{1,0,1}$ or $\mathbb{P}^{0,1,1}$, then $F_{\Phi}$ must be
constant (and hence null) since there is only one null point in $L$. However,
$F$ is assumed to be not null and $F_{\Phi}$ is obtained from $F$ by composing
a number of projections which are standard each time, thus $F_{\Phi}$ cannot
be null neither. So we can only have $L\cong\mathbb{P}^{1,1}$. Consequently,
$F_{\Phi}$ can be regarded as a (non-null) local orthogonal map from
$\mathbb{P}^{r,s}$ to $\mathbb{P}^{1,1}$ and by Theorem 3.8, we get that
$\min\\{r,s\\}=1$. If $\max\\{r,s\\}\geq 2$, for any point
$p\in\mathbb{P}^{r,s}$ at which $F_{\Phi}$ is defined, we have
$\dim(p^{\perp})=\max\\{r,s\\}-1\geq 1$. However, we also have
$\dim(F_{\Phi}(p)^{\perp})=0$ since $F_{\Phi}(p)\in\mathbb{P}^{1,1}$, so by
orthogonality $F_{\Phi}$ maps lines to points. Therefore, $F_{\Phi}$ is
constant and hence null, contradicting our assumption at the beginning that
$F$ is not null. Therefore, we must have $\max\\{r,s\\}=1$ and hence $r=s=1$.
This is again a contradiction since $\dim(\mathbb{P}^{r,s})\geq 2$.

For case $(i)$, if the image of $F$ is contained in $\Phi$, then $F$ is just
$F_{\Phi}$ and is linear. Otherwise, consider the map
$F_{\Phi^{\perp}}:=\pi_{\Phi^{\perp}}\circ F$, where
$\pi_{\Phi^{\perp}}:\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}\dasharrow\Phi^{\perp}$
is the canonical projection. Then, $F_{\Phi^{\perp}}$ is a local orthogonal
map from $\mathbb{P}^{r,s}$ to
$\Phi^{\perp}\cong\mathbb{P}^{r^{\prime}-r^{\prime\prime},s^{\prime}-s^{\prime\prime},t^{\prime}}$
by Proposition 3.4. If $F_{\Phi^{\perp}}$ is null, then $F$ is quasi-linear.
If not, we can repeat the entire argument above on $F_{\Phi^{\perp}}$ and use
induction to conclude that $F$ is quasi-linear.

Finally, if $F$ maps some positive (resp. negative) point to a positive (resp.
negative) point, then the linear part of $F$ is standard by Lemma 3.7 and so
$F$ is quasi-standard. ∎

######  Theorem 3.12.

Let $F$ be a local orthogonal map from $\mathbb{P}^{r,s}$ to
$\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$. If

$\min\\{r^{\prime},s^{\prime}\\}\leq 2\min\\{r,s\\}-2,$

then $F$ is either null or quasi-linear. If in addition $F$ preserves the sign
of any single positive or negative point, then $F$ is quasi-standard.

###### Proof.

The theorem is trivial if $\min\\{r,s\\}\leq 1$ since it would imply
$\min\\{r^{\prime},s^{\prime}\\}=0$. Suppose $\min\\{r,s\\}\geq 2$. As before,
we just need to prove the case $t^{\prime}=0$. By Proposition 2.6, $F$ maps
${(\min\\{r,s\\}-1)}$-planes to $(\min\\{r^{\prime},s^{\prime}\\}-1)$-planes.
By hypotheses $\min\\{r^{\prime},s^{\prime}\\}-1\leq 2(\min\\{r,s\\}-1)-1$,
and since $\dim(\mathbb{P}^{r,s})=\min\\{r,s\\}-1+\max\\{r,s\\}$, we deduce
from Proposition 3.5$(ii)$ that the image of $F$ is contained in a linear
subspace $\Xi\subset\mathbb{P}^{r^{\prime},s^{\prime}}$ such that
$\dim(\Xi)\leq\min\\{r^{\prime},s^{\prime}\\}-1+\max\\{r,s\\}$. If we write
$\Xi\cong\mathbb{P}^{a_{1},b_{1},c_{1}}$ for some non-negative integers
$a_{1},b_{1},c_{1}$, then we can regard $F$ as a local orthogonal map from
$\mathbb{P}^{r,s}$ to $\mathbb{P}^{a_{1},b_{1},c_{1}}$. Note that

$a_{1}+b_{1}\leq\dim(\Xi)+1\leq\min\\{r^{\prime},s^{\prime}\\}+\max\\{r,s\\}\leq
2\min\\{r,s\\}-2+\max\\{r,s\\}.$

Thus, $a_{1}+b_{1}<2(r+s)-3=2\dim(\mathbb{P}^{r,s})-1$ and now the desired
results follows directly from Theorem 3.11. ∎

######  Theorem 3.13.

Let $F$ be a local orthogonal map from $\mathbb{P}^{1,s}$ to
$\mathbb{P}^{1,s^{\prime},t^{\prime}}$. If $s^{\prime}\leq 2s-2$, then $F$
either null or quasi-standard. If in addition if $t^{\prime}=0$, then $F$ is
either constant or standard.

###### Proof.

As usual, we just need to prove the theorem for $t^{\prime}=0$. Let
$F:U\subset\mathbb{P}^{1,s}\rightarrow\mathbb{P}^{1,s^{\prime}}$ be a non-
constant orthogonal map and $s^{\prime}\leq 2s-2$. Take a null point $x\in U$.
Since we are in $\mathbb{P}^{1,s}$, it follows that $x^{\perp}$ is a semi-
negative hyperplane and $x$ is the only null point in $x^{\perp}$. By
orthogonality, $F(x^{\perp}\cap U)\subset(F(x))^{\perp}$ and $F(x)$ is also
the only null point in the semi-negative hyerplane
$F(x)^{\perp}\subset\mathbb{P}^{1,s^{\prime}}$. As $F$ is non-constant, we can
always choose $x$ such that $F$ is not constant on $x^{\perp}\cap U$ and hence
$F$ preserves the sign of some negative point on $x^{\perp}$.

Now since $1+s^{\prime}\leq 2\dim(\mathbb{P}^{1,s})-1$, from Theorem 3.11 we
have that $F$ is quasi-standard. Finally, as the orthogonal complement of a
$(1,s)$-subspace in $\mathbb{P}^{1,s^{\prime}}$ is a
$(0,s^{\prime}-s)$-subspace which does not contain any null point, we see from
the definition of quasi-standard maps that $F$ is actually standard. ∎

##  4 Proper maps between generalized balls

On $\mathbb{P}^{r,s,t}$, the set of positive points
$\mathbb{B}^{r,s,t}\subset\mathbb{P}^{r,s,t}$ (or
$\mathbb{B}^{r,s}\subset\mathbb{P}^{r,s}$) has been called a generalized ball
in the literature since $\mathbb{B}^{1,s}$ is just the ordinary
$s$-dimensional complex unit ball $\mathbb{B}^{s}$ embedded in
$\mathbb{P}^{s}$. In additon, the boundary $\partial\mathbb{B}^{r,s,t}$ of
$\mathbb{B}^{r,s,t}$ is simply the set of null points on $\mathbb{P}^{r,s,t}$.

A local holomorphic map
$f:U\subset\mathbb{P}^{r,s,t}\rightarrow\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$,
defined on a connected open set $U$ such that
$U\cap\partial\mathbb{B}^{r,s,t}\neq\varnothing$, is called a local proper
holomorphic map from $\mathbb{B}^{r,s,t}$ to
$\mathbb{B}^{r^{\prime},s^{\prime},t^{\prime}}$ if
$f(U\cap\mathbb{B}^{r,s,t})\subset\mathbb{B}^{r^{\prime},s^{\prime},t^{\prime}}$
and
$f(U\cap\partial\mathbb{B}^{r,s,t})\subset\partial\mathbb{B}^{r^{\prime},s^{\prime},t^{\prime}}$.

The following statement is a direct consequence of Proposition 2.5.

######  Proposition 4.1.

By shrinking the domain of definition if necessary, a local proper holomorphic
map from $\mathbb{B}^{r,s,t}$ to
$\mathbb{B}^{r^{\prime},s^{\prime},t^{\prime}}$ is a local orthogonal map from
$\mathbb{P}^{r,s,t}$ to $\mathbb{P}^{r^{\prime},s^{\prime},t^{\prime}}$.

Consequently, our results for local orthogonal maps in the previous sections
also hold for local proper holomorphic maps among generalized balls, from
which we can obtain and generalize many well-known results in the literature.

######  Theorem 4.2.

When $r,s\geq 2$, every local proper holomorphic map from $\mathbb{B}^{r,s,t}$
to $\mathbb{B}^{r,s^{\prime},t^{\prime}}$ is quasi-standard (standard for
$t^{\prime}=0$).

###### Proof and remarks..

The statement follows from Theorem 3.10. Here we note that although Theorem
3.10 is for local sign-preserving maps, its proof actually only assumes that
positive points are mapped to positive points.

The case for $r\leq s$ and $t=t^{\prime}=0$ has been obtained by Baouendi-
Huang [BH] (Theorem 1.2 therein) while the case for $r\leq s$ has been
obtained by Ng-Zhu [NZ] (Theorem 1.2 therein). ∎

######  Theorem 4.3.

When $r^{\prime}+s^{\prime}\leq 2(r+s)-3$, every local proper holomorphic map
from $\mathbb{B}^{r,s}$ to $\mathbb{B}^{r^{\prime},s^{\prime},t^{\prime}}$ is
quasi-standard (standard for $t^{\prime}=0$). In particular, every local
proper holomorphic map from $\mathbb{B}^{s}$ to $\mathbb{B}^{s^{\prime}}$ is
standard if $s^{\prime}\leq 2s-2$.

###### Proof and remarks..

It follows from Theorem 3.11. The special case for $r+1=r^{\prime}=2$ and
$t^{\prime}=0$ has been proven by Xiao-Yuan [XY] (Theorem 3.2 therein). The
special case for the ordinary complex unit balls is a classical theorem by
Faran [Fa1]. ∎

######  Theorem 4.4.

When $r\leq s$ and $r^{\prime}\leq 2r-2$, every local proper holomorphic map
from
$\mathbb{B}^{r,s}\rightarrow\mathbb{B}^{r^{\prime},s^{\prime},t^{\prime}}$ is
quasi-standard.

###### Proof and remarks..

It is a direct consequence of Theorem 3.12. The case $t^{\prime}=0$ has been
obtained by Baouendi-Ebenfelt-Huang [BEH] (Theorem 1.1 and Corollary 1.6
therein) ∎

Acknowledgements. The authors would like to thank Prof. Xiaojun Huang, Ming
Xiao, Wanke Yin and Yuan Yuan for their comments on the first version of this
article. The first author was partially supported by Institute of Marine
Equipment, Shanghai Jiao Tong University. The second author was partially
supported by Science and Technology Commission of Shanghai Municipality
(STCSM) (No. 13dz2260400).

## References

  * [BH] Baouendi, M. S., Huang X.,: Super-rigidity for holomorphic mappings between hyperquadrics with positive signature. J. Diff. Geom. 69 (2005), 379-398.
  * [BEH] Baouendi, M. S., Ebenfelt, P., Huang, X.: Holomorphic mappings between hyperquadrics with small signature difference. Amer. Math. J. 133 (2011), 1633-1661.
  * [Da] D’Angelo, J. P.: Polynomial proper maps between balls. Duke Math. J. 57 (1988), 211-219.
  * [DL] D’Angelo, J. P., Lebl, J.: On the complexity of proper mappings between balls. Complex Var. Elliptic Equ. 54(3) (2009), 187–204.
  * [Fa1] Faran, J.: The linearity of proper holomorphic maps between balls in the low codimension case. J. Diff. Geom. 24 (1986), 15-17.
  * [Fa2] Faran, J.: Maps from the 2 ball to the 3 ball. Invent. Math. 68 (1982), 441-475.
  * [Fo] Forstnerič, F.: Proper holomorphic maps between balls. Duke Math. J. 53 (1986) 427–440.
  * [Ga] Gao, Y.: Orthogonal pair and a rigidity problem for Segre maps between hyperquadrics. arXiv. arXiv:2109.09340
  * [GN] Gao, Y., Ng, S.-C.: A hyperplane restriction theorem for holomorphic mappings and its application for the gap conjecture. Preprint.
  * [HJ] Huang, X., Ji, S.: Mapping $B^{n}$ into $B^{2n-1}$. Invent. Math. 145 (2001), 219-250.
  * [Ng1] Ng, S.-C.: Proper holomorphic mappings on flag domains of $SU(p,q)$-type on projective spaces . Michigan Math. J. 62 (2013), 769-777.
  * [Ng2] Ng, S.-C.: Cycle spaces of flag domains on Grassmannians and rigidity of holomorphic mappings. Math. Res. Lett. 19(6) (2012), 1219-1236.
  * [Ng3] Ng, S.-C.: Holomorphic double fibration and the mapping problems of classical domains. Int. Math. Res. Not. 215 Issue 2, (2015), 291-324.
  * [NZ] Ng, S.-C., Zhu, Y.: Rigidity of proper holomorphic maps among generalized balls with Levi-degenerate boundaries. To appear in J. Geom. Anal. https://doi.org/10.1007/s12220-021-00698-x
  * [XY] Xiao, M., Yuan, Y.: Holomorphic maps from the complex unit ball to Type IV classical domains. J. Math. Pures Appl.  133 (2020), 139-166.
  * [Za1] Zaitsev, D.: Obstructions to embeddability into hyperquadrics and explicit examples. Math. Ann., 342(3) (2008), 695-726.
  * [Za2] Zaitsev, D.: Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces. Acta. Math. 183 (1999), 273-305.