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SPHERES IN THE SHILOV BOUNDARIES OF BOUNDED SYMMETRIC DOMAINS
SPHERES IN THE SHILOV BOUNDARIES OF BOUNDED SYMMETRIC DOMAINS
Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics. 2015. Feb, 22(1): 35-56
Copyright © 2015, Korean Society of Mathematical Education
  • Received : September 26, 2014
  • Accepted : January 06, 2015
  • Published : February 28, 2015
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SUNG-YEON KIM

Abstract
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 Sp,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.
Keywords
1. INTRODUCTION
The rigidity of holomorphic maps between open pieces of a sphere was first studied by Poincaré [13] in 2-dimensional 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
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and
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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
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and
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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 : Sp,q Sp′,q′ under the assumption that q ≥ 2 and ( p′ − q′ ) < 2( p q ). Here, Sp,q and Sp′,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 Sp,q and Sp′,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 Sp,q and S 2p−1,2q−1 . Therefore, to classify all CR maps between Sp,q and Sp′,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
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be the set of all complex p × q matrices. A map fθ : S n,1 Sp,q for a fixed 0 < θ π /2, is called a D’Angelo map if fθ is equivalent to the following map
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up to automorphisms of S n,1 and Sp,q , where Wθ ( z ) is a map from S n,1 to S 3n−3,1 defined by
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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 Sp,q . For q = 1 and θ = π /2, this is the classical Whitney map between unit balls in
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and
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respectively. In this paper, we classify all locally defined CR maps from a sphere S n,1 with n > 3 into the Shilov boundary Sp,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 Sp,q . Assume that n > 3 and p − q < 3 n − 4. Then after composing with suitable automorphisms of S n,1 and Sp,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 Sp,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 , , … run over {1,…, q } and {1,…, p − q }, respectively. For q = 1, i.e., sphere case, we omit Greek indices.
A Hermitian symmetric domain Dp,q of Cartan type I has a standard realization in the space
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of p × q matrices, given by
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where Iq is the q × q identity matrix and
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The Shilov boundary of Dp,q is given by
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In particular, Sp,q is a CR manifold of CR dimension ( p q ) × q . For q = 1, S p,1 is the unit sphere in
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. We shall always assume p > q so that Sp,q has positive CR dimension.
Let Aut ( Sp,q ) be the Lie group of all CR automorphisms of Sp,q . By [10, Theorem 8.5], every f Aut ( Sp,q ) extends to a biholomorphic automorphism of the bounded symmetric domain Dp,q . Consider the standard linear inclusion
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Then we may regard Sp,q as a real submanifold in the Grassmanian Gr ( q, p + q ) of all q -planes in
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and Aut ( Sp,q )(= Aut ( Dp,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
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, define a Hermitian inner product by
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A Grassmannian frame adapted to Sp,q , or simply Sp,q-frame is a frame { Z 1 , …, Z p+q } of
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with det ( Z 1 , …, Z p+q ) = 1 such that scalar product <·, ·> in basis ( Z 1 , …, Z p+q ) is given by the matrix
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Now let
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be the set of all Sp,q -frames. Then
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is identified with SU ( p, q ) by the left action. The Maurer-Cartan form
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on
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is given by the equation
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where π satisfies the trace-free condition
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and the structure equation
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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 …, Zp+q )
so that the Maurer-Cartan form with respect to the basis ( Z, X, Y ) can be written as
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with the symmetry relations
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By abuse of notation, we also denote by Z the q -dimensional subspace of
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spanned by Z 1 , …, Zq . Then the defining equations of Sp,q can be written as
  • Sp,q= {Z∈Gr(q, p+q) : <·,·>|z= 0}
and hence their differentiation yields
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By substituting
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into (1, 0) component of (2.3) we obtain, in particular,
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when restricted to the (1, 0) tangent space. Comparing the dimensions, we conclude that
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span the space of contact forms on Sp,q , i.e.,
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is the complex tangent space of Sp,q . The structure equation is given by
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Moreover, since
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we conclude that θαj 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
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such that
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and
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where
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The new frame
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is an Sp,q -frame and the related 1-forms
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remain the same, while
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change to
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3.Sp,q-FRAMES ADAPTED TO CR MAPPINGS
Let f : S n,1 Sp,q be a (germ of a) smooth CR mapping. We shall identify S n,1 and its image f ( S n,1 ) ⊂ Sp,q . We consider the connection forms φ , θj , 𝜓 , ωjk , σj , ξ with j, k = 1, …, n − 1 on S n,1 and denote by capital letters Φ αβ , Θ αj , Ψ αβ , Ω JK Kβ , Ξ αβ with α, β = 1,…, q and J, K = 1,…, p − q , their corresponding counterparts on Sp,q . We also define one forms φαβ , θαJ 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 / TcS n,1 to a rank r vector in TSp,q / TcSp,q .
For a map f of contact rank r , we define φαβ , θαJ 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 Sp,q with p − q < 3( n −1), there exist r ∈ {1, 2} and a choice of Sp,q-frames such that f is of contact rank r and the forms φαβ , θαJ adapted to f satisfy
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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 αβ ) are contact forms on S n,1 and Sp,q , respectively, the pull back of Φ f is a span of φ . Choose a diagonal contact form of Sp,q and say Φ 1 1 . Then we can write
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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
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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 αα vanishes identically for all α . Then we obtain
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Since each Θ αJ is a (1, 0) form, it follows that
  • ΘαJ= 0 modφ,
i.e., f ( S n,1 ) is a totally real submanifold. Since S n,1 is Levi-nondegenerate, 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
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Therefore after a suitable rotation of Sp,q , we may assume that
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Write
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for some smooth functions λ α . Then by differentiating (3.6) and using (2.4) together with (3.4), (3.5), we obtain
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Choose a suitable change of position that leaves Θ 1 J invariant and replaces Θ αJ with Θ αJ − 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
  • Φα1= 0,α≥ 2,
and (3.7) becomes
  • Θαj∧θj1= 0 modφ,α≥ 2.
Sincec Θ αj are (1, 0) but θj are (0, 1) and linearly independent, it follows that
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Next for each α ≥ 2, let
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for another smooth function λ α . If λ α ≡ 0 for all α ≥ 2, then by differentiation, we obtain
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which yields
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In this case, by considering the differentiation of
  • Φαβ= λαβφ
and substituting (3.10), we conclude that
  • Φαβ= 0, (α, β) ≠ (1, 1),
which implies that df ( T ) modulo TcSp,q is a rank 1 vector for any T TS n,1 transversal to TcS 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
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Hence after a suitable rotation
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where ( UKJ ) is unitary matrix leaving Θ αj , j = 1,…, n −1, invariant, we may assume that
  • Θ2n−1+j=θjmodφ,j= 1,…,n−1
and
  • Θ2J= 0 modφ
otherwise. Write
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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
  • ΘαJ= 0 modφ,α≥ 2
still holds and (3.11) becomes
  • Φα2= 0,α> 2.
By differentiating this we obtain
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which yields
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Write
  • Φαα= λαφ,α> 2
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.,
  • Φ33=φ.
By differentiating this, we obtain
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hen by (3.8) and (3.12), we have at most p − q − 2( n −1) linearly independent (1, 0) forms on the left-hand side, while on the right-hand 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
  • ΘαJ=θαJ.
Write
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for some ηαJ . Consider the equations obtained by differentiating (3.13):
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where
  • ψαα=ψ,α= 1, …,r,ψαβ= 0 otherwise
and
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Let α > r . Then left-hand side of (3.14) contains at most one (1, 0) form, while the right-hand side contains ( n − 1) linearly independent (1, 0) forms with n − 1 > 1 unless ηαJ = 0. Therefore we conclude that
  • ηαJ= 0,α>r
or equivalently
  • ΘαJ= 0,α>r.
Finally, define a matrix ( BαJ ) by
  • BαJ:=ηαJ,
where ηαJ satisfies
  • ΘαJ−θαJ=ηαJφ.
Consider the change of frame of Sp,q discussed after Definition 2.1, given by
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such that
  • CJα:= −BJα
and Aαβ satisfies
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Since the sum here is hermitian, one can always choose Aαβ with this property. Then Φ αβ remain the same while
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change to
  • ΘαJ− ΦαβBβJ.
Therefore the new Θ α J satisfies
  • ΘαJ=θαJ.
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4. SECOND FUNDAMENTAL FORMS AND GAUSS EQUATIONS FOR CR EMBEDDINGS
In this section, we determine second fundamental forms given by Ω JK . Then we determine Ψ αβ and Σ αJ . 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
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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
  • Ψα1∧θj= 0,α> 1.
By Cartan Lemma we obtain
  • Ψα1= 0 modθj
for fixed j . Since Ψ is independent of j = 1, … , n −1 and we assumed n −1 > 1, we obtain
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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
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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
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forming a Sp,q -frame at a fixed reference point of f ( M ) and let
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be an adapted Sp, q -frame along f ( M ). Write
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so that (4.3) - (4.5) take the form
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Since Z, X, Y form an adapted frame at a reference point of M , we may assume that
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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
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Next, it follows from Lemma 3.2 and (4.2) that
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in particular, the span of
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is independent of the point in M . Hence together with (4.3) and (4.7), we conclude
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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
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Now with (4.9) taken into account, (4.10) becomes
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Thus each of the vector valued functions
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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
  • ζβ= 0 ,β> 1
Hence (4.3) implies
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Now setting
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we still have
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whereas (4.11) becomes
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implying
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Then we conclude that
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where
  • V1= span {Z1,X1,…,Xp−q,Y1},V2= span {Z2,…,Zq}.
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4.2. Contact rank 2 map Choose α > 2 and J = j or J = n − 1 + j . Then (4.1) takes the form
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Since Ψ is independent of j = 1,…, n − 1 and we assumed n − 1 > 1, by Cartan Lemma we obtain
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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
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By Cartan’s Lemma, we obtain
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where θ is an ideal generated by θ 1 ,…, θ n−1 . Since
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by using (4.16), we conclude that
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Moreover, since Ψ is independent of j , substituting (4.18) into (4.13) and (4.14), we obtain
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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
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Then by Cartan Lemma, we obtain
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By symmetry relation for ­Ω, we obtain
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Furthermore, differentiation of
  • Φ11−φ= 0
by using the structure equations yields
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or equivalently
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Therefore we obtain
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for some pure imaginary function g .
Similar computation for (4.1) with α = 2 and J = n − 1 + j together with the relation
  • Φ22−φ= 0
yields
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and
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for a pure imaginary function h .
Take a real vector change of Sp,q defined by
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for a smooth function μ satisfying
  • Ψ12=μφ
in (4.19) and fixing the rest. Then after the frame change, we obtain
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By differentiating (4.20),(4.21),(4.22) and substituting (4.12) and (4.19), we obtain
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Then by Cartan Lemma, we obtain
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By (4.16) and symmetry relation for Σ, we obtain
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Now let
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Then (4.1) implies
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Write
  • ΩjJ=hjJℓθℓmodφ,K> 2(n− 1).
Differentiate (4.24) and substitute (4.18) to obtain
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which implies
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If p−q < 3( n −1), then (4.26) has trivial solution only.(See [3] .) Therefore we obtain
  • hkJℓ= 0
or equivalently
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Similar computation for
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using (4.25) yields
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By (4.18) and (4.27), we can write
  • ΩkJ=ηkJφ,J>n− 1.
By differentiating this, we obtain
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By (4.17) and (4.23) we can show that the left-hand side of (4.28) contains at most one (0, 1) form, while the right-hand side contains ( n − 1) linearly independent (0, 1) forms unless ηkJ = 0. Hence we conclude that
  • ηkJ= 0
or equivalently
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and therefore by substituting (4.17) and (4.23) into (4.28), we obtain
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Similar computation for
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implies
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Furthermore, by substituting (4.29) to(4.15) with J = j , we obtain
  • Σ2J= 0 modφ,j≤n− 1.
Finally we will determine Ψ and Σ. By (4.19), we can write
  • Ψ21=μφ.
By differentiating this and substituting (4.12) and (4.19), we obtain
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By (4.30), this implies
  • μ= 0
or equivalently
  • Ψ21= 0.
Let
  • Σ1J=μJφ,J> (n− 1).
By differentiation, we obtain
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which yield
  • μJ= 0
or equivalently
  • Σ1J= 0.
Since Ξ 1 2 is independent of j , we obtain
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Similar computation for Σ 2 J yields
  • Σ2j= Σ2J= 0,j 2(n− 1).
Summing up we obtain the following:
For any contact rank 2 local CR embedding f from S n,1 into Sp,q , there is a choice of frames such that
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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
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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
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be as in Lemma 4.1.
It follows from Lemma 3.2 and (4.31) that
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in particular, the span of
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is independent of the point in M . Hence as in Lemma 4.1, we conclude
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Furthermore, (4.8) implies
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In particular, restricting to α = 1 and J = j n with (4.31)-(4.34) and (4.37) taken into account, we obtain
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Repeating the above argument for λ and ζ instead of η , we obtain
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and
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Thus each of the vector valued functions
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for a fixed K and
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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
  • λ12= 0
and
  • ηK=ζβ= 0,K>n,β> 1.
Hence (4.35) implies
PPT Slide
Lager Image
Similar computation for
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implies
PPT Slide
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Now setting
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we still have
PPT Slide
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whereas (4.38), (4.39) become
PPT Slide
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implying
PPT Slide
Lager Image
Then together with (4.36) we conclude that
PPT Slide
Lager Image
where
  • V1= span {Z1,X1,…,Xn−1,Y1},V2= span {Z2,Xn,…,X2n−2,Y2},V3= span {Z3,…,Zq}
PPT Slide
Lager Image
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 ) =
PPT Slide
Lager Image
. 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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