In this paper, we classify all nonconstant smooth CR maps from a sphere
with
n
> 3 to the Shilov boundary
of a bounded symmetric domain of Cartan type I under the condition that
p − q
< 3
n
− 4. We show that they are either linear maps up to automorphisms of
S
_{n,1}
and
S_{p,q}
or D’Angelo maps. This is the first classification of CR maps into the Shilov boundary of bounded symmetric domains other than sphere that includes nonlinear maps.
1. INTRODUCTION
The
rigidity of holomorphic maps
between open pieces of a sphere was first studied by Poincaré
[13]
in 2dimensional case and later by Alexander
[1]
and Chern and Moser
[2]
for general dimensions. Then Webster
[16]
obtained rigidity for holomorphic maps between open pieces of spheres of different dimension, proving that any such map between spheres in
and
extends as a totally geodesic map between balls with respect to the Bergman metric. Later, Huang
[6]
generalized Webster’s result for CR maps between open pieces of spheres in
and
under the assumption
n′
− 1 < 2(
n
− 1). Beyond this bound, the rigidity fails as illustrated by the Whitney map.
Unit ball is a bounded symmetric domain of Cartan type I with rank 1 and sphere is its Shilov boundary. However, comparing with rigidity of holmorphic maps between spheres mentioned above, holomorphic rigidity for maps between bounded symmetric domains
D
and
D′
of higher rank remains much less understood. If the rank
r′
of
D′
does not exceed the rank
r
of
D
and both ranks
r
,
r′
≥ 2, the
rigidity of proper holomorphic maps f
:
D
→
D′
was conjectured by Mok
[12]
and proved by Tsai
[15]
, showing that
f
is necessarily totally geodesic (with respect to the Bergmann metric).
For the case
r
<
r′
, in
[11]
, Zaitsev and author showed the
rigidity of CR maps f
:
S_{p,q}
→
S_{p′,q′}
under the assumption that
q
≥ 2 and (
p′ − q′
) < 2(
p
−
q
). Here,
S_{p,q}
and
S_{p′,q′}
are the Shilov boundaries of a bounded symmetric domains of Cartan type I((See §1 for definition) and
q
and
q′
are the ranks of
S_{p,q}
and
S_{p′,q′}
, respectively. When (
p′ − q′
) = 2 (
p − q
), then the rigidity fails to hold, as authors introduced the generalized Whitney map as a counterexample in the same paper.
Recently, in
[14]
, A. Seo introduced a nonlinearizable proper holomorphic maps between
S_{p,q}
and
S
_{2p−1,2q−1}
. Therefore, to classify all CR maps between
S_{p,q}
and
S_{p′,q′}
when
p′ − q′
≥ 2(
p − q
), one should consider nonlinear maps. In
[9]
, Huang, Ji and Xu classified all locally defined CR maps between
S
_{n,1}
and
S
_{n′,1}
under the assumption that 3 <
n
≤
n′
< 3
n
− 3. It is proved that such map is either a linear map or a D’Angelo map.
In this paper, we generalize the result of Huang, Ji and Xu. We define D’Angelo map from a sphere into the Shilov boundary of bounded symmetric domains of type I as follows:
Definition 1.1.
Let
be the set of all complex
p
×
q
matrices. A map
f_{θ}
:
S
_{n,1}
→
S_{p,q}
for a fixed 0 <
θ
≤
π
/2, is called a
D’Angelo map
if
f_{θ}
is equivalent to the following map
up to automorphisms of
S
_{n,1}
and
S_{p,q}
, where
W_{θ}
(
z
) is a map from
S
_{n,1}
to
S
_{3n−3,1}
defined by
and
I
_{q−1}
is the identity matrix of size (
q
−1).
This map is not linear after composing with any automorphisms of
S
_{n,1}
and
S_{p,q}
. For
q
= 1 and
θ
=
π
/2, this is the classical Whitney map between unit balls in
and
respectively. In this paper, we classify all
locally defined CR maps
from a sphere
S
_{n,1}
with
n
> 3 into the Shilov boundary
S_{p,q}
of a general Cartan type I bounded symmetric domain of higher rank. We showed
Theorem 1.2.
Let f be a nonconstant smooth CR map from an open piece of
S
_{n,1}
into S_{p,q}
.
Assume that n
> 3
and p − q
< 3
n
− 4.
Then after composing with suitable automorphisms of
S
_{n,1}
and S_{p,q}
,
f is either a linear embedding or D’Angelo map
.
Note that our basic assumption
p − q
< 3
n
−4 corresponds precisely to the
optimal bound
n′
− 1 < 3(
n
− 1) in the rank 1 case (
q
= 1) of maps between spheres, where
n
−1 and
n′
− 1 are the CR dimensions of the spheres.
Throughout this paper we adopt the Einstein summation convention unless mentioned otherwise.
2. PRELIMARIES
In this section, we review CR structure and Grassmannian frames adapted to
S_{p,q}
. For details, we refer
[2]
and
[11]
as references. In this section, we let Greek indices
α
,
β
,
γ
, … and Latin indices
j
,
k
,
ℓ
, … run over {1,…,
q
} and {1,…,
p − q
}, respectively. For
q
= 1, i.e., sphere case, we omit Greek indices.
A Hermitian symmetric domain
D_{p,q}
of Cartan type I has a standard realization in the space
of
p
×
q
matrices, given by
where
I_{q}
is the
q
×
q
identity matrix and
The
Shilov boundary
of
D_{p,q}
is given by
In particular,
S_{p,q}
is a CR manifold of CR dimension (
p
−
q
) ×
q
. For
q
= 1,
S
_{p,1}
is the unit sphere in
. We shall always assume
p
>
q
so that
S_{p,q}
has positive CR dimension.
Let
Aut
(
S_{p,q}
) be the Lie group of all CR automorphisms of
S_{p,q}
. By [10, Theorem 8.5], every
f
∈
Aut
(
S_{p,q}
) extends to a biholomorphic automorphism of the bounded symmetric domain
D_{p,q}
. Consider the standard linear inclusion
Then we may regard
S_{p,q}
as a real submanifold in the Grassmanian
Gr
(
q, p
+
q
) of all
q
planes in
and
Aut
(
S_{p,q}
)(=
Aut
(
D_{p,q}
)) becomes a subgroup of the automorphism group of
Gr
(
q, p
+
q
).
For column vectors
u
= (
u
_{1}
, … ,
u
_{p+q}
)
^{t}
and
v
= (
v
_{1}
, … ,
v
_{p+q}
)
^{t}
in
, define a Hermitian inner product by
A
Grassmannian frame adapted to S_{p,q}
, or simply
S_{p,q}frame
is a frame {
Z
_{1}
, …,
Z
_{p+q}
} of
with det (
Z
_{1}
, …,
Z
_{p+q}
) = 1 such that scalar product <·, ·> in basis (
Z
_{1}
, …,
Z
_{p+q}
) is given by the matrix
Now let
be the set of all
S_{p,q}
frames. Then
is identified with
SU
(
p, q
) by the left action. The MaurerCartan form
on
is given by the equation
where
π
satisfies the tracefree condition
and the structure equation
where the capital Greek indices Λ, Γ, Ω etc. run from 1 to
p
+
q
.
From now, we will use the notation
Z
:= (
Z
_{1}
, … ,
Z
_{q}
),
X
= (
X
_{1}
, … ,
X
_{p−q}
) := (
Z
_{q+1}
, … ,
Z
_{p}
),
Y
= (
Y
_{1}
, … ,
Y
_{q}
) := (
Z
_{p+1}
…,
Z_{p+q}
)
so that the MaurerCartan form with respect to the basis (
Z, X, Y
) can be written as
with the symmetry relations
By abuse of notation, we also denote by
Z
the
q
dimensional subspace of
spanned by
Z
_{1}
, …,
Z_{q}
. Then the defining equations of
S_{p,q}
can be written as

Sp,q= {Z∈Gr(q, p+q) : <·,·>z= 0}
and hence their differentiation yields
By substituting
into (1, 0) component of (2.3) we obtain, in particular,
when restricted to the (1, 0) tangent space. Comparing the dimensions, we conclude that
span the space of contact forms on
S_{p,q}
, i.e.,
is the complex tangent space of
S_{p,q}
. The structure equation is given by
Moreover, since
we conclude that
θ_{α}^{j}
form a basis in the space of (1, 0) forms.
There are several types of frame changes.
Definition 2.1.
We call a change of frame

i)change of positionif


whereW= (Wαβ) andV= (Vαβ) areq × qmatrices satisfyingV*W=Iq;

ii)change of real vectorsif


whereH= (Hαβ) is a hermitian matrix;

iii)dilationif


whereλα> 0;

iv)rotationif


where (Ujk) is a unitary matrix.
Finally, we shall use the change of frame given by
such that
and
where
The new frame
is an
S_{p,q}
frame and the related 1forms
remain the same, while
change to
3.Sp,qFRAMES ADAPTED TO CR MAPPINGS
Let
f
:
S
_{n,1}
→
S_{p,q}
be a (germ of a) smooth CR mapping. We shall identify
S
_{n,1}
and its image
f
(
S
_{n,1}
) ⊂
S_{p,q}
. We consider the connection forms
φ
,
θ^{j}
,
𝜓
,
ω_{j}^{k}
,
σ_{j}
,
ξ
with
j, k
= 1, …,
n
− 1 on
S
_{n,1}
and denote by capital letters Φ
_{α}^{β}
, Θ
_{α}^{j}
, Ψ
_{α}^{β}
, Ω
_{J}^{K}
,Σ
_{K}^{β}
, Ξ
_{α}^{β}
with
α, β
= 1,…,
q
and
J, K
= 1,…,
p − q
, their corresponding counterparts on
S_{p,q}
. We also define one forms
φ_{α}^{β}
,
θ_{α}^{J}
adapted to
f
as follows:
Definition 3.1.
We say that
f
is of
contact rank r
if
f
sends any nonzero vector in
TS
_{n,1}
/
T^{c}S
_{n,1}
to a rank
r
vector in
TS_{p,q}
/
T^{c}S_{p,q}
.
For a map
f
of contact rank
r
, we define
φ_{α}^{β}
,
θ_{α}^{J}
for
α
= 1, …,
q
and
J
= 1, …,
p − q adapted to f
by

φ11= ⋯φrr=φ,

θ1j= ⋯ =θr(r−1)(n−1)+j=θj,j= 1, …,n− 1
and 0 otherwise.
In this section we show the following lemma.
Lemma 3.2.
For any nonconstant local CR map f
:
S
_{n,1}
→
S_{p,q}
with p − q
< 3(
n
−1),
there exist r
∈ {1, 2}
and a choice of S_{p,q}frames such that f is of contact rank r and the forms
φ_{α}^{β}
,
θ_{α}^{J}
adapted to f satisfy
Proof is a slight modification of the proof of Lemma 4.2 and argument in §.5 of
[11]
. We refer
[11]
for details.
Proof
. Since
φ
and Φ = (Φ
_{α}^{β}
) are contact forms on
S
_{n,1}
and
S_{p,q}
, respectively, the pull back of Φ
f
is a span of
φ
. Choose a diagonal contact form of
S_{p,q}
and say Φ
_{1}
^{1}
. Then we can write
for some smooth function λ. At generic points, we may assume that either λ ≡ 0 or λ never vanishes. By differentiating (3.2) and using (2.4) we obtain
Arguing similar to
[11]
we conclude λ ≥ 0 and, after dilation of Φ
_{1}
^{1}
, we may assume that λ = 1 if λ ≢ 0.
Suppose that Φ
_{α}^{α}
vanishes identically for all
α
. Then we obtain
Since each Θ
_{α}^{J}
is a (1, 0) form, it follows that
i.e.,
f
(
S
_{n,1}
) is a totally real submanifold. Since
S
_{n,1}
is Levinondegenerate, this implies that
f
is a constant map, which contradicts our assumption. Hence there exists at least one diagonal term of Φ whose pullback does not vanish identically.
Choose such a diagonal term of Φ, say Φ
_{1}
^{1}
. Then (3.3) yields
Therefore after a suitable rotation of
S_{p,q}
, we may assume that
Write
for some smooth functions λ
_{α}
. Then by differentiating (3.6) and using (2.4) together with (3.4), (3.5), we obtain
Choose a suitable change of position that leaves Θ
_{1}
^{J}
invariant and replaces Θ
_{α}^{J}
with Θ
_{α}^{J}
− λ
_{α}
Θ
_{1}
^{J}
for
α
≥ 2. This change of position leaves Φ
_{1}
^{1}
invariant and transforms Φ
_{α}
^{1}
into Φ
_{α}
^{1}
− λ
_{α}
Φ
_{1}
^{1}
for
α
≥ 2. After performing such change of position, (3.6) becomes
and (3.7) becomes
Sincec Θ
_{α}^{j}
are (1, 0) but
θ_{j}
are (0, 1) and linearly independent, it follows that
Next for each
α
≥ 2, let
for another smooth function λ
_{α}
. If λ
_{α}
≡ 0 for all
α
≥ 2, then by differentiation, we obtain
which yields
In this case, by considering the differentiation of
and substituting (3.10), we conclude that
which implies that
df
(
T
) modulo
T^{c}S_{p,q}
is a rank 1 vector for any
T
∈
TS
_{n,1}
transversal to
T^{c}S
_{n,1}
. That is to say,
f
is of contact rank 1 and the forms adapted to
f
satisfy

Φαβ−φαβ= 0,

ΘαJ−θαJ= 0 modφ.
Suppose there exists
α
such that λ
_{α}
≢ 0. We may assume
α
= 2. After a dilation of Φ
_{2}
^{2}
, we may assume that at generic points, λ
_{2}
= 1. By differentiating (3.9) for
α
= 2 and substituting (3.8) we obtain
Hence after a suitable rotation
where (
U_{K}^{J}
) is unitary matrix leaving Θ
_{α}^{j}
,
j
= 1,…,
n
−1, invariant, we may assume that

Θ2n−1+j=θjmodφ,j= 1,…,n−1
and
otherwise. Write
for some smooth function λ
_{α}
. Then as before, we can choose a suitable change of position that leaves Θ
_{1}
^{J}
and Θ
_{2}
^{J}
invariant and replaces Θ
_{α}
^{J}
with Θ
_{α}
^{J}
− λ
_{α}
Θ
_{2}
^{J}
for
α
> 2, which also leaves Φ
_{1}
^{1}
, Φ
_{2}
^{1}
and Φ
_{2}
^{2}
invariant and transforms Φ
_{α}
^{2}
into Φ
_{α}
^{2}
− λ
_{α}
Φ
_{2}
^{2}
for
α
> 2. By (3.8), after performing such change of position, the following property
still holds and (3.11) becomes
By differentiating this we obtain
which yields
Write
for some smooth functions λ
_{α}
. Suppose that λ
_{α}
≡ 0 for all
α
. Then as before, we can obtain

ΘαJ= 0 modφ,α> 2, ∀J,

Φαβ= 0,α> 2 orβ> 2,
i.e.,
f
is of contact rank 2 and the forms adapted to
f
satisfy

Φαβ−φαβ= 0,

ΘαJ−θαJ= 0 modφ.
Suppose there exists
α
such that λ
_{α}
≠ 0 We may assume
α
= 3. After a dilation of Φ
_{3}
^{3}
, we may assume that at generic points, λ
_{3}
= 1, i.e.,
By differentiating this, we obtain
hen by (3.8) and (3.12), we have at most
p − q
− 2(
n
−1) linearly independent (1, 0) forms on the lefthand side, while on the righthand side we have
n
− 1 linearly independent (1, 0) forms. Since we assumed that
p
−
q
< 3(
n
− 1), this is a contradiction.
Next we will show that there exists a choice of frames such that
Write
for some
η_{α}^{J}
. Consider the equations obtained by differentiating (3.13):
where

ψαα=ψ,α= 1, …,r,ψαβ= 0 otherwise
and
Let
α
>
r
. Then lefthand side of (3.14) contains at most one (1, 0) form, while the righthand side contains (
n
− 1) linearly independent (1, 0) forms with
n
− 1 > 1 unless
η_{α}^{J}
= 0. Therefore we conclude that
or equivalently
Finally, define a matrix (
B_{α}^{J}
) by
where
η_{α}^{J}
satisfies
Consider the change of frame of
S_{p,q}
discussed after Definition 2.1, given by
such that
and
A_{α}^{β}
satisfies
Since the sum here is hermitian, one can always choose
A_{α}^{β}
with this property. Then Φ
_{α}^{β}
remain the same while
change to
Therefore the new Θ
_{α}
^{J}
satisfies
4. SECOND FUNDAMENTAL FORMS AND GAUSS EQUATIONS FOR CR EMBEDDINGS
In this section, we determine second fundamental forms given by Ω
_{J}^{K}
. Then we determine Ψ
_{α}^{β}
and Σ
_{α}^{J}
. By using these forms, we construct a linear subspace of
Gr
(
q, p+q
) that contains the image of a given embedding(Lemma 4.1, Lemma 4.2). Their proofs are slight modification of the proof of Proposition 7.1 in
[11]
.
Let
f
be a CR map of contact rank
r
with
r
∈ {1, 2}. Differentiate (3.1) using the structure equations to obtain
where

σα(α−1)(n−1) +j=σj,α= 1,…,r,j= 1,…,n− 1
and 0 otherwise.
4.1. Contact rank 1 map
Choose
α
> 1 and
J = j
. Then (4.1) takes the form
By Cartan Lemma we obtain
for fixed
j
. Since Ψ is independent of
j
= 1, … ,
n
−1 and we assumed
n
−1 > 1, we obtain
We will show the following lemma.
Lemma 4.1.
There exists
(
p−q
+2)
dimensional subspace
V
_{1}
and
(
q
−1)
dimensional subspace
V
_{2}
in
orthogonal to each other such that Gr
(1,
V
_{1}
) ⊕
V
_{2}
contains the image f
(
S
_{n,1}
).
Proof
. Choose an open set
M
⊂
S
_{n,1}
where
f
is defined. Let
Z
,
X
,
Y
be constant vector fields of
forming a
S_{p,q}
frame at a fixed reference point of
f
(
M
) and let
be an adapted
S_{p, q}
frame along
f
(
M
). Write
so that (4.3)  (4.5) take the form
Since
Z, X, Y
form an adapted frame at a reference point of
M
, we may assume that
at the reference point. Since
Z, X, Y
are constant vector fields, i.e.,
dZ
=
dX
=
dY
= 0, differentiating (4.6) and using (2.1) we obtain
Next, it follows from Lemma 3.2 and (4.2) that
in particular, the span of
is independent of the point in
M
. Hence together with (4.3) and (4.7), we conclude
Furthermore, (4.8) for
α
= 1 together with Lemma 3.2 and (4.2) (and with the symmetry relations analogous to (2.2)) we obtain
Now with (4.9) taken into account, (4.10) becomes
Thus each of the vector valued functions
for a fixed
β
satisfies a complete system of linear first order differential equations. Then by the initial condition (4.7) and the uniqueness of solutions, we conclude, in particular, that
Hence (4.3) implies
Now setting
we still have
whereas (4.11) becomes
implying
Then we conclude that
where

V1= span {Z1,X1,…,Xp−q,Y1},V2= span {Z2,…,Zq}.
4.2. Contact rank 2 map
Choose
α
> 2 and
J
=
j
or
J
=
n
− 1 +
j
. Then (4.1) takes the form
Since Ψ is independent of
j
= 1,…,
n
− 1 and we assumed
n
− 1 > 1, by Cartan Lemma we obtain
Use (4.1) for either
α
= 1 and
J
=
n
− 1 +
J
or
α
= 2 and
J
=
j
or
α
= 1, 2 and
J
> 2(
n
−1) to obtain
By Cartan’s Lemma, we obtain
where
θ
is an ideal generated by
θ
^{1}
,…,
θ
^{n−1}
. Since
by using (4.16), we conclude that
Moreover, since Ψ is independent of
j
, substituting (4.18) into (4.13) and (4.14), we obtain
Next we will determine second fundamental forms of
f
as in
[16]
. We will show that it has a trivial solution only. For details, we refer
[16]
.
Use (4.1) for
α
= 1 and
J
=
j
≤ (
n
− 1) to obtain
Then by Cartan Lemma, we obtain
By symmetry relation for Ω, we obtain
Furthermore, differentiation of
by using the structure equations yields
or equivalently
Therefore we obtain
for some pure imaginary function
g
.
Similar computation for (4.1) with
α
= 2 and
J
=
n
− 1 +
j
together with the relation
yields
and
for a pure imaginary function
h
.
Take a real vector change of
S_{p,q}
defined by
for a smooth function
μ
satisfying
in (4.19) and fixing the rest. Then after the frame change, we obtain
By differentiating (4.20),(4.21),(4.22) and substituting (4.12) and (4.19), we obtain
Then by Cartan Lemma, we obtain
By (4.16) and symmetry relation for Σ, we obtain
Now let
Then (4.1) implies
Write

ΩjJ=hjJℓθℓmodφ,K> 2(n− 1).
Differentiate (4.24) and substitute (4.18) to obtain
which implies
If
p−q
< 3(
n
−1), then (4.26) has trivial solution only.(See
[3]
.) Therefore we obtain
or equivalently
Similar computation for
using (4.25) yields
By (4.18) and (4.27), we can write
By differentiating this, we obtain
By (4.17) and (4.23) we can show that the lefthand side of (4.28) contains at most one (0, 1) form, while the righthand side contains (
n
− 1) linearly independent (0, 1) forms unless
η_{k}^{J}
= 0. Hence we conclude that
or equivalently
and therefore by substituting (4.17) and (4.23) into (4.28), we obtain
Similar computation for
implies
Furthermore, by substituting (4.29) to(4.15) with
J
=
j
, we obtain
Finally we will determine Ψ and Σ. By (4.19), we can write
By differentiating this and substituting (4.12) and (4.19), we obtain
By (4.30), this implies
or equivalently
Let
By differentiation, we obtain
which yield
or equivalently
Since Ξ
_{1}
^{2}
is independent of
j
, we obtain
Similar computation for Σ
_{2}
^{J}
yields
Summing up we obtain the following:
For any contact rank 2 local CR embedding
f
from
S
_{n,1}
into
S_{p,q}
, there is a choice of frames such that
We will show the following lemma.
Lemma 4.2.
There exist
(
n
+1)
dimensional subspaces V
_{1}
,
V
_{2}
and
(
q
−2)
dimensional subspace
V
_{3}
in
orthogonal to each other such that Gr
(1,
V
_{1}
) ⊕
Gr
(1,
V
_{2}
) ⊕
V
_{3}
contains the image f
(
S
_{n,1}
).
Proof
. We use the same method in Lemma 4.1. Let
M
⊂
S
_{n,1}
,
Z, X, Y
and
be as in Lemma 4.1.
It follows from Lemma 3.2 and (4.31) that
in particular, the span of
is independent of the point in
M
. Hence as in Lemma 4.1, we conclude
Furthermore, (4.8) implies
In particular, restricting to
α
= 1 and
J
=
j
≤
n
with (4.31)(4.34) and (4.37) taken into account, we obtain
Repeating the above argument for λ and
ζ
instead of
η
, we obtain
and
Thus each of the vector valued functions
for a fixed
K
and
for a fixed
β
satisfies a complete system of linear first order differential equations. Then as in Lemma 4.1 we conclude, in particular, that
and
Hence (4.35) implies
Similar computation for
implies
Now setting
we still have
whereas (4.38), (4.39) become
implying
Then together with (4.36) we conclude that
where

V1= span {Z1,X1,…,Xn−1,Y1},V2= span {Z2,Xn,…,X2n−2,Y2},V3= span {Z3,…,Zq}
5. PROOF OF THEOREM 1.2
Suppose
f
is of contact rank 1. Then by Lemma 4.1, there exist (
p
−
q
+ 2)dimensional subspace
V
_{1}
and (
q
− 1)dimensional subspace
V
_{2}
such that the image of
f
is contained in
Gr
(1,
V
_{1}
) ⊕
V
_{2}
. The
V
_{2}
component of
f
is a constant map. Therefore it is enough to show that
Gr
(1,
V
_{1}
)component of
f
is either a linear map or Whitney map. But
Gr
(1,
V
_{1}
) =
. Therefore by the result of
[9]
under the condition
n
> 3 and (
p
−
q
) < 3
n
− 4, we conclude that
Gr
(1,
V
_{1}
)component of
f
is either a flat embedding or D’Angelo map.
Suppose
f
is of contact rank 2, then by Lemma 4.2, there exist (
n
+1)dimensional subspaces
V
_{1}
,
V
_{2}
and (
q
− 2)dimensional subspace
V
_{3}
such that the image of
f
is contained in
Gr
(1,
V
_{1}
)⊕
Gr
(1,
V
_{2}
)⊕
V
_{3}
. As before, it is enough to show that
Gr
(1,
V
_{1}
) and
Gr
(1,
V
_{2}
)components of
f
are linear. Since
V
_{1}
and
V
_{2}
are of dimension (
n
+1), each component of
f
is a CR automorphism of
S
_{n,1}
. Therefore, it is projective linear, which completes the proof.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology (grant number 2012R1A1B5003198).
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