Classification of Moebius homogeneous Wintgen ideal submanifolds

Classification of Möbius Homogeneous Wintgen Ideal Submanifolds

arXiv:1402.3430v1 [math.DG] 14 Feb 2014

Tongzhu Li

Xiang Ma

Changping Wang

Zhenxiao Xie

Abstract A submanifold f : M m → Qm+p (c) in a real space form attaining equality in the DDVV inequality at every point is called a Wintgen ideal submanifold. They are invariant objects under the Möbius transformations. In this paper, we classify those Wintgen ideal submanifolds of dimension m ≥ 3 which are Möbius homogeneous. There are three classes of non-trivial examples, each related with a famous class of homogeneous minimal surfaces in S n or CP n : the cones over Veronese surfaces S 2 → S n , the cones over homogeneous flat minimal surfaces R2 → S n , and the Hopf bundle over the Veronese embeddings CP 1 → CP n .

2000 Mathematics Subject Classification: 53A30, 53A55; Key words: Möbius homogeneous submanifolds, DDVV inequality, Wintgen ideal submanifolds, Veronese surfaces, reduction theorem

1

Introduction

A central theme in geometry is to find and characterize those best shapes. It often means to find the optimally immersed submanifolds in a fixed ambient space. Two widely used optimality criteria are the minimization of certain functional(s), and the existence of many symmetries. From this viewpoint, homogeneous minimal surfaces in real space forms are best submanifolds, which include the Veronese surfaces S 2 → S 2k and Clifford type surfaces R2 → S 2k+1 [3, 15]. In complex space forms there are similar examples [1, 7]. In this paper we will consider Möbius homogeneous, Wintgen ideal submanifolds in Möbius geometry, which might be regarded also as best submanifolds according to both criteria. To our happy surprise, the classification shows that they are closely related with those homogeneous minimal surfaces mentioned above. To explain what is a Wintgen ideal submanifold, note that an optimality criterion is to consider a universal inequality and find out all the cases when the equality is achieved, which is somewhat similar to the minimization criterion. In this spirit, we are interested in the equality case in the so-called DDVV inequality [10] for a generic submanifold f : M m → Qm+p (c) in a real space form. This inequality is remarkable because it relates the most important intrinsic and extrinsic quantities at an arbitrary point x ∈ M , without any restriction on the dimension/codimension or any further geometric/topological assumptions. This universal inequality was a difficult conjecture in [10, 11], and was finally proved in [12] and [22]. By the suggestion of [6, 24] and the characterization of [12] about the equality case at an arbitrary point, we make the following definition. 0

T. Li, X. Ma, C.P. Wang are partially supported by the grant No. 11171004 of NSFC.

1

Möbius homogeneous Wintgen ideal submanifolds

2

Definition 1.1. We denote f : M m −→ Qm+p (c) a submanifold of dimension m and

codimension p in a real space form of constant sectional curvature c. It is a Wintgen ideal submanifold if the equality is attained at every point of M m in the DDVV inequality. This happens if, and only if, at every point x ∈ M there exists an orthonormal basis {e1 , · · · , em } of the tangent plane Tx M m and an orthonormal basis {n1 , · · · , np } of the normal plane Tx⊥ M m , such that the shape operators {Ani , i = 1, · · · , p} take the form as below [12]:     λ1 µ 0 0 · · · 0 λ2 + µ0 0 0 ··· 0 µ0 λ1 0 · · · 0   0 λ2 − µ0 0 · · · 0       0 0 λ1 · · · 0   0 0 λ2 · · · 0  (1.1) An1 =   , An2 =  ,  ..  .. .. . . ..  .. .. .. . . ..  .   . . . . . . . . . 0 0 0 · · · λ1 0 0 0 · · · λ2 An3 = λ3 Im ,

Anr = 0, r ≥ 4.

Wintgen ideal submanifolds are abundant. Wintgen first proved the DDVV inequality for surfaces in S4 , and characterized the equality case [26]. More general, a surface f : M 2 −→ Q2+p of arbitrary codimension p is Wintgen ideal exactly when the curvature c ellipse is a circle at every point [13], which is also equivalent to the Hopf differential being isotropic. For more examples see [2, 8, 9, 17, 19]. An important observation [11, 9] is that the DDVV inequality as well as the equality case are invariant under Möbius transformations of the ambient space. Thus it is appropriate to put the study of Wintgen ideal submanifolds in the framework of Möbius geometry. It follows that Wintgen ideal submanifolds in the sphere Sm+p or hyperbolic space Hm+p are the pre-image of a stereographic projection of Wintgen ideal submanifolds in Rm+p . For the same reason it is no restriction when we describe them in the Euclidean space. Since there are still many possible examples of Wintgen ideal submanifolds up to Möbius transformatons, it is natural to restrict to the best examples with many symmetries, called Möbius homogeneous submanifolds. This means that for f : M m −→ Qm+p (c) and arbitrary two points x1 , x2 ∈ M m , there exists a Möbius transformation φ of Qm+p (c) satisfying φ ◦ f (x1 ) = f (x2 ) and φ ◦ f (M m ) = f (M m ). Such a submanifold is an orbit of a subgroup in the Möbius transformation group. Our goal in this paper is to classify Möbius homogeneous Wintgen ideal submanifolds of dimension m ≥ 3 in Rm+p . Below are some examples.

Example 1.2. Let f : S 2 → S 2m (m ≥ 2) be one of the Veronese surfaces mentioned

at the beginning. As is well-known, such examples are totally isotropic and homogeneous with respect to the isometry group of S 2m . So they are Möbius homogeneous Wintgen ideal submanifolds. They come from the irreducible orthogonal representations of SO(3).

Example 1.3. Let f : R2 → S 2m−1 be a Clifford-type surface. That means it is homogeneous, flat, and minimal. It comes from a subgroup of the maximal torus group T m → SO(2m). Following [3], it is given by f = (f 1 , · · · , f 2m )

(1.2)

f 2k−1 (x, y) = rk cos(x cos θk + y sin θk ), f 2k (x, y) = rk sin(x cos θk + y sin θk ), (1 ≤ k ≤ m)

where (rk , θk ) are m real numbers satisfying rk > 0, {θk } are distinctive modulo kπ, and 2 r12 + · · · + rm = 1, e2iθ1 r21 + · · · + e2iθm r2m = 0.


3

Clearly, the flat minimal surface f : R2 → S 2m−1 is an orbit of a 2-dimensional abelian subgroup of SO(2m). By direct computation we know that f : R2 → S 2m−1 is Wintgen ideal if, and only if, e4iθ1 r21 + · · · + e4iθm r2m = 0.

Example 1.4. Let f : CP 1 → CP m be a Veronese 2-sphere [1]. Let π : S 2m+1 → CP m

be the projection map of the Hopf bundle. Then π −1 ◦ f : CP 1 → S 2m+1 is a Möbius homogeneous Wintgen ideal submanifold [10]. It comes from the irreducible unitary representations of SU (2). When m is an even number, this submanifold factors as an embedded SO(3) = RP 3 ; otherwise it is an embedded SU (2) = S 3 .

Example 1.5. The cone over an immersed submanifold u : M r −→ S r+p ⊂ Rr+p+1 is f : R+ × Rm−r−1 × M r −→ Rm+p , f (t, y, u) = (y, tu),

It is a Wintgen ideal submanifold if (and only if) u is a minimal Wintgen ideal submanifold in Sr+p . (See Section 4 for the proof.) Our main theorem is as below.

Theorem 1.6. Let f : M m −→ Rm+p (m ≥ 3) be a Möbius homogeneous Wintgen ideal

submanifold. Then locally f is Möbius equivalent to (i) a cone over a Veronese surface in S 2k , (ii) a cone over a Clifford-type surface in S 2k+1 , (iii) or a cone over π −1 ◦ f : CP 1 → S 2k−1 , (iv) or an affine subspace in Rm+p .

Our conclusions do not extend to Möbius homogeneous Wintgen ideal surfaces, i.e., when m = 2. In S 4 , any of them is Möbius equivalent to part of the Veronese surface. This follows from the classification of Willmore surfaces with constant Möbius curvature [23], or from an unpublished old manuscript by H. Li, F. Wu and the third author which gave a classification of all Möbius homogeneous surfaces in S 4 ). On the other hand, we can modify Example 1.3 to obtain homogeneous, Wintgen ideal (i.e., isotropic), isometric immersions R2 in S 5 which are not minimal. Whether there exist other kinds of examples are still unknown to us. In the rest part of this introduction, we give an overview of the proof and the whole structure of this paper. We start by reviewing the submanifold theory in Möbius geometry in Section 2. In Section 3 we restrict to Wintgen ideal submanifolds. Due to the specific, simple structure of the (Möbius) second fundamental form, we derive the explicit expressions of the Möbius invariants. From the statement of the main theorem one can see the importance of the construction by cones (Example 1.5). This is described in detail in Section 4. In particular, we show that the cone f is Wintgen ideal if and only if the original submanifold u in the sphere is minimal and Wintgen ideal. In Section 5, we start to utilize the assumption of Möbius homogeneity. The first structure result is that when the dimension m ≥ 3, the Möbius form Φ of a Möbius homogeneous


4

Wintgen ideal submanifold vanishes. This is proved by contradiction and detailed analysis of the Möbius invariants using the integrable equations. One crucial ingredient in the discussions of Section 5 and 6 is to re-choose the tangent and normal frames so that the normal connection takes an elegant form. This is the main content of Lemma 5.3, Proposition 6.1 and 6.2. In Section 7 we prove a somewhat surprising reduction result, namely, if the dimension m ≥ 3, the Möbius homogeneous Wintgen ideal submanifold is a cone over a surface or a three dimensional submanifold in Sm+p . In Section 8, we give the proof of our classification theorem. In particular, the only three dimensional Möbius homogeneous and Wintgen ideal examples which are not cones over surfaces must be given by the Hopf bundle over the Veronese surfaces in CP n .

2

Submanifolds theory in Möbius geometry

In this section we briefly review the theory of submanifolds in Möbius geometry. For details we refer to [25] and [18]. Let R1m+p+2 be the Lorentz space with inner product h·, ·i defined by hY, Zi = −Y0 Z0 + Y1 Z1 + · · · + Ym+p+1 Zm+p+1 , where Y = (Y0 , Y1 , · · · , Ym+p+1 ), Z = (Z0 , Z1 , · · · , Zm+p+1 ) ∈ Rm+p+2 . Let f : M m → Rm+p be a submanifold without umbilics and assume that {ei } is an orthonormal basis with respect to the induced metric I = df · df with {θi } the dual basis. Let {nα |1 ≤ α ≤ p} be a local orthonormal basis for the normal bundle. As usual we denote the second fundamental form and the mean curvature of f as II =

X ij,α

hαij θi ⊗ θj nα , H =

X 1 X α hjj nα = H α nα . m α j,α

We define the Möbius position vector Y : M m → R1m+p+2 of f by 2 1 + |f |2 1 − |f |2 1 m 2 . Y =ρ II − , ,f , ρ = tr(II)I 2 2 m−1 m

It is known that Y is a well-defined canonical lift of f . Two submanifolds f, f¯ : M m → Rm+p are Möbius equivalent if there exists T in the Lorentz group O(m + p + 1, 1) in R1m+p+2 such that Y¯ = Y T. It follows immediately that g = hdY, dY i = ρ2 df · df is a Möbius invariant, called the Möbius metric of f . Let ∆ be the Laplacian with respect to g. Define N =−

1 1 ∆Y − h∆Y, ∆Y iY, m 2m2

which satisfies hY, Y i = 0 = hN, N i, hN, Y i = 1 .

Let {E1 , · · · , Em } be a local orthonormal basis for (M m , g) with dual basis {ω1 , · · · , ωm }. Write Yj = Ej (Y ). Then we have hYj , Y i = hYj , N i = 0, hYj , Yk i = δjk , 1 ≤ j, k ≤ m.

Möbius homogeneous Wintgen ideal submanifolds We define ξα = H

α

1 + |f |2 1 − |f |2 , ,f 2 2

5

+ (f · nα , −f · nα , nα ) .

Then {ξ1 , · · · , ξp } be the orthonormal basis of the orthogonal complement of Span{Y, N, Yj |1 ≤ j ≤ m}. And {Y, N, Yj , ξα } form a moving frame in R1m+p+2 along M m .

Remark 2.1. Geometrically, ξα corresponds to the unique sphere tangent to M m at one

point x with normal vector nα and the same mean curvature H α (x). We call {ξα } the mean curvature spheres of M m .

We will use the following range of indices in this section: 1 ≤ i, j, k ≤ m; 1 ≤ α, β ≤ p. We can write the structure equations as below: X dY = ω i Yi , i

dN =

X

Aij ωi Yj +

ij

dYi = − dξα = −

X

Ciα ωi ξα ,

i,α

X j

X i

Aij ωj Y − ωi N + Ciα ωi Y −

X

X

ωij Yj +

j

X

α ω j ξα , Bij

j,α

α ωi Bij Yj +

ij

X

θαβ ξβ ,

β

where ωij are the connection 1-forms of the Möbius metric g and θαβ the normal connection 1-forms. The tensors X X X α A= Aij ωi ⊗ ωj , B = Bij ω i ⊗ ω j ξα , Φ = Cjα ωj ξα ij

ijα

jα

are called the Blaschke tensor, the Möbius second fundamental form and the Möbius form α are defined by of x, respectively. The covariant derivatives of Ciα , Aij , Bij X

α Ci,j ωj = dCiα +

j

X

X j

Aij,k ωk = dAij +

k

X

Cjα ωji +

X

X β

Aik ωkj +

k

α α ωk = dBij + Bij,k

k

X

Cjβ θβα ,

X

Akj ωki ,

k

α ωkj + Bik

k

X

α ωki + Bkj

k

X

β Bij θβα .

β

The integrability conditions for the structure equations are given by X α α α α Aij,k − Aik,j = (2.3) (Bik Cj − Bij Ck ), α

(2.4)

α Ci,j

(2.5)

α Bij,k

−

α Cj,i

X α α = Aki ), Akj − Bjk (Bik k

−

(2.6)

Rijkl =

(2.7)

⊥ Rαβij

α Bik,j

X α

= δij Ckα − δik Cjα ,

α α α ) + δik Ajl + δjl Aik − δil Ajk − δjk Ail , Bjl − Bilα Bjk (Bik

X β α α β = (Bik Bkj − Bik Bkj ). k


6

Here Rijkl denote the curvature tensor of g. Other restrictions on tensors A, B are X

(2.8)

X m−1 α 2 , (Bij ) = m

α Bjj = 0,

j

ijr

trA =

(2.9)

X

Ajj =

j

1 (1 + m2 κ). 2m

1 Where κ = n(n−1) obius scalar curvature. We know that all ij Rijij is its normalized M¨ coefficients in the structure equations are determined by {g, B} and the normal connection {θαβ }. Coefficients of Möbius invariants and the isometric invariants are also related by [25]

P

α Bij = ρ−1 (hαij − H α δij ), X Ciα = −ρ−2 [H,iα + (hαij − H α δij )ej (ln ρ)].

(2.10) (2.11)

j

3

Möbius invariants on Wintgen ideal submanifolds

A submanifold f : M m → Rm+p is a Wintgen ideal submanifold if and only if, at each point of M m , there is a suitable frame such that the second fundamental form has the form (1.1). If µ0 = 0 in (1.1), then the Wintgen ideal submanifold is totally umbilical submanifold. Next we consider non-umbilical Wintgen ideal submanifolds, that is µ0 6= 0 on M m and m ≥ 3. Since µ0 6= 0, we can choose a local orthonormal basis {E1 , · · · , Em } of T M m with respect to the Möbius metric g and a local orthonormal basis {ξ1 , · · · , ξp } of T ⊥ M m , such that the coefficients of the Möbius second fundamental form B have the form     0 µ 0 ··· 0 µ 0 0 ··· 0 µ 0 0 · · · 0  0 −µ 0 · · · 0      0 0 0 · · · 0  0 0 0 · · · 0 1 2 (3.12) B =  , B =   ; B α = 0, α ≥ 3.  .. .. .. . .   . . .. . .. . . . . .  .. . ..  . ..  . . 0

0 0 ···

0

0

0

0 ···

0

q m−1 By (2.8), the norm of B is constant and µ = 4m . Clearly the distribution D = span{E1 , E2 } is well-defined. For convenience we adopt the convention below on the range of indices: 1 ≤ i, j, k, l ≤ m, 3 ≤ a, b, c ≤ m, 1 ≤ α, β, γ ≤ p.

α . Since the M¨ First we compute the covariant derivatives of Bij obius second fundamental α , we have form B has the form (3.12), using the definition of the covariant derivatives of Bij

(3.13)

(3.14)

α α δ = 0, 1 ≤ δ ≤ p, 1 ≤ k ≤ m; B1a,i = 0, B2a,i = 0, α ≥ 3, Bab,k α α α α B12,1 B12,2 B11,1 B11,2 θ1α = ω1 + ω2 , θ2α = ω1 + ω2 , α ≥ 3. µ µ µ µ

ω2a =

1 X B1a,i i

µ

ωi = −

2 X B2a,i i

µ

ωi , ω1a =

1 X B2a,i i

µ

ωi =

2 X B1a,i i

µ

ωi .


2ω12 + θ12 =

(3.15)

1 X −B11,i

µ

i

ωi =

1 2 2 B12,i = 0, B11,i = B22,i = 0.

7

1 X B22,i i

µ

ωi =

2 X B12,i i

µ

ωi ,

It follows from (2.5) and (3.13) that, when α ≥ 3, α α α α C1α = Baa,1 − Ba1,a = 0, C2α = Baa,2 − Ba2,a = 0; α α α α α α Caα = B11,a − B1a,1 = B11,a , Caα = B22,a − B2a,2 = B22,a .

Since

P

δ i Bii,k

= 0, 1 ≤ δ ≤ p, 1 ≤ k ≤ m, we have Ciα = 0, α ≥ 3.

From (3.14) and (3.15), we obtain 1 1 2 2 2 B2a,2 = B22,a = B1a,2 , B1a,1 = 0, B2a,2 = 0. 1 1 = 0. Similarly Ca2 = 0. − B2a,2 This implies that Ca1 = B22,a r The other coefficients of {Cj } are obtained similarly as below:

(3.16)

2 1 = −µω2a (ea ), C22 = −B2a,a = µω2a (ea ), C11 = −B1a,a

1 2 C21 = −B2a,a = −µω1a (ea ), C12 = −B1a,a = −µω1a (ea ).

In particular we have C11 = −C22 , C21 = C12 .

(3.17)

Lemma 3.1. In the sub-bundles Span{E1 , E2 } and Span{ξ1 , ξ2 }, we can always choose new orthonormal basis {E1 , E2 } and {ξ1 , ξ2 } such that the Möbius second fundamental form B still takes the form (3.12), and the coefficients of the Möbius form satisfy C11 = −C22 , C21 = C12 = 0, Ca1 = Ca2 = 0, Ciα = 0, α ≥ 3. Proof. Under a new basis given as below: ˜1 = cos θE1 + sin θE2 , E ξ˜1 = cos ϕξ1 + sin ϕξ2 , ˜2 = − sin θE1 + cos θE2 , E ξ˜2 = − sin ϕξ1 + cos ϕξ2 , we have

C˜11 C˜ 2 1

=

sin(2θ + ϕ)µ cos(2θ + ϕ)µ cos(2θ + ϕ)µ − sin(2θ + ϕ)µ

˜2 ˜2 B cos(2θ + ϕ)µ B 12 11 = ˜2 B ˜2 − sin(2θ + ϕ)µ B 21 22 cos(θ + ϕ)C11 + sin(θ + ϕ)C21 C˜21 = 2 cos(θ + ϕ)C21 − sin(θ + ϕ)C11 C˜2

˜1 B ˜1 B 11 12 ˜1 B ˜1 B 21 22

− sin(2θ + ϕ)µ − cos(2θ + ϕ)µ

,

,

cos(θ + ϕ)C21 − sin(θ + ϕ)C11 − cos(θ + ϕ)C11 − sin(θ + ϕ)C21

Let ϕ = −2θ, then the coefficients of the Möbius second fundamental form B satisfy (3.12). Clearly there exists a value θ such that C˜21 = C˜12 = 0.

Lemma 3.2. We can choose a local orthonormal basis basis {E3 , · · · , Em } such that 1 1 1 B11,4 = B11,5 = · · · = B11,m = 0.

.


8

Pm 1 Proof. Let E = a=3 B11,a Ea . If E = 0,, then the Lemma is true. If E 6= 0, then ˜3 , · · · , E˜m } in Span{E3 , · · · , Em } such that we can choose a local orthonormal basis {E E 1 1 ˜3 = E |E| . Clearly, under this basis B11,4 = · · · = B11,m = 0 as desired. From (3.13), (3.14), (3.15), Lemma3.1 and Lemma3.2, we write out the connection forms with respect to the local orthonormal basis {E1 , E2 , · · · , Em }: 2ω12 + θ12 = (3.18)

ω13 ω23

1 B11,3 C11 ω1 − ω3 , µ µ

1 B11,3 ω2 , ω1i = 0, i ≥ 4, =− µ 1 B11,3 C1 C1 = ω1 − 1 ω3 , ω2i = − 1 ωi , i ≥ 4. µ µ µ

α , we have Combining C21 = 0 and the definition of Ci,j

C11 (ω12 + θ12 ) =

X

1 C2,k ωk .

k

Combining (3.18), we obtain that C11 ω12

(3.19)

1 X C11 B11,3 (C11 )2 1 ωk . ω1 − ω3 − C2,k = µ µ k

Using dωij − equations

1P

P

ωik ∧ωkj = − 2

X

R13kl ωk ∧ ωl =

k

k 0.

(B 1

)2

11,3 1 6= 0, from Proposition 5.6, A1 = 2µ Proof. If B11,3 > 0. Thus we need only to 2 1 consider the case when B11,3 = 0. 1 = 0, ω12 = 0, then R1212 = 0, i.e.,−2µ2 + 2A1 = 0. Thus A1 = µ2 > 0. If B11,3 1 = 0, ω12 6= 0. From Proposition 6.1, we have Consider the case B11,3

θ12 − (k + 1)ω12 = θ(2k+3)(2k+4) .

(7.80) From (5.53), we have (7.81)

2ω12 + θ12 = 0,

ω1i = 0, i ≥ 3,

ω2i = 0, i ≥ 3.

Thus, we have θ(2k+3)(2k+4) = −(k + 3)ω12 .

(7.82)

Using Proposition (6.1) and (7.82), we have X dθ(2k+3)(2k+4) = θ(2k+3)τ ∧ θτ (2k+4) = 2a2k ω1 ∧ ω2 = (k + 3)R1212 ω1 ∧ ω2 , τ

which implies (7.83)

2a2k = (k + 2)R1212 = (k + 3)(−2µ2 + 2A1 ).

Thus A1 > 0.

Proposition 7.2. Let f : M m −→ Rm+p (m ≥ 3) be a Möbius homogeneous Wintgen

1 = 0, then locally f is Möbius equivalent to a cone over a ideal submanifold. If B11,3 2 2+p surface u : M → S .


23

Proof. From (5.53), we have (7.84)

2ω12 + θ12 = 0,

ω1i = 0, i ≥ 3,

ω2i = 0, i ≥ 3.

Since dωa ≡ 0, mod{ω3 , · · · , ωm }, a ≥ 3, D = span{E1 , E2 } is integrable. Using (7.84) and Proposition 5.6, we have X dξ1 = −µω1 Y2 − µω2 Y1 + θ1α ξα , α=1

(7.85)

dξ2 = −µω1 Y1 + µω2 Y2 + dξα = −θ1α ξ1 − θ2α ξ2 +

(7.86)

X

θ2α ξα ,

α=1

X

θαβ ξβ .

β

dY1 = ω12 Y2 − ω1 (A1 Y + N ) + µω2 ξ1 + µω1 ξ2 ,

dY2 = −ω12 Y1 − ω2 (A1 Y + N ) + µω1 ξ1 − µω2 ξ2 ,

d (A1 Y + N ) = 2A1 [ω1 Y1 + ω2 Y2 ] From (7.85) and (7.86), we know that the subspace

V = span{(A1 Y + N ), Y1 , Y2 , ξ1 , ξ2 , · · · , ξp } is parallel along M m . The orthogonal complement V ⊥ also is parallel along M m . In fact, V ⊥ = span{(A1 Y − N ), Y3 , · · · , Ym }. Using (7.84), we can obtain (7.87)

d(A1 Y − N ) = 2A1

X a≥3

ωa Ya , dYa = ωa (A1 Y − N ) +

X

ωab Yb ,

b≥3

a ≥ 3.

Since dω1 = ω12 ∧ ω1 , dω2 = −ω12 ∧1 , the distribution D⊥ = span{E3 , · · · , Em } also is integrable. From (7.85) and (7.86), we know that the mean curvature spheres ξ1 , ξ2 induce 2-dimensional submanifolds in the de sitter space S1m+p+1 ξ1 , ξ2 : M 2 = M m /F −→ S1m+p+1 , where fibers F are integral submanifolds of distribution D⊥ . In other words, ξ1 , ξ2 form 2-parameter family of (m + p − 1)-spheres enveloped by f : M m −→ Rm+p . Since h(A1 Y + N ), (A1 Y + N )i = 2A1 > 0, V is a fixed space-like subspace, V ⊥ is a fixed Lorentz subspace in R1m+p+2 . We can assume that V = R3+p , V ⊥ = Rm−1 . From 1 (7.85) and (7.86), we know 1 u = √ (A1 Y + N ) : M 2 → S2+p . A1 On the other hand, the equation (7.87) implies that 1 φ = √ (A1 Y − N ) : Hm−2 → Rm−1 1 A1 is the standard embedding of the hyperbolic space Hm−2 in Rm−1 . Then 1 p Y = 2 A1 (u, φ) : M 2 × Hm−2 → S2+p × Hm−2 ⊂ R1m+p+2 ,

where φ : Hm−2 → Hm−2 is the identity map. From Proposition 4.3, we know that f is a cone over u : M 2 → S2+p . We complete the proof of Proposition 7.2.


24


1 6= 0, then locally f is Möbius equivalent to a cone over a three ideal submanifold. If B11,3 dimensional Möbius homogeneous Wintgen ideal submanifold in S3+p .

Proof. From (5.53) and Proposition 5.6, we have

(7.88)

1 1 1 −B11,3 B11,3 −B11,3 ω3 , ω13 = ω2 , ω23 = ω1 , µ µ µ ω1i = 0, i ≥ 4, ω2i = 0, i ≥ 4, ω3i = 0, i ≥ 4.

2ω12 + θ12 =

Since dωa ≡ 0, mod{ω4 , · · · , ωm }, a ≥ 4, D = span{E1 , E2 , E2 } is integrable, Using (7.88) and Proposition 5.6, we have X dξ1 = −µω1 Y2 − µω2 Y1 + θ1α ξα , α=1

(7.89)

dξ2 = −µω1 Y1 + µω2 Y2 + dξα = −θ1α ξ1 − θ2α ξ2 +

X

θ2α ξα ,

α=1

X

θαβ ξβ .

β

dY1 = −ω1 (A1 Y + N ) + ω12 Y2 + ω13 Y3 + µω2 ξ1 + µω1 ξ2 ,

(7.90)

dY2 = −ω2 (A1 Y + N ) − ω12 Y1 + ω23 Y3 + µω1 ξ1 − µω2 ξ2 ,

dY3 = −ω3 (A1 Y + N ) − ω13 Y1 − ω23 Y2 ,

d (A1 Y + N ) = 2A1 [ω1 Y1 + ω2 Y2 + ω3 Y3 ].

From (7.89) and (7.90), we know that the subspace V = span{(A1 Y + N ), Y1 , Y2 , Y3 , ξ1 , ξ2 , · · · , ξp } is parallel along M m . The orthogonal complement V ⊥ also is parallel along M m . In fact, V ⊥ = span{(A1 Y − N ), Y4 , · · · , Ym }. Using (7.84), we can obtain (7.91)

d(A1 Y − N ) = 2A1

X a≥4

ωa Ya , dYa = ωa (A1 Y − N ) +

X b≥4

ωab Yb , a ≥ 4.

Since h(A1 Y + N ), (A1 Y + N )i = 2A1 > 0, V is a fixed space-like subspace. Like as the proof of Proposition 7.2, we can prove that f is Möbius equivalent to a cone over a three dimensional Wintgen ideal submanifold u : M 3 → S3+p . Since f is Möbius homogeneous, clearly u : M 3 → S3+p is also Möbius homogeneous.

8

Proof of the Main theorem


1 = 0, then locally f is Möbius equivalent to ideal submanifold. If B11,3

(i) a cone over a Veronese surface in S2k , (ii) a cone over a flat minimal surface in S2k+1 .


25

Proof. From Proposition 7.2, we know that f is a cone over u : M 2 → S2+p . Since the Möbius form of f vanishes, from Proposition 4.4, we know that the Möbius form of the surface u vanishes. The surfaces with vanishing Möbius form is classified in [14]. We complete the proof to Proposition 8.1. 1 If B11,3 6= 0, by Proposition 7.3 we need only to consider three dimensional Möbius homogeneous Wintgen ideal submanifolds in S 3+p .

Proposition 8.2. Let x : M 3 −→ S3+p be a Möbius homogeneous Wintgen ideal sub-

1 manifold. If B11,3 6= 0, then locally x is Möbius equivalent to the Möbius homogeneous Wintgen ideal submanifold given by Example 1.4.

Proof. Let σ : S3+p → R3+p the stereographic projection. From [18], we know that the submanifolds x : M 3 −→ S3+p and f = σ ◦ x : M 3 −→ R3+p have the same Möbius invariants, especially, the normal connection. From Proposition 6.2, we can assume f : M 3 → S2n+3 and there exists basis {ξ1 , ξ2 , · · · , ξ2n } such that the normal connection under this frame has the following forms θ(2k+1)(2k+3) = ak ω2 , θ(2k+1)(2k+4) = ak ω1 , θ(2k+1)α = 0, 2k + 5 ≤ α ≤ 2n, θ(2k+2)(2k+3) = −ak ω1 , θ(2k+2)(2k+4) = ak ω2 , θ(2k+2)α = 0, 2k + 5 ≤ α ≤ 2n, θ34 = θ12 − ω12 −

1 1 B11,3 B11,3 ω3 , θ(2k+3)(2k+4) = θ(2k+1)(2k+2) − ω12 − ω3 , µ µ

where k = 0, 1, 2, · · · , n − 2. 1 6= 0 From the preceding discussion, we know that when B11,3 ω3i = 0, i ≥ 4; (Aij ) = diag(A1 , A1 , A1 , −A1 , · · · , −A1 ); (B 1

)2

11,3 . Without lost of generality, we assume where A1 = 2µ 2 following, we define A1 + N η= √ . 2A1

√

2A1 =

1 B11,3 µ

. = L. In the

In fact we have η : M 3 → S2n+3 , this follows from the following structure equations (8.92). Combining (7.89) and (7.90), with respect to the frame {η, Y3 , Y1 , Y2 , ξ1 , ξ2 , · · · , ξ2n−1 ξ2n } we can write out the structure equations as follows: 

(8.92)

η Y3 Y1 Y2 ξ1 ξ2 .. .





η Y3 Y1 Y2 ξ1 ξ2 .. .



                            d  = Θ ,                     ξ2n−1  ξ2n−1  ξ2n ξ2n


26

where Θ =  0 Lω3 Lω1 Lω2 0 0 0 0  −Lω3 0 −Lω2 Lω1 0 0 0 0   −Lω Lω2 0 ω12 µω2 −µω1 0 0 1   0 µω1 µω2 0 0  −Lω2 −Lω1 −ω12  0 −µω2 −µω1 0 θ12 a0 ω2 a0 ω1  0   0 0 µω1 −µω2 −θ12 0 −a0 ω1 a0 ω2   0 0 0 0 −a0 ω2 a0 ω1 0 θ34   0 0 0 0 −a ω −a ω −θ 0 0 1 0 2 34   .. .. .. .. .. .. .. ..  . . . . . . . . ~0 ~0 ~0 ~0 ~0 ~0 ~0 ~0

where

B=

~0 ~0 ~0 ~0 ~0 ~0 ~0 ~0

··· ··· ··· ··· ··· ··· ··· ··· .. .



                ~0  B

···

−an−2 ω2 an−2 ω1 0 θ2n−1 2n . −an−2 ω1 −an−2 ω2 −θ2n−1 2n 0

Denote the frame as a matrix T : M 3 → SO(2n + 4) with respect to a fixed basis {ek }2n+4 k=1 of R2n+4 , we can rewrite (8.92) as dT = ΘT.

(8.93)

The algebraic form of Θ motivates us to introduce a complex structure J on R2n+4 = SpanR {η, Y3 , Y1 , Y2 , ξ1 , ξ2 , · · · , ξ2n−1 , ξ2n } as below:      η 0 −1 η   Y3   Y3   1 0        Y1   Y1   0 −1        Y2   Y2   1 0         ξ1   ξ1   0 −1 J . =    ξ2   ξ2   1 0       .   .   .. .     ..   .      .  ξ2n−1   0 −1  ξ2n−1  1

ξ2n

0

ξ2n

Denote the diagonal matrix at the right hand side as J0 . Then the matrix representation of operator J under {ek }2n+4 k=1 is: J = T −1 J0 T.

Using dT = ΘT and the fact that J0 commutes with Θ, it is easy to verify dJ = −T −1 dT T −1 J0 T + T −1 J0 dT = −T −1 ΘJ0 T + T −1 J0 ΘT = 0.

So J is a well-defined complex structure on this R2n+4 . Another way to look at the structure equations (8.92) is to consider the complex version. We define Z1 = η + iY3 , Z2 = Y1 + Y2 , ζ1 = ξ1 − iξ2 , · · · , ζn = ξ2n−1 − iξ2n . Then the complex version of the equation (8.92) is dZ1 = −iLω3 Z1 + L(ω1 − iω2 )Z2 ,

dZ2 = −L(ω1 + iω2 )Z1 − iω12 Z2 + iµ(ω1 − iω2 )ζ1 ,

(8.94)

dζ1 = iµ(ω1 + iω2 )Z2 + iθ12 ζ1 + ia0 (ω1 − iω2 )ζ2 ,

dζk = iak−2 (ω1 + iω2 )ζk−1 + iθ2k−1 2k ζk

+ iak−1 (ω1 − iω2 )ζk+1 , 2 ≤ k ≤ n − 1,

dζn = ian−2 (ω1 + iω2 )ζn−1 + iθ2n−1 2n ζn .


27

Geometrically, this implies that Cn+2 = SpanC {Z1 , Z2 , ζ1 , ζ2 , · · · , ζn }, is a fixed n+2 dimensional complex vector space endowed with the complex structure i, which is identified with (R2n+4 , J) via the following isomorphism between complex linear spaces: v ∈ Cn+2 7→ Re(v) ∈ R2n+4 . For example, η + iY3 7→ η, iη − Y3 7→ −Y3 and so on. The second geometrical conclusion is an interpretation of (??) that [η + iY3 ] defines 2 a holomorphic mapping from the quotient surface M = M 3 /Γ to the projective space CP n+1 . Moreover, the unit circle in SpanR {η, Y3 } = SpanC {η + iY3 } is a fiber of the Hopf fibration of S2n+3 ⊂ (R2n+4 , J). In fact it corresponds to the subspace SpanR {Y, Yˆ , Y3 }, which is geometrically a leave of the foliation (M 3 , Γ). To see this, it follows from the equations of d(ξ2k−1 − iξ2k ) that {ξ1 , ξ2 , · · · , ξ2n , dξ1 , dξ2 , · · · , dξ2n } span a (2n+2)-dimensional spacelike subspace, the corresponding 2-parameter family of 3-dimensional mean curvature sphere congruence has an envelop M 3 , whose points correspond to the light-like directions in the orthogonal complement Span{Y, Yˆ , Y3 }. In particular [Y ], [Yˆ ] are two points on this circle and such circles form a 2-parameter family, with M as the parameter space, they give a foliation of M 3 which is also a circle fibration. Thus 2 the whole M 3 is the Hopf lift of M → CP n+1 . In other words we have the following commutative diagram η ⊂ / Cn+2 / S2n+3 ❄❄ ⑧ ❄❄ ⑧⑧ ⑧ ❄❄ ⑧ ❄❄ ⑧⑧ ⑧ ❄ ⑧ π ⑧ ❄❄ π M 3 /Γ ⑧⑧ [η+iY3 ] ❄❄ ⑧ ❄❄ ⑧ ❄❄ ⑧⑧ ❄❄ ⑧⑧ ⑧ ⑧ 2 / CP n+1 . M

M 3❄

2

Next we prove M → CP n+1 is the Veronese surface of CP n+1 . Note that any fibre of 2 the Hopf fibration has the S1 homogenous. So M → CP n+1 must be homogenous and hence has constant curvature. The conclusion follows from the classical results of Calabi([7] [1]). Combining Proposition 8.1, Proposition 7.3 and Proposition 8.2, we finish the proof of our main Theorem1.6.

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[22] Z. Lu, Normal Scalar Curvature Conjecture and its applications, Journal of Functional Analysis 261, 1284-1308(2011). [23] X. Ma, C. Wang, Willmore Surfaces of Constant Möbius Curvature, Ann. Glob. Anal. Geom. 32, no.3, 297-310 (2007). [24] M. Petrovié-torga˘sev, L.Verstraelen, On Deszcz symmetries of Wintgen ideal Submanifolds, Arch. Math., 44,57-67(2008). [25] C. P. Wang, Möbius geometry of submanifolds in S n , Manuscripta Math., 96,517534(1998). [26] P. Wintgen, Sur l’inégalité de Chen-Willmore, C. R. Acad. Sci. Paris, 288, 993995(1979). [27] Z. X. Xie, T. Z. Li, X. Ma, C. P. Wang, Möbius geometry of three dimensional Wintgen ideal submanifolds in S5 , Science China Mathematics, to appear, (2013).

Tongzhu Li, Department of Mathematics, Beijing Institute of Technology, Beijing 100081, People’s Republic of China. e-mail:[email protected] Xiang Ma, LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China. e-mail: [email protected] Changping Wang, School of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350108, People’s Republic of China. e-mail: [email protected]

Zhenxiao Xie, School of Mathematical Sciences, Peking University, Beijing 100871, People’s Republic of China. e-mail: [email protected]