Ann Glob Anal Geom https://doi.org/10.1007/s10455-017-9581-1

Wintgen ideal submanifolds: reduction theorems and a coarse classification Zhenxiao Xie1 · Tongzhu Li2 · Xiang Ma3 Changping Wang4

·

Received: 8 July 2017 / Accepted: 23 October 2017 © Springer Science+Business Media B.V. 2017

Abstract Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. As conformal invariant objects, they are suitable to study in the framework of Möbius geometry. This paper continues our previous work in this program, showing that Wintgen ideal submanifolds can be divided into three classes: the reducible ones, the irreducible minimal ones in space forms (up to Möbius transformations), and the generic (irreducible) ones. The reducible Wintgen ideal submanifolds have a specific low-dimensional integrable distribution, which allows us to get the most general reduction theorem, saying that they are Möbius equivalent to cones, cylinders, or rotational surfaces generated by minimal Wintgen ideal submanifolds in lower-dimensional space forms. Keywords Wintgen ideal submanifolds · DDVV inequality · Möbius geometry · Conformal Gauss map · Minimal submanifolds

Supported by the Fundamental Research Funds for Central Universities. Supported by NSFC 11601513, 11571037, 11471021 and 11331002.

B

Xiang Ma [email protected] Zhenxiao Xie [email protected] Tongzhu Li [email protected] Changping Wang [email protected]

1

Department of Mathematics, China University of Mining and Technology (Beijing), Beijing 100083, China

2

Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China

3

LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China

4

School of Mathematics and Computer Science and FJKLMAA, Fujian Normal University, Fuzhou 350108, China

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Mathematics Subject Classification 53A30 · 53A55 · 53C42

1 Introduction For a m-dimensional submanifold x : M m −→ Qm+ p (c) immersed in a real space form of dimension m + p with constant sectional curvature c, there holds a pointwise inequality (the celebrated DDVV inequality in [7]): s ≤ c + || H ||2 − s N

(1.1)

between its (normalized) scalar curvature s, the mean curvature || H || and the normal scalar curvature s N . In retrospect, the DDVV inequality first appeared for surfaces M 2 in S4 [23] (see also [11]). In this paper, Wintgen not only proved inequality (1.1) in this special situation, but also characterized the equality case. These extremal surfaces are also known as superconformal surfaces, coming from the projection of complex curves in the twistor space CP 3 of S4 [1]. Later, according to the suggestion in [2,20], those submanifolds attaining the equality in DDVV inequality (1.1) at every point are called Wintgen ideal submanifolds. (For different notions of ideal submanifolds, see [21] and references therein.) The significance of Wintgen ideal submanifolds was immediately recognized [6,7,17] after DDVV inequality (1.1) appeared in 1999 [7]. This topic became more important after Ge and Tang gave a complete proof to (1.1) in 2008 [10]. (See also an independent proof by Zhiqin Lu in [3,17].) Ge and Tang also showed that the equality holds at q ∈ M m if and only if there exist an orthonormal basis {e1 , . . . , em } of Tq M m and an orthonormal basis {n 1 , . . . , n p } ◦

◦

◦

of Tq⊥ M m such that the traceless second fundamental form { I In 1 , I In 2 , I Inr , 3 ≤ r ≤ p} has the simple form ⎞ ⎛ ⎛ ⎞ 0 0 μ0 μ0 ◦ ◦ ◦ ⎠ , I In 2 = ⎝ 0 −μ0 ⎠ , I Inr = 0. I In 1 = ⎝μ0 0 (1.2) 0 0 Here 0 denotes zero matrix. The examples of Wintgen ideal submanifolds include superconformal surfaces and totally umbilic submanifolds. See [5–7,12,14] for more examples. So far there are only partial results on their classification. In 2009, by an equivalent formulation of the DDVV inequality in [8], Dajczer and Tojeiro observed in [6] that the Wintgen ideal condition is an Möbius invariant property. This put the study of Wintgen ideal submanifolds in the most suitable framework, i.e., the Möbius geometry, and the classification should be up to Möbius transformations. In terms of Möbius invariant conditions using the Möbius submanifold theory established by the fourth author [22], we carried out this research program [12,13,15,18,25] in recent years. It is interesting to notice that, although Wintgen ideal submanifolds are generally not minimal submanifolds in space forms, they might be minimal up to Möbius transformations, or coming from specific constructions using minimal submanifolds in other spaces as building blocks [5,6,8]. Similar phenomenon is typical in the study of submanifolds in Möbius geometry [1,14,16]. In particular, two of such characterization theorems for Wintgen ideal submanifolds have been obtained in our work [15,25]. In this paper, we shall give a coarse classification of all Wintgen ideal submanifolds into three classes:

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Class 1 The reducible cases: M m is a cone, a cylinder or a rotational submanifold over a lowdimensional minimal Wintgen ideal submanifold Mˆ l in the space form Sl+ p , Rl+ p or Hl+ p , respectively. Class 2 The irreducible minimal cases: M m is Möbius equivalent to a minimal Wintgen ideal submanifold in a space form; at the same time it is not of Class 1. Class 3 The generic case: M m is neither Möbius equivalent to a minimal Wintgen ideal submanifold nor reducible to such an example of lower dimension. As a consequence, the study of Wintgen ideal submanifolds is now reduced to the minimal ones (a special case of the austere submanifolds [4]) and the generic (irreducible) ones. This classification is meaningful only if one can provide explicit criterion (in terms of Möbius invariant properties) to distinguish these three classes. Here we establish two characterization theorems for Wintgen ideal submanifolds of the first and the second classes separately. Along the way we build a canonical tangent and normal frame field on any Wintgen ideal submanifold, which would be very useful for further investigation. To state the reduction theorems, assume that there is no umbilic points on M m , and the dimension m ≥ 3, the codimension p ≥ 2. For such a Wintgen ideal submanifold M m , suppose the second fundamental form takes the desired form in (1.2) with respect to a specific frame field; then the distribution D := Span{e1 , e2 }

(1.3)

and the normal subbundle Span{n 1 , n 2 } are both well defined. We call D the canonical distribution of M m . D is not necessarily integrable in general. Let D denote the minimal integrable distribution containing D. Here minimal means that the rank is smallest possible. It is clear that D is the unique common subbundle of any integrable distribution generated by D, well defined at least on an open dense subset. In our previous exploration [15] we discovered that the information of D and D greatly determines the geometry of a Wintgen ideal submanifold. Below is the most general form of this reduction theorem, which is the main theorem in this paper. Theorem A Let x : M m → Rm+ p (m ≥ 3, p ≥ 2) be a Wintgen ideal submanifold without umbilic points, if dimD = l < m, then x is Möbius equivalent to (1) a cone over a l-dimensional minimal Wintgen ideal submanifold in Sl+ p , or (2) a submanifold of revolution over a l-dimensional minimal Wintgen ideal submanifold in Hl+ p , or (3) a cylinder over a l-dimensional minimal Wintgen ideal submanifold in Rl+ p . In [15], we obtained this theorem when l = 2 or 3, and conjectured that this is true for arbitrary l < m. To prove this tantalizing conjecture, the main difficulty is how to deal with various possibilities of rank(D) together with different choices of tangent frame vectors. Only after many attempts we find a unified treatment and simplify the tedious case-by-case discussions. The basic idea is as follows. Via the light-cone model of Möbius geometry, the submanifold x : M m → Sm+ p can m+ p+2 of dimension m + p + 2; at the be lifted to a submanifold in the Lorentz space R1 same time, the normal bundle Span{n 1 , . . . , n p } corresponds to Span{ξ1 , . . . , ξ p } where {ξr } m+ p+2 are orthonormal spacelike vectors in R1 . The geometric meaning of this spacelike pdimensional subspace at each point q ∈ M m is the unique p-sphere in Sm+ p tangent to M m at q, and sharing with M m the same mean curvature vector H (q). This construction was introduced by Blaschke and Bryant independently in the study of the conformal geometry of (Willmore) surfaces, also known as the mean curvature sphere.

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Definition 1.1 The mapping from q ∈ M m to the spacelike subspace Span{ξ1 (q), . . . , ξ p (q)} m+ p+2 is called the conformal Gauss map. The target space is the Grassmannian Gr p (R1 ) consisting of spacelike p-planes. Like the classical Gauss map, this conformal Gauss map is expected to encode some important information of the original submanifold x : M m → Sm+ p . To obtain these knowledge, we differentiate the frame vectors {ξ1 , . . . , ξ p } repeatedly. This helps to construct a canonical frame and define some nice invariants. A crucial consequence of the Wintgen ideal condition is that the conformal Gauss map has its image as a two-dimensional surface M [13,18,25]. Moreover, we obtain a pair of new frame vectors after each differentiation. By taking higher and higher-order derivatives of {ξr }, we obtain the jet bundle(s) V j ( j > 0) associated with the conformal Gauss map, which is spanned by {ξ1 , ξ2 , . . . , ξ p } and their partial derivatives up to order j. We denote the highest order jet bundle V :=

∞ 

Vj.

j=0

It is clearly a trivial bundle over M m . Moreover, we show that rank(V) = 1 + p + k for some integer k (2 ≤ k < m) if and only if rank(D) = k (Proposition 6.1). The building process of the jet bundle together with the decomposition into constant subspaces V ⊕ V⊥ leads to the proof of reduction theorem A. A Wintgen ideal submanifold is called irreducible when rank(D) = m, or equivalently, rank(V) ≥ 1 + p + m. The jet bundle V now has a natural stratification and a canonically chosen new frame, which allow us to define a Möbius invariant 1-form ω. See Sect. 5 for the construction of this new frame and the definition of this connection 1-form ω. Then we have the following theorem characterizing the second class. Theorem B Let x : M m → Sm+ p be an irreducible Wintgen ideal submanifold without umbilic points (hence dimD = m). Then the 1-form ω is closed if and only if x is Möbius equivalent to a minimal submanifold in a space form. In particular, in this case we have rank(V) = 1 + p + m. Previously we discussed the specific case of m = 3 and p = 2 in [25] and proved Theorem B. This paper completes the proof in the general case, which follows from the construction of the canonical frame in Sect. 5 in a straightforward manner. To better appreciate the geometric meaning of our results, we emphasize two structure results on any Wintgen ideal submanifold M m established already in our previous work [18,25]: Fact 1 M m has a Riemannian submersion structure over a Riemann surface M whose horizontal distribution is exactly the canonical distribution D. Fact 2 Every vertical fiber of this submersion is part of a round sphere of dimension m − 2. The union of these round spheres extends the original M m to be a larger Wintgen ideal submanifold. The integrability problem of the horizontal distribution D now seems a very natural question on the submersion structure based on Fact 1. On the other hand, the reduction to cones, cylinders and rotational surfaces in Theorem A clearly illustrates the round sphere fiber structure in Fact 2. This paper is organized as follows. In Sect. 2, we briefly review the submanifold theory in Möius geometry. Section 3 gives the information on the Möbius invariants and

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the structure equations of Wintgen ideal submanifolds. The higher-order derivatives of Span{ξ1 , ξ2 , . . . , ξ p } are calculated in Sect. 4. This uncovers a stratification structure of the jet bundle V, based on which we introduce a new moving frame for any Wintgen ideal submanifold in Sect. 5. We prove Theorems A and B in Sects. 6 and 7, respectively. Then we conclude this paper by a discussion on generic Wintgen ideal submanifolds in Sect. 8.

2 Submanifold theory in Möbius geometry In this section we briefly review the theory of submanifolds in Möbius geometry. For details we refer to [22], [16]. Recall that in the classical light-cone model, the light-like (space-like) directions in the m+ p+2 Lorentz space R1 correspond to points (hyperspheres) in the round sphere Sm+ p , and the Lorentz orthogonal group correspond to conformal transformation group of Sm+ p . The Lorentz metric is written out explicitly as Y, Z = −Y0 Z 0 + Y1 Z 1 + · · · + Ym+ p+1 Z m+ p+1 , m+ p+2

. for Y = (Y0 , Y1 , . . . , Ym+ p+1 ), Z = (Z 0 , Z 1 , . . . , Z m+ p+1 ) ∈ R1 Let x : M m → Sm+ p ⊂ Rm+ p+1 be a submanifold without umbilics. Take {ei |1 ≤ i ≤ m} as the tangent frame with respect to the induced metric I = d x · d x, and {θi } as the dual 1-forms. Let {nr |1 ≤ r ≤ p} be an orthonormal frame for the normal bundle. The second fundamental form and the mean curvature of x are   1  r II = h ri j θi ⊗ θ j nr , H = h j j nr = H r nr , (2.1) m r i j,r

j,r

m+ p+2

respectively. We define the Möbius position vector Y : M m → R1 Y = ρ(1, x),

ρ2 =

of x by

2   1 m   . tr (I I )I I I −   m−1 m

(2.2)

Y is called the canonical lift of x [22]. Two submanifolds x, x¯ : M m → Sm+ p are Möbius equivalent if there exists T in the Lorentz group O(m + p + 1, 1) such that Y¯ = Y T. It follows immediately that g = dY, dY = ρ 2 d x · d x

(2.3)

is a Möbius invariant, called the Möbius metric of x. Let  be the Laplacian with respect to g. Define N =−

1 1 Y, Y Y, Y − m 2m 2

(2.4)

which satisfies Y, Y = 0 = N , N , N , Y = 1 . Let {E 1 , . . . , E m } be a local orthonormal frame for (M m , g) with dual 1-forms {ω1 , . . . , ωm }. Write Y j = E j (Y ). Then we have Y j , Y = Y j , N = 0, Y j , Yk = δ jk , 1 ≤ j, k ≤ m.

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We define ξr = (H r , nr + H r x). Then {ξ1 , . . . , ξ p } form the orthonormal frame of the orthogonal complement of Span{Y, N , m+ p+2 Y j |1 ≤ j ≤ m}. And {Y, N , Y j , ξr } is a moving frame in R1 along M m . Remark 2.1 Geometrically, at one point x, ξr corresponds to the unique sphere tangent to  is a M m with normal vector nr and the same mean curvature H r = ξr , g where g = (1, 0) constant time-like vector. Remark 2.2 More generally, a k-dimensional spacelike subspace U k (i.e., the inner prodm+ p+2 is positive definite on U k ) corresponds to a codimension-k uct induced from R1 round sphere; the points of this corresponding sphere are nothing else but the light-like lines in the orthogonal complement of this U k . In particular, the conformal normal space Span{ξ1 , · · · , ξ p } is called the mean curvature sphere of the submanifold at one point, which corresponds to the unique sphere tangent to M m with the same mean curvature vector H . This is the conformal Gauss map as in Definition 1.1. We fix the range of indices in this section as follows: 1 ≤ i, j, k ≤ m; 1 ≤ r, s ≤ p. The structure equations are:  ωi Yi , dY = i

dN =

 ij

dYi = −

Ai j ωi Y j +



 i,r

Ai j ω j Y − ωi N +

j

dξr = −



Cir ωi ξr ,

Cir ωi Y

−



i



ωi j Y j +

j

ωi Birj Y j

+





Birj ω j ξr ,

(2.5)

j,r

θr s ξs ,

s

ij

where ωi j are the connection 1-forms of the Möbius metric g, and θr s are the normal connection 1-forms. The tensors    A= Ai j ωi ⊗ ω j , B = Birj ωi ⊗ ω j ξr ,  = C rj ω j ξr (2.6) ij

i jr

jr

are called the Blaschke tensor, the Möbius second fundamental form and the Möbius form of x, respectively. The covariant derivatives Ai j,k , Birj,k , Ci,r j are defined as usual. For example,    Ci,r j ω j = dCir + C rj ω ji + C sj θsr , j



s

j

Birj,k ωk

=

d Birj

k

+



r Bik ωk j

+

k

The integrability conditions are:  r r Ai j,k − Aik, j = Bik C j − Birj Ckr ,

 k

Bkr j ωki +



Bisj θsr .

s

(2.7)

r

Ci,r j − C rj,i =

 r (Bik Ak j − B rjk Aki ), k

123

(2.8)

Ann Glob Anal Geom r r r Birj,k − Bik, j = δi j C k − δik C j ,  r r Ri jkl = Bik B jl − Bilr B rjk + δik A jl + δ jl Aik − δil A jk − δ jk Ail ,

(2.9) (2.10)

r

Rr⊥si j =



r s s r Bik Bk j − Bik Bk j .

(2.11)

k

Here Ri jkl denote the curvature tensor of g. We recognize that the first three groups of equations correspond to the Codazzi equation; the last two (2.10) and (2.11) correspond to the Gauss and Ricci equation, respectively. Other restrictions on the tensor B are   m−1 B rj j = 0, (Birj )2 = . (2.12) m j

i jr

All coefficients in the structure equations are determined by {g, B} and the normal connection {θαβ }. Coefficients of Möbius invariants and the isometric invariants are related as follows. (We omit the formula for Ai j since it will not be used later.) Birj = ρ −1 (h ri j − H r δi j ),  (h ri j − H r δi j )e j (ln ρ)]. Cir = −ρ −2 [H,ir +

(2.13) (2.14)

j m+ p+2

Remark 2.3 For x : M m → Rm+ p , the Möbius position vector Y : M m → R1 mean curvature sphere {ξ1 , . . . , ξ p } are given by

1 + |x|2 1 − |x|2 Y =ρ , ,x , 2 2 ξr =

1 + |x|2 1 − |x|2 , ,x 2 2

and the

H r + (x · nr , −x · nr , nr ).

 is a constant light-like vector. For x : M m → Note that H r = ξr , g where g = (−1, 1, 0) m+ p+1 m+ p ⊂ R1 (the hyperboloid model of the hyperbolic space), the corresponding forH mulae are Y = ρ(x, 1), ξr = (nr + H r x, H r ), r = 1, . . . , p.  1) is a constant space-like vector. The Möbius In this case H r = ξr , g where g = (0, invariants are related to the isometric invariants still by (2.13) and (2.14). It is easy to verify the following Möbius characterization of minimal submanifolds in space forms by Remark 2.1 and Remark 2.3. This classical result will be repeatedly used in Sect. 6 and 7. Theorem 2.4 Let x : M m → Sm+ p be an immersed submanifold, then x is Möbius equivalent to a minimal submanifold in the space form Qm+ p (c) if and only if there exists a m+ p+2 constant vector g ∈ R1 so that g is orthogonal with the mean curvature spheres of x, i.e., g, ξr = 0, 1 ≤ r ≤ p (see Remark 2.1 and 2.2). Moreover, we have (i) Qm+ p (c) = Sm+ p if and only if g is a constant time-like vector; (ii) Qm+ p (c) = Rm+ p if and only if g is a constant light-like vector; (iii) Qm+ p (c) = Hm+ p if and only if g is a constant space-like vector.

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3 Möbius invariant of Wintgen ideal submanifolds From now on, we assume that x : M m → S m+ p be an umbilic-free Wintgen ideal submanifolds unless it is stated otherwise. According to (1.2), (2.13) and (2.12), we can choose a suitable tangent frame {E 1 , E 2 , . . . , E m } and Möbius normal frame {ξ1 , ξ2 , . . . , ξ p } such that the Möbius second fundamental form B takes the form ⎛ ⎞ ⎛ ⎞  0 μ μ 0 1 2 ⎠ , B = ⎝ 0 −μ ⎠ , μ = m − 1 ; B α = 0, α ≥ 3. (3.1) B = ⎝μ 0 4m 0 0 Remark 3.1 The canonical distribution D = Span{E 1 , E 2 } and the plane-bundle Span{ξ1 , ξ2 } are well defined if (3.1) holds. This frame will be fixed up to the following simultaneous rotations:



cos t sin t cos 2t − sin 2t 1 , E 2 ) = (E 1 , E 2 ) (E , ( ξ1 , ξ2 ) = (ξ1 , ξ2 ) . (3.2) − sin t cos t sin 2t cos 2t Convention: We will adopt the convention below on the range of indices: 1 ≤ i, j, k, l ≤ m, 3 ≤ a, b ≤ m;

1 ≤ r, s ≤ p, 3 ≤ α, β ≤ p.

We compute the covariant derivatives of Birj and obtain

ω2a =

1  B1a,i

μ

i

r α α Bab,i = 0, B1a,i = B2a,i = 0,

(3.3)

1 1 2 2 = B21,i = 0, B11,i = B22,i = 0, B12,i

(3.4)

ωi = −

2ω12 + θ12 =

2  B2a,i

μ

i

1  −B11,i

μ

i

θ1α =

α  B12,i i

μ

ωi , ω1a =

ωi =

1  B2a,i i

1  B22,i i

ωi , θ2α =

μ

μ

ωi =

α  B11,i i

μ

ωi =

2  B1a,i

2  B12,i i

μ

i

μ

ωi , (3.5)

ωi ,

ωi .

(3.6)

(3.7)

By (2.9), Birj,k is symmetric for distinctive i, j, k. It follows from (3.3) to (3.6) that 1 1 μω1a (E b ) = B2a,b = Bab,2 = 0,

1 1 μω2a (E b ) = B1a,b = Bab,1 = 0 (a  = b);

2 1 1 = B2a,1 = B21,a = 0, μω1a (E 1 ) = B1a,1

2 1 1 μω2a (E 2 ) = −B2a,2 = B1a,2 = B21,a = 0;

2 1 1 1 = μω1a (E 2 ) = −μω2a (E 1 ) = μ(2ω12 + θ12 )(E a ) = B2a,2 = B22,a = −B11,a . B1a,2

Based on this information, we use (2.9) to compute Ci,r j as follows: 1 1 1 C11 = B22,1 − B21,2 = B22,1 ,

C11 C12

= =

123

1 Baa,1 2 Baa,1

− −

1 B1a,a 2 B1a,a

= =

1 −B1a,a , 2 −B1a,a ,

1 1 1 C21 = B11,2 − B12,1 = B11,2 ,

C21 C22

= =

1 Baa,2 2 Baa,2

− −

1 B2a,a 2 B2a,a

= =

1 −B2a,a , 2 −B2a,a ,

(3.8) (3.9) (3.10)

Ann Glob Anal Geom 1 1 Ca1 = B22,a − B2a,2 = 0,

2 2 Ca2 = B11,a − B1a,1 = 0,

C1α Caα

C2α Caα

= =

α Baa,1 α B11,a

−

α Ba1,a α B1a,1

= 0, α B11,a ,

=

α Baa,2 α B22,a

−

α Ba2,a α B2a,2

(3.11)

= 0,

(3.12)

α B22,a .

− = = − = (∀ a, α) (3.13)

α α Utilizing the fact i Bii,k = 0, we deduce from (3.3) that Ca = 0. By (3.4), (3.5) and (3.8)–(3.13), the final result is C11 = −C22 = −μω2a (ea ), C21 = C12 = −μω1a (ea ),

(3.14)

Ca1 = Ca2 = 0, Ciα = 0.

(3.15)

For similar reasons, (3.12) and (3.13) imply α α − B11,2 = −C2α = 0, μ[θ1α (E 1 ) − θ2α (E 2 )] = B12,1 α α α α μ[θ1α (E 2 ) + θ2α (E 1 )] = (B21,2 − B22,1 ) + (B22,1 + B11,1 ) = −C1α = 0.

Remark 3.2 From Remark 3.1, we can choose appropriately {E 1 , E 2 } and {ξ1 , ξ2 } so that C21 = C12 = 0. 1 =0 It is easy to see that we can also choose a suitable frame {E 3 , . . . , E m } so that B11,a when a ≥ 4. This is called an adapted frame, under which we have

2ω12 + θ12 = ω13 = − ω23 =

1 B11,3

μ

1 B11,3

μ

1 B11,3 C11 ω1 − ω3 , μ μ

(3.16)

ω2 ,

(3.17)

ω1 −

ω1σ = 0, σ ≥ 4, C11 C1 ω3 , ω2σ = − 1 ωσ , σ ≥ 4. μ μ

(3.18)

To calculate the Blaschke tensor A, note that C21 = 0, we have  1 C2,k ωk . C11 (ω12 + θ12 ) = k

It follows from (3.16) that C11 ω12 =

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

(3.19)

k

Taking differentiation to (3.17) and (3.18) we can get that 

R13kl ωk ∧ ωl =

1 d B11,3

μ

k
+ 

R1σ kl ωk ∧ ωl =

k
+

∧ ω2 +

1  C2,k k

1 )2 − (C 1 )2 (B11,3 1

μ2

μ

ωk ∧ ω3

ω1 ∧ ω3 ,

1 C11 B11,3 −(C11 )2 ω ∧ ω + ω3 ∧ ωσ 1 σ μ2 μ2 1  C2,k k

μ

ωk ∧ ωσ −

1 B11,3

μ

(3.20)

ω2 ∧ ω3σ , σ ≥ 4,

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R23kl ωk ∧ ωl =

1 −d B11,3

μ

k
+  k
∧ ω1 +

1  C1,k

μ

k

1 )2 − (C 1 )2 (B11,3 1

μ2

ωk ∧ ω3

ω2 ∧ ω3 −

1 2C11 B11,3

μ2

ω1 ∧ ω2 ,

1  C1,k −(C11 )2 R2σ kl ωk ∧ ωl = ω ∧ ω + ωk ∧ ωσ 2 σ μ2 μ

(3.21)

k

+

1 B11,3

μ

ω1 ∧ ω3σ ,

σ ≥ 4.

Combining these equations with Gauss equation (2.10), we can get that the Blaschke tensor A takes the form as follows ⎛ ⎞ A11 A12 A13 A14 · · · · · · A1m ⎜ A12 A22 A23 A24 · · · · · · A2m ⎟ ⎜ ⎟ ⎜ A13 A23 A33 0 · ·· ··· 0 ⎟ ⎜ ⎟ ⎜ A14 A24 0 A4 0 ··· 0 ⎟ ⎜ ⎟, (3.22) A=⎜ . ⎟ .. .. .. ⎜ .. ⎟ . 0 . . 0 A 4 ⎜ ⎟ ⎜ . ⎟ .. .. .. .. .. ⎝ .. . . 0 ⎠ . . . 0 0 ··· 0 A4 A1m A2m where A33 −

A1σ

μ2 1 C2,1

= Aσ σ = A4 , σ ≥ 4,

1 C1,2 (C11 )2 (C11 )2 − A , A = − A4 , − 4 22 μ μ2 μ μ2 1 1 ) 1 1 1 E 3 (B11,3 C11 B11,3 C1,1 C2,3 C1,3 + , A13 = + , = , A = 23 μ μ μ μ2 μ 1 1 C2,σ C1,σ = , A2σ = , σ ≥ 4. μ μ

A11 = A12

1 )2 (B11,3

−

(3.23)

In the following, for convenience, we denote L=−

1 B11,3

μ

, U=

α α B11,2 B11,1 C22 C1 = − 1 , Sα = , Tα = . μ μ μ μ

(3.24)

We summarize the most important information on the connection 1-forms as follows: Proposition 3.3 For a Wintgen ideal submanifold, we have ω13 = Lω2 , ω1σ = 0, σ ≥ 4;

(3.25)

ω23 = −Lω1 + U ω3 , ω2σ = U ωσ , σ ≥ 4;

(3.26)

2ω12 + θ12 = −U ω1 + Lω3 ;

(3.27)

θ1α = Sα ω1 − Tα ω2 , θ2α = Tα ω1 + Sα ω2 .

(3.28)

At the end of this section, taking (3.1), (3.15) and (3.24) into (2.5), we write down the structure equations of Wintgen ideal submanifold under the adapted frame {Y, N , Y1 , Y2 , . . . Ym , ξ1 , ξ2 , . . . , ξ p }

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as follows: dY =



ωk Y k ,

(3.29)

A jk ω j Yk − μU ω1 ξ1 + μU ω2 ξ2 ,

(3.30)

k

dN =

 j,k

dY1 = −



A1k ωk Y − ω1 N +

k

dY2 = −

 

ω1k Yk + μω2 ξ1 + μω1 ξ2 ,

(3.31)

ω2k Yi + μω1 ξ1 − μω2 ξ2 ,

(3.32)

ωak Yk ,

(3.33)

k

A2k ωk Y − ω2 N +

k

dYa = −

  k

Aak ωk Y − ωa N +

k

 k

dξ1 = −μ(Y2 − U Y )ω1 − μY1 ω2 + θ12 ξ2 +



θ1α ξα ,

(3.34)

θ2α ξα ,

(3.35)

α

dξ2 = −μY1 ω1 + μ(Y2 − U Y )ω2 − θ12 ξ1 + dξα = −θ1α ξ1 − θ2α ξ2 +



 α

θαβ ξβ .

(3.36)

β

4 The jet bundle of the conformal Gauss map The last three structure equations above show that the conformal Gauss map q ∈ M m → Spanq {ξ1 , . . . , ξ p } (see Definition 1.1) has its image as a 2-dimensional surface when M m is Wintgen ideal (see [18]). In this section we will study its geometry in more detail by moving frame method. Structure equation (3.36) tells us that dξα yields no new frame vectors. Thus we focus on the derivatives of ξ1 , ξ2 . Note that the plane-bundle Span{ξ1 , ξ2 } defines a second Gauss map from M m into the real m+ p+2 2-plane Grassmannian Gr 2 (R1 ) (consists of only spacelike 2-planes). The following m+ p+2 ) to the complex quadric: bijection is also well known, from this Grassmannian Gr 2 (R1 Qm+ p := {[ξ ] = [ξ1 − iξ2 ]|[ξ ] ∈ CP m+ p+1 , ξ, ξ = 0}. (Here we use the complex bilinear extension of the Lorentz inner product.) Inspired by this complex representation, we rewrite (3.34), (3.35) as follows: d(ξ1 − iξ2 ) = iμ(ω1 + iω2 )(η1 + iη2 ) + iθ12 (ξ1 − iξ2 ) +



(θ1α − iθ2α )ξα , (4.1)

α≥3

with η1 = Y1 , η2 = Y2 − U Y,

(4.2)

as the tangent frame vectors of either of the Gauss maps aforementioned. Taking one more differentiation, it follows from (3.22), (3.23), (3.30), (3.32) and (3.33) that

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d(η1 + iη2 ) = (ω1 + iω2 )[A4 Y − N − U η2 − i(LY3 − E 3 (L)Y )] −i(ω12 + U ω1 )(η1 + iη2 ) + iμ(ω1 − iω2 )(ξ1 − iξ2 ) ≡ (ω1 + iω2 )(A4 Y − N − i Lη3 ), mod(ξ1 , ξ2 , . . . , ξ p , η1 , η2 ) with

 η3 := Y3 − λ3 Y, wher e λ3 =

0, E 3 (L) L ,

L = 0, L  = 0.

(4.3)

(4.4)

Formulas (4.1) and (4.3) are interesting in the following sense. By computing the first and the second derivatives of Span{ξ1 , ξ2 }, we are investigating the geometric properties of the mean curvature sphere (congruence) which is an invariant object. In particular, we are deriving new frames and defining convenient invariants, which are both canonically defined and geometrically meaningful. We will compute higher-order derivatives of Span{ξ1 , ξ2 } in an inductive way. Before that let us introduce some notations. Definition 4.1 (1) x ∗ R1 is the pull back of the trivial bundle to M m . m+ p+2 . (2) V0 = Span{ξ1 , ξ2 , . . . , ξ p } is a subbundle of x ∗ R1 m+ p+2 ∗ (3) V j ( j > 0) is the subbundle of x R1 spanned by {ξ1 , ξ2 , . . . , ξ p } and their partial derivatives up to order j (the so-called jet bundle associated with the conformal Gauss map. (4) m+ p+2

V=

∞ 

Vj.

j=0

Remark 4.2 On an open dense subset of M m , these jet bundles are well defined (i.e., each of them has constant rank and is independent to the choice of the frames). In particular, V is a m+ p+2 m+ p+2 parallel subbundle of x ∗ R1 . So it describes a constant subspace of R1 containing {ξ1 , ξ2 , . . . , ξ p } at every point. In summary we have a filtration of the jet bundle as follows: V0  V1  V2  · · ·  V j = V j+1 = V. For j = 1, it follows from (4.1) and (3.36) that V1 = Span{ξ1 , ξ2 , . . . , ξ p , η1 , η2 }. Similarly, from (4.3) we have that if L = 0 , V2 = Span{ξ1 , ξ2 , . . . , ξ p , η1 , η2 , A4 Y − N }; Otherwise, when L  = 0 there is V2 = Span{ξ1 , ξ2 , . . . , ξ p , η1 , η2 , A4 Y − N , η3 }. Remark 4.3 The case L = 0 is special. Differentiating both sides of (4.3), we have 0 = d 2 (η1 + iη2 ) ≡ (ω1 + iω2 ) ∧ d(A4 Y − N ), (mod V2 ), it is easy to verify that this will yield no new frame and V = V2 . On the other hand, one can show that L = 0 if and only if the canonical distribution D itself is integrable [15]. This is the first hint of the close relationship between the jet bundles of the conformal Gauss map and the integrable distribution generated by D. See Proposition 6.1.)

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Convention. In the following section, we assume that L  = 0 and the dimension m ≥ 4. We denote VC k = Vk ⊗ C as the complexified bundle of Vk for any k ≥ 0. Lemma 4.4 There always exist sections {ψ j | j = 1, 2, . . .} of x ∗ R1 η1 , ψ2 = η2 , ψ3 = A4 Y − N , ψ4 = −Lη3 , and ∀ k ≥ 1,

m+ p+2

d(ψ2k−1 + iψ2k ) = (ω1 + iω2 )(ψ2k+1 + iψ2k+2 ),

such that ψ1 =

(mod VC k ).

(4.5)

In particular, that implies Vk+1 = Span{Vk , ψ2k+1 , ψ2k+2 } and 0 ≤ dimVk − dimVk−1 ≤ 2, ∀ k ≥ 1. Proof Note that the conclusion is true when k = 1 by (4.3). Suppose that for 1 ≤ j ≤ k + 1, the conclusion is true. This implies (4.5) and Vk+1 = Span{Vk , ψ2k+1 , ψ2k+2 }. Differentiating both sides of (4.5) will yield 0 = d 2 (ψ2k−1 + iψ2k ) ≡ d(ω1 + iω2 ) · (ψ2k+1 + iψ2k+2 ) − (ω1 + iω2 ) ∧ d(ψ2k+1 + iψ2k+2 ) (mod VC k+1 ) ≡ −(ω1 + iω2 ) ∧ d(ψ2k+1 + iψ2k+2 ) (mod VC k+1 ). Thus there must be d(ψ2k+1 + iψ2k+2 ) = (ω1 + iω2 )(ψ2k+3 + iψ2k+4 ),

(mod VC k+1 )

for some sections ψ2k+3 , ψ2k+4 . This completes the proof by induction on k.

 

Lemma 4.5 Suppose dimV j − dimV j−1 = 2 holds true for j = 1, 2, . . . , k − 1 and dimVk − dimVk−1 ≤ 1. Then Vk = V. Proof Using the frame and structure equations described in Lemma 4.4, we may write d(ψ2k−3 + iψ2k−2 ) = (ω1 + iω2 )(ψ2k−1 + iψ2k ) = (ω1 + iω2 ) · λψ2k−1 ,

(mod VC k−1 )

by the assumption dimVk − dimVk−1 = 1, where λ is a nonzero complex function locally defined. Differentiating at both sides, we get 0 = d 2 (ψ2k−3 + iψ2k−2 ) = −λ(ω1 + iω2 ) ∧ dψ2k−1 , (mod VC k ). Note that dψ2k−1 is a real vector-valued 1-form, which means that there must be dψ2k−1 ≡ 0 (mod Vk ). Hence Vk = Vk+1 = V.   Remark 4.6 When m ≥ 4, k ≥ 2 and dimVk − dimVk−1 ≥ 1, the sections ψ2k−1 , ψ2k can be chosen so that they are orthogonal to {ψ1 , ψ2 , Y, ψ4 , . . . , ψ2k−2 }. (These 2k − 2 sections are linearly independent by Lemma 4.5.) In particular, except the case j = 3 we always take ψ j ∈ Span{Y1 , Y2 , Y3 , . . . , Ym , Y }. To rewrite ψ j ’s under a new frame, we should take care of the degenerate case when Span{ψ2k−1 , ψ2k } = Span{Yn , Y } or Span{ψ2k−1 , ψ2k } = Span{Y } for certain tangent vector field Yn . This is clarified as follows. / (Vk−1 ). Lemma 4.7 Suppose there exists an integer k ≥ 3 such that Y ∈ (Vk ) and Y ∈ Then there must be Vk = V.

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Proof We prove this lemma by reduction to absurdity. Suppose the conclusion is false. Without loss of generality we may assume that the orthonormal tangent basis {Y1 , . . . , Ym } is chosen so that Y1 , Y2 , Y3 are fixed as before, Vk = Span{Y, A4 Y − N , Y1 , Y2 , Y3 , . . . , Yn , ξ1 , ξ2 , . . . , ξ p } with dimension n + p + 2 (n < m), and Vk = Vk−1 ⊕ Span{Yn , Y }. (The case Span{ψ2k−1 , ψ2k } = Span{Y } has been treated in Lemma 4.5.) Obviously Ym ⊥Vk . As before, there exist sections ψ2k−3 , ψ2k−2 ∈ (Vk−1 ) and nonzero complex functions a, b such that d(ψ2k−3 + iψ2k−2 ) = (ω1 + iω2 )(aYn + bY ) (mod VC k−1 ). Note that a, b  = 0 and c = a/b is not real-valued; otherwise there will be contradiction with the assumption Span{ψ2k−1 , ψ2k } = Span{Yn , Y }. Let ψ = b1 (ψ2k−3 + iψ2k−2 ). Then dψ = (ω1 + iω2 )(cYn + Y ) (mod VC k−1 ).

(4.6)

Differentiating both sides, we have 0 = d 2 ψ ≡ −(ω1 + iω2 ) ∧ (c · dYn + dY )(mod VC k ). Take inner product with Ym at both sides and denote dYn , Ym = ωnm , dY = The result is

m

j=1 ω j Y j .

0 = (ω1 + iω2 ) ∧ (cωnm + ωm ). So (cωnm + ωm ) = f (ω1 + iω2 ) for some complex function f . Pairing with the tangent vector E m will cancel the right-hand side and leave us with cωnm (E m ) + 1 = 0. But then c = −1/ωnm (E m ) is a real function. This is impossible as we pointed out after (4.6). This completes the proof.   Based on these lemmas and Remark 4.6, we can construct a new frame as follows in an inductive manner. Corollary 4.8 Suppose m + p + 2 > dimVk+1 ≥ dimVk + 1. Then ∀ 3 ≤ j ≤ k, by applying the Gram–Schmidt orthogonalization procedure to space-like vectors {ψ2 j−1 , ψ2 j }, one obtains two new orthonormal frame vectors η2 j−2 = Y2 j−2 − λ2 j−2 Y, η2 j−1 = Y2 j−1 − λ2 j−1 Y, So that



ψ2 j−1 = a2 j−2 η2 j−2 , a2 j−2  = 0, ψ2 j = b2 j−2 η2 j−2 + b2 j−1 η2 j−1 , b2 j−1  = 0.

(Notice that starting from ψ4 = −Lη3 , the subscripts of the corresponding frame vectors differ by 1 and ψ2 j−1  η2 j−2 .) We also have an orthogonal direct sum V j = V j−1 ⊕ Span{η2 j−2 , η2 j−1 }. Here Y2 j−2 , Y2 j−1 are unit tangent vectors orthogonal to {Y1 , Y2 , Y3 , . . . , Y2 j−3 }; λ2 j−2 , λ2 j−1 are real coefficients. Corollary 4.9 In case that V is the total space of the pull back bundle with dimension m + p + 2, there can be two possibilities: 1. If m + p + 2 = dimV = dimVk = dimVk−1 + 1, then Vk = Vk−1 ⊕ {Y }. Moreover we have dimVk = dimVk−1 + 1 = dimVk−2 + 3 = · · · = dimV0 + 2k − 1 = p + 2k − 1 and the dimension m = 2k − 3 is odd.

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2. If m + p + 2 = dimV = dimVk = dimVk−1 + 2, then similarly as Corollary 4.8, we can find a space-like vector ηm = Ym + λm Y so that Vk = Vk−1 ⊕ {ηm , Y },here Ym are unit tangent vectors orthogonal to {Y1 , Y2 , Y3 , . . . , Ym−1 } and λm is a real coefficient. Moreover we have dimVk = dimVk−1 + 2 = dimVk−2 + 4 = · · · = dimV0 + 2k = p + 2k, and the dimension m = 2k − 2 is even. Corollary 4.10 Given a Wintgen ideal submanifold x : M m → Sm+ p , the structure of the jet bundle V has the following possibilities when m ≥ 4: 1. dimV ≤ m + p, and V = Span{ξ1 , ξ2 , . . . , ξ p , A4 Y − N , η1 , η2 , η3 , . . . , ηl } with l ≤ m − 1. (This is the so-called reducible case. See Sect. 6.) 2. dimV = m + p + 1, and V = Span{ξ1 , ξ2 , . . . , ξ p , A4 Y − N , η1 , η2 , η3 , . . . , ηm }. (This is the irreducible minimal case. See Sect. 7.) m+ p+2 3. dimV = m + p + 2, and V = x ∗ R1 = Span{ξ1 , . . . , ξ p , N , η1 , . . . , ηm , Y }. (This is the so-called generic case. See discussions in Sect. 8. ) Remark 4.11 If dimV ≥ m + p + 1 , then we have a well-defined decomposition of the tangent bundle Span{E 1 , E 2 } ⊕ Span{E 3 } ⊕ Span{E 4 , E 5 } ⊕ · · · ⊕ Span{E 2[ m2 ] , E m }. In particular, under the additional condition that the Möbius form   = 0, we can find a canonical orthonormal tangent frame {E 1 , E 2 , E 3 , . . . , E m }. Similarly, we can differentiate ξ1 − iξ2 repeatedly and take the coefficient of the term ω1 − iω2 instead (the (0, 1)-component as in complex geometry). Under suitable conditions (e.g., the Möbius form  does not vanish), we can also get a decomposition of the conformal normal bundle V0 = Span{ξ1 , ξ2 , . . . , ξ p } Span{ξ1 , ξ2 } ⊕ Span{ξ3 , ξ4 } ⊕ · · · ⊕ Span{ξ2[ p+1 ]−1 , ξ p } 2

and a canonical frame {ξ1 , ξ2 , . . . , ξ p }. Remark 4.12 On the other hand, Wintgen ideal submanifolds with vanishing Möbius form can be reduced to those ones of dimension two or three with a nice description. See [24].

5 A new moving frame of Wintgen ideal submanifold In this section we will rewrite the structure equations of any Wintgen ideal submanifold x : M m → Sm+ p with respect to a new moving frame {Y, Yˆ ; ξ1 , ξ2 , . . . , ξ p ; η1 , η2 , η3 , . . . , ηm }.

(5.1)

Here {η j = Y j − λ j Y } are given as in Corollary 4.8 and Corollary 4.9, and instead of N we take  1 2 λk Y + λk Yk , Yˆ := N − 2 m

m

k=1

k=1

(5.2)

so that it is orthogonal to {η1 , . . . , ηm } and it satisfies Y, Yˆ = 1, Yˆ , Yˆ = Y, Y = 0.

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Ann Glob Anal Geom

This natural choice encodes the structure of the jet bundle V, and it can be defined in a canonical way. To unify our treatment in the first case in Corollary 4.10 where dimV ≤ m + p, we make the convention that η j = Y j − λ j Y, with λ j = 0 for any l + 1 ≤ j ≤ m

(5.3)

In the following, we always denote λ1 = 0, λ2 = U. Using (5.2), we can rewrite A4 Y − N as follows ˆ + Yˆ ) + A4 Y − N = −( FY

m 

λjηj,

(5.4)

j=1

where Fˆ = −A4 −

1 2

m

2 k=1 λk .

Remark 5.1 1. For any Wintgen ideal submanifold x : M m → Sm+ p , Yˆ is a well-defined  m of M m by vector along M m . In [18], we have introduced the Wintgen ideal extension M m taking the envelope of the mean curvature spheres. For any q ∈ M , [Yˆ (q)] is just another  m (in fact, located on the same spherical fiber; point dual to [Y (q)] located on the same M for more detail see [18]). Also note that Span{Y, Yˆ } defines a plane-bundle over M m . 2. It is easy to see that Fˆ is a well-defined Möbius scalar invariant. Using (3.29)–(3.36), the structure equations of x under this new frame are as follows dY = ωY +

m 

ωk ηk ,

k=1 m 

d Yˆ = −ωYˆ +

(5.5)

ωˆ k ηk .

(5.6)

k=1

dξ1 = −μω2 η1 − μω1 η2 + θ12 ξ2 + dξ2 = −μω1 η1 + μω2 η2 − θ12 ξ1 +

p  α=3 p 

θ1α ξα ,

(5.7)

θ2α ξα ,

(5.8)

α=3

dξα = −θ1α ξ1 − θ2α ξ2 +

p 

θαβ ξβ , 3 ≤ α ≤ p,

(5.9)

β=3

dη1 = −ωˆ 1 Y − ω1 Yˆ + dη2 = −ωˆ 2 Y − ω2 Yˆ + dη j = −ωˆ j Y − ω j Yˆ +

m  k=1 m  k=1 m 

1k ηk + μω2 ξ1 + μω1 ξ2 ,

(5.10)

2k ηk + μω1 ξ1 − μω2 ξ2 ,

(5.11)

 jk ηk , 3 ≤ j ≤ m;

(5.12)

λk ωk

(5.13)

k=1

where ω=

m  k=1

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Ann Glob Anal Geom

is the connection 1-form of the plane-bundle Span{Y, Yˆ } over M m and hence is well defined; {ωk } is the orthonormal coframe of [Yˆ ] : M m → Sm+ p ; i j = dηi , η j is the new connection 1-form. Comparing (4.3) with (5.9)–(5.10) we have ˆ 1 , ωˆ 2 = Fω ˆ 2. ωˆ 1 = Fω

(5.14)

By the definition and (3.25)–(3.27) we have 12 = ω12 + U ω1 ,

(5.15)

13 = λ3 ω1 + Lω2 , 23 = −Lω1 + λ3 ω2 ,

(5.16)

1σ = λσ ω1 , 2σ = λσ ω2 , λσ ≥ 4,

(5.17)

θ12 + 212 = U ω1 + Lω3 .

(5.18)

Differentiating (5.5)–(5.11), we can get the following integrability equations: dω1 = −(θ12 + 12 ) ∧ ω2 ;

(5.19)

dω2 = (θ12 + 12 ) ∧ ω1 ;

(5.20)

dω j = ω ∧ ω j + d ωˆ j = ωˆ j ∧ ω +

m  k=1 m 

 jk ∧ ωk , 1 ≤ j ≤ m;

(5.21)

 jk ∧ ωˆ k , 1 ≤ j ≤ m;

(5.22)

k=1 m 

d12 =

k=1 m 

d jb = dθ12 = dθ jβ =

1k ∧ k2 − ω1 ∧ ωˆ 2 − ωˆ 1 ∧ ω2 + 2μ2 ω1 ∧ ω2 ;

(5.23)

 jk ∧ kb − ω j ∧ ωˆ b − ωˆ j ∧ ωb , 1 ≤ j ≤ m, 3 ≤ b ≤ m;

(5.24)

k=1 p  k=1 p 

θ1k ∧ θk2 + 2μ2 ω1 ∧ ω2 ;

(5.25)

θ jk ∧ θkβ ; 1 ≤ j ≤ p, 3 ≤ β ≤ p;

(5.26)

k=1 m 

dω = −

ωk ∧ ωˆ k .

(5.27)

k=1

For the expressions of ωˆ j ( j ≥ 3), we have the following two propositions. Proposition 5.2 Let x : M m → Sm+ p be a Wintgen ideal submanifold with dimension m ≥ 3. If dimV = p + l + 1 and l ≤ m, then we have the following equations ˆ = 0; d Fˆ + 2 Fω

(5.28)

ˆ j , 1 ≤ j ≤ l; ωˆ j = Fω

(5.29)

ˆ σ , l + 1 ≤ σ ≤ m. ωˆ σ = − Fω

(5.30)

Proof By Corollary 4.10 and (5.4), we can get that ˆ + Yˆ , ξ1 , . . . , ξ p , η1 , η2 , . . . , ηl }. V = Span{ FY

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Ann Glob Anal Geom

It is easy to see that the orthogonal complement of V is ˆ − Yˆ , ηl+1 , ηl+2 , . . . , ηm }. V⊥ = Span{ FY Both V and V⊥ are trivial bundle over M m , i.e., they are constant along M m as subspaces of m+ p+2 . R1 It follows from (5.5) and (5.6) that ˆ + Yˆ ) ≡ (d Fˆ + 2 Fω)Y ˆ d( FY +

m 

ˆ j + ωˆ j )η j , (mod FY ˆ + Yˆ , η1 , · · · , ηl ) ( Fω

j=l+1

ˆ − Yˆ ) ≡ (d Fˆ + 2 Fω)Y ˆ d( FY +

l 

ˆ j − ωˆ j )η j , (mod FY ˆ − Yˆ , ηl+1 , · · · , ηm ). ( Fω

j=1

Using the fact that V and V⊥ are constant along M m , we can get (5.28)–(5.30).

 

Remark 5.3 It is easy to see from (5.27) and Proposition (5.2) that if dimV < p + m + 2, then the connection 1-form ω of the plane-bundle Span{Y, Yˆ } on M m is closed, and the  m (for definition see [18]) is a correspondence Y → Yˆ of the enveloping submanifold M ˆ conformal map. (Y might degenerate.) In the following, we discuss the expressions of ωˆ j in the generic case dimV = m + p + 2. Proposition 5.4 Let x : M m → Sm+ p be a Wintgen ideal submanifold with dimV = m + p + 2 and m ≥ 3. We have the following expressions ˆ j, ωˆ j = Fω

(5.31)

ˆ τ +  τ ω1 + ωˆ τ = Fω  τ ω2 .

(5.32)

The range of indices j, τ depends on the dimension m as follows 1. When m = 3 or m = 4, we take j = 1, 2 and τ = 3. 2. When m ≥ 5 is an odd number, we take 1 ≤ j ≤ m − 2, m − 1 ≤ τ ≤ m. 3. When m ≥ 6 is an even number, we take 1 ≤ j ≤ m − 3, m − 2 ≤ τ ≤ m − 1. τ } are real-valued functions on M m with τ2 + τ2  = 0 at the given point In all cases, {τ , under consideration. Proof The proposition is obvious for m = 3. In the following, we assume m > 3. Since dimV = m + p + 2, we have L  = 0. When m = 4, V = V2 ⊕ Span{η4 , Y }, it follows from the definition of η4 that d[A4 Y − N − i Lη3 ] ≡ (ω1 + iω2 )[(a4 + ib4 )η4 + (c4 + i c˜4 )Y ] − i Lω3 (A4 Y − N ), (mod ξ1 , . . . , ξ p , η1 , . . . , ηm ),

(5.33)

where the coefficient of the term A4 Y − N on the right side coming from d[A4 Y − N − i Lη3 ], Y = −A4 Y − N − i Lη3 , dY = i Lω3 . Here {a4 , b4 , c4 , c˜4 } are four real-valued functions on M m with a42 +b42  = 0 and c42 + c 4 2  = 0 at the fixed point under consideration. Using (5.33) and (5.4), we can get that ˆ + Yˆ ), Ldη3 ≡ −(c˜4 ω1 + c4 ω2 )Y + Lω3 (A4 Y − N ) ≡ −(c˜4 ω1 + c4 ω2 )Y − Lω3 ( FY (mod ξ1 , . . . , ξ p , η1 , . . . , ηm ).

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Ann Glob Anal Geom

Combining this with (5.11), we find that ωˆ 3 has form (5.32). So the conclusion for m = 4 is proved. When m ≥ 5, to determine the expression of ωˆ j , according to (5.11) we shall compute dη j , modulo the components {ξ1 , . . . , ξ p ; η1 , . . . , ηm }, and find out the coefficient of Y . It follows from Corollaries 4.8 and 4.9 that when j ≤ m − 2 and m ≥ 5 is odd, dη j is contained in V m+1  V m+3 = V. It should take values in Span{ψ3 = A4 Y − N , η1 , η2 , η3 , . . . , ηm }. 2 2 (Otherwise there will be a contradiction with Lemma 4.7.) Because dη j , Y = −η j , dY = −ω j , we know ˆ + Yˆ ), (mod ξ1 , . . . , ξ p , η1 , . . . , ηm ). dη j ≡ ω j (A4 Y − N ) ≡ −ω j ( FY

(5.34)

When j ≤ m − 3 and m ≥ 6 is even, dη j is contained in V m2  V m+2 = V, the conclusion 2 follows similarly. This proves (5.31). When m ≥ 5 is odd, by assumption there should be V m+1  V m+3 = V. According to 2 2 Lemmas 4.4 and 4.7, there must exists nonzero complex function λ such that d(ψm + iψm+1 ) ≡ (ω1 + iω2 )λY (mod {ξ1 , . . . , ξ p , η1 , . . . , ηm , A4 Y − N }).

(5.35)

Since {ηm−1 , ηm } com forms the orthogonalization of {ψm , ψm+1 }, dηm−1 and dηm can be expressed as linear combinations of dψm and dψm+1 modulo the components {ηm−1 , ηm }. It follows from (5.35) that ˆ + Yˆ ), (mod {ξ1 , . . . , ξ p , η1 , . . . , ηm }), τ = m −1, m, dητ ≡ (−τ ω1 − ˜τ ω2 )Y − ωτ ( FY ˆ + Yˆ ) are found as before by taking inner product with Y . Now where the coefficients of ( FY (5.32) follows by comparison with (5.11). Note that the coefficients τ , ˜τ cannot vanish at the same time whether τ = m − 1 or τ = m; otherwise d(ψm + iψm+1 ) will be a complex ˆ + Yˆ }. multiple of a Y -valued real 1-form modulo the components {ξ1 , . . . , ξ p , η1 , . . . , FY Comparing with (5.35) one would get ω1 + iω2 is a complex multiple of a real 1-form, which is impossible. The proof of (5.32) is similar when m ≥ 6 is even.   Combining Remark 5.3 and Proposition 5.4, we can get the following theorem. Theorem 5.5 Let x : M m → Sm+ p be a Wintgen ideal submanifold with dimension m ≥ 3. Then the following conditions are equivalent: (1) dimV < p + m + 2; (2) dω = 0, where ω is the connection 1-form of the plane-bundle Span{Y, Yˆ };  m (for definition see [18]) (3) The correspondence Y → Yˆ of the enveloping submanifold M ˆ is a conformal map (Y might degenerate).

6 The reducible Wintgen ideal submanifolds In this section, we will expose a close relationship between the jet bundle V and the minimal integrable distribution D generated by the canonical distribution D = Span{E 1 , E 2 }. It follows from Remark 4.3 that dimD = 2 if and only if dimV = p + 1 + 2. This can be generalized to the following proposition. Proposition 6.1 Let x : M m → Sm+ p be a Wintgen ideal submanifold with dimension m ≥ 3. Then for any integer 2 ≤ k < m, dimD = k if and only if dimV = p + 1 + k.

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Proof When L = 0, this proposition is obvious from Remark 4.3. So we assume that L  = 0 in the following. We just need to prove that if dimV = p + 1 + l with l < m, then there must be dimD = l. From Corollary 4.10, we have V = Span{A4 Y − N , ξ1 , . . . , ξ p , η1 , η2 , . . . , ηl }. Note that Span{E 1 , E 2 , . . . , El } is well defined now. Claim 6.2 D = Span{E 1 , E 2 , . . . , El }. From the convention made in (5.3), in moving frame (5.1) {Y, Yˆ } ∪ {ξ1 , ξ2 , . . . , ξ p } ∪ {η1 , η2 , . . . , ηm }, there are λσ = 0 and ησ = Yσ for any l + 1 ≤ σ ≤ m. By the definition of {η j , 1 ≤ j ≤ l}, we have dη j ≡ 0, (mod Y, Yˆ , ξ1 , . . . , ξ p , η1 , . . . , ηl ). Hence there must be  jσ = 0, 1 ≤ j ≤ l, l + 1 ≤ σ ≤ m. It follows from (5.21) that dωσ ≡ 0, (mod ωl+1 , ωl+2 , . . . , ωm ),

l + 1 ≤ σ ≤ m.

Hence Span{E 1 , E 2 , . . . , El } is integrable; this implies that D ⊂ Span{E 1 , E 2 , . . . , El }. To get the inverse inclusion relation, we discuss by induction. Obviously, E 1 , E 2 ∈ D. It follows from (5.16) and (5.21) that dω3 ≡ 2Lω1 ∧ ω2 , (mod ω3 , . . . , ωm ), which means that E 3 ∈ D since L  = 0. Suppose there is a j so that 3 ≤ j ≤ l − 1 and E 1 , E 2 , . . . , E j ∈ D. By the definition of η j+1 , we know that {E 1 (ηk ), η j+1 , E 2 (ηk ), η j+1 | 3 ≤ k ≤ j} cannot be zero at the same time; otherwise, there will be a contradiction with η j+1 ∈ V. This means that { j+1 k (E 1 ),  j+1 k (E 2 ) | 3 ≤ k ≤ j} cannot be zero at the same time. Hence dω j+1 =

m 

 j+1 k ∧ ωk  ≡ 0, (mod ω j+1 , ω j+2 , . . . , ωm ),

k=1

which implies E j+1 ∈ D since D is integrable. By induction, we can get that Span{E 1 , E 2 , . . . , El } ⊂ D. This finishes the proof.   Remark 6.3 It is easy to verify that if dimV = p + 1 + k with k < m, then D⊥ = Span{E k+1 , E k+2 , . . . , E m } is also integrable.

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In [15], the authors proved that if dimD = 2 < m, then x can be reduced to a cone, a cylinder or a revolution submanifold over a superminimal surface (minimal Wintgen ideal submanifold with dimension 2). The case of dimD = 3 < m was also characterized similarly in [15]. The general reduction theorem when dimD = k < m is Theorem A in the introduction. With the help of Proposition 6.1, we can now give a proof. Proof to Theorem A From Proposition 6.1, we have dimV = p + 1 + k. It follows from (5.7)–(5.6) and (5.28)–(5.30) that . ˆ + Yˆ ) = −ω( FY ˆ + Yˆ ) + 2 Fˆ d( FY

k 

ωjηj;

(6.1)

j=1

ˆ + Yˆ ) + dη1 = −ω1 ( FY

k 

1i ηi + μω2 ξ1 + μω1 ξ2 ;

(6.2)

2i ηi + μω1 ξ1 − μω2 ξ2 ;

(6.3)

 ji ηi , 3 ≤ j ≤ k;

(6.4)

i=1

ˆ + Yˆ ) + dη2 = −ω2 ( FY

k  i=1

ˆ + Yˆ ) + dη j = −ω j ( FY

k  i=1

dξ1 = −μω2 η1 − μω1 η2 + θ12 ξ2 + θ1α ξα ;

(6.5)

dξ2 = −μω1 η1 + μω2 η2 − θ12 ξ1 + θ2α ξα ;  dξα = −θ1α ξ1 − θ2α ξ2 + θαβ ξβ ;

(6.6) (6.7)

β m 

ˆ − Yˆ ) = −ω( FY ˆ − Yˆ ) + 2 Fˆ d( FY

ωσ ησ ;

(6.8)

σ =k+1

ˆ − Yˆ ) + dησ = ωσ ( FY

m 

σ τ ητ , k + 1 ≤ σ ≤ m.

(6.9)

τ =k+1

From Proposition 6.1 and Claim 6.2, we know that D = Span{E 1 , E 2 , . . . , E k } and D⊥ = Span{E k+1 , . . . , E m } are both integrable. Their integral submanifolds are denoted by ϒ and , respectively. Locally, we can define M k = M m / , M m−k = M m /ϒ. It follows from (5.28) that there must be Fˆ ≡ 0 or Fˆ > 0 or Fˆ < 0 on a connected open set. We will prove this theorem separately by these three cases. Case 1, Fˆ > 0. It is obvious that V is a constant space-like subspace and V⊥ is a constant Lorentz subspace m+ p+2 along M m . Without less of generality, we can assume that V = Rk+ p+1 and in R1 m−k+1 ⊥ . Define V = R1 f =

ˆ + Yˆ FY  , 2 Fˆ

g=

ˆ − Yˆ FY  . 2 Fˆ

(6.10)

Then  f, f = 1, g, g = −1.

(6.11)

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It follows from (6.1)–(6.9) that df =

k   2 Fˆ ωjηj,

(6.12)

j=1 k   ˆ 1f + 1 j η j + μω2 ξ1 + μω1 ξ2 , dη1 = − 2 Fω

(6.13)

j=1 k   ˆ 2f + dη2 = − 2 Fω 2 j η j + μω1 ξ1 − μω2 ξ2 ,

(6.14)

j=1 k   ˆ af + a j η j , 3 ≤ a ≤ k, dηa = − 2 Fω

(6.15)

j=1

dg =

m   2 Fˆ ωσ ησ .

(6.16)

σ =k+1

So f can define an immersion f : M k → Sk+ p , and it is minimal and Wintgen ideal by (6.13–6.15). On the other hand, g can define an immersion g : Mm−k → R1m−k+1 and it can be seen as the standard embedding of Hm−k in R1m−k+1 since g, g = −1. Note that  ˆ (6.17) f + g = 2 FY. Combining with Proposition (4.1) in [15], we conclude that x is Möbius equivalent to a cone over the minimal Wintgen submanifold f : M l → Sl+ p . Case 2, Fˆ < 0. In this case, V is a constant Lorentz subspace and V⊥ is a constant space-like subspace m+ p+2 k+ p+1 along M m . Without less of generality, we can assume that V = R1 and in R1 ⊥ m−k+1 V =R . Define ˆ + Yˆ FY f =  , −2 Fˆ

ˆ − Yˆ FY g=  . −2 Fˆ

(6.18)

Then  f, f = −1, g, g = 1.

(6.19)

It follows from (6.1)–(6.9) that  df =

−2 Fˆ

k 

ωjηj,

(6.20)

j=1

 k  ˆ 1f + 1 j η j + μω2 ξ1 + μω1 ξ2 , dη1 = − −2 Fω

(6.21)

j=1

 k  ˆ 2f + dη2 = − −2 Fω 2 j η j + μω1 ξ1 − μω2 ξ2 , j=1

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(6.22)

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 k  ˆ af + dηa = − −2 Fω a j η j , 3 ≤ a ≤ k,

(6.23)

j=1

dg =

 m  −2 Fˆ ωσ ησ .

(6.24)

σ =k+1

So f defines an immersion f : M k → Hk+ p which is minimal and Wintgen ideal by (6.21– 6.23). On the other hand, g defines an immersion g : Mm−k → R m−k+1 , which can be seen as the standard embedding of Sm−k in Rm−k+1 since g, g = 1. Note that  ˆ f + g = 2 FY. Combining with Proposition(4.1) in [15], we know that x is Möbius equivalent to a rotational submanifold over the minimal Wintgen submanifold f : M k → Hk+ p . Case 3, Fˆ ≡ 0. Now ωˆ i ≡ 0 for any i = 1, 2, . . . , m and Yˆ is a constant light-like direction since d Yˆ = −ωYˆ . Since dω = 0, we can write ω as ω = dτ , where τ ∈ C ∞ (M m ). Without loss of generality, we assume that Yˆ = e−τ (−1, 1, 0, 0, 0, 0, 0). On the other hand, V, V⊥ are two constant subspaces endowed with a degenerate inner p+k+1 and V⊥ = product along M m . Without less of generality, we can assume that V = R0 m+ p+2 m−k+1 ⊥ R0 . Moreover, we assume the embedding of V and V in R1 is defined as follows. k+ p+1 m−k+1 For any vectors v ∈ R0 and w ∈ R0 , we denote them as m+ p+2

v = (v0 , −v0 , v1 , . . . , v p+k , 0, . . . , 0) ∈ R1 w = (w0 , −w0 , 0, . . . , 0, w1 , . . . , wm−k ) ∈

,

m+ p+2 R1 .

(6.25)

Using the fact that Y, Y = 0 and Y, Yˆ = 1 we can get that

1 + φ2 1 − φ2 Y = eτ , ,φ , (6.26) 2 2 . where φ = ( f 1 , . . . , f p+k , g1 , . . . , gm−k ) is a vector-valued function on M m . Define two new vector-valued functions as follows: f = ( f 1 , . . . , f p+k ) ∈ Rk+ p , g = (g1 , . . . , gm−k ) ∈ Rm−k . Taking differentiation to (6.26), we get that dY = dτ Y + eτ (dφ · φ, −dφ · φ, d f, dg). In the sense of (6.25), we have that d f can be spanned by {ξ1 , ξ2 , . . . , ξ p , η1 , η2 , . . . , ηk } and dg can be spanned by {ηk+1 , ηk+2 , . . . , ηm }. On the other hand, it follows from (5.5) that dY = ωY +

m 

ωjηj.

j=1

So there must be d f = e−τ

k  j=1

ω j η j , dg = e−τ

m 

ωσ ησ .

σ =k+1

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f and g define two immersion f = ( f 1 , . . . , f p+k ) : M k → Rk+ p , g = (g1 , . . . , gm−k ) : Mm−k → Rm−k . Here g can be seen as an identity map. It follows from (6.2) ∼ (6.7) that f is a minimal Wintgen ideal submanifold. Combining (6.26) with the Proposition 4.1 in [15], we see that x is Möbius equivalent to a cylinder over the minimal Wintgen submanifold f : M k → Rk+ p .  

7 The irreducible minimal Wintgen ideal submanifolds In view of Proposition 6.1, we introduce the following definition. . Definition 7.1 Let x : M m → Sm+ p be a Wintgen ideal submanifold with dimension m. If dimD = m, then we say that x is irreducible. In this section, we will generalize the result in [25] and give a Möbius characterization of the irreducible minimal Wintgen ideal submanifolds. Combining Theorem 2.4 and Proposition 6.1, we get the following theorem. . Theorem 7.2 Let x : M m → Sm+ p be an irreducible Wintgen ideal submanifold with dimension m. Then dimV = m + p + 1 if and only if x is Möbius equivalent to a minimal Wintgen ideal submanifold in either of the three space forms Sm+ p , Hm+ p , Rm+ p . ˆ − Yˆ orthogonal to V and Proof If dimV = m + p + 1, then there is a constant vector FY the mean curvature sphere represented by {ξ1 , ξ2 , . . . , ξ p }. It follows from Theorem 2.4 that x is Möbius equivalent to a minimal submanifold in Sm+ p if Fˆ > 0, or Hm+ p if Fˆ < 0, or Rm+ p if Fˆ ≡ 0. Conversely, assume that x is Möbius equivalent to a minimal Wintgen ideal submanifold m+ p+2 in space forms. Then there is a nonzero constant vector g ∈ R1 which is orthogonal with {ξ1 , ξ2 , . . . , ξ p } at every point q ∈ M m . Since V is spanned by the derivatives of {ξ1 , ξ2 }, g is orthogonal with V as well. On the other hand, since x is irreducible, it follows from Proposition 6.1 that dimV ≥ m + p + 1. It follows from Corollary 4.10 that {ξ1 , ξ2 , . . . , ξ p , η1 , η2 , . . . , ηm } ⊂ V. Using (5.12) we get that −ωˆ j g, Y − ω j g, Yˆ = 0. Note that if g, Y = 0, it follows from the above equation that g, Yˆ = 0; this means that g = 0, which is a contradiction. Hence g, Y  = 0 and Y ∈ / V. So there must be dimV = m + p + 1.   In [25], we used the flatness of plane-bundle {Y, Yˆ } to give a characterization of the irreducible minimal Wintgen submanifolds of dimension 3. From Theorems 5.5 and 7.2, we can easily generalize this characterization to irreducible minimal Wintgen ideal submanifolds of arbitrary dimension as the following theorem. Theorem B in the introduction is part of its conclusions.

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Theorem 7.3 Let x : M m → Sm+ p be an irreducible Wintgen ideal submanifold with dimension m ≥ 3. Then the following conditions are equivalent: (1) dω = 0, where ω is the connection of plane-bundle Span{Y, Yˆ }. (See the structure equations under the new frame in Sect. 5.)  m is a conformal map. (2) The correspondence Y → Yˆ of the enveloping submanifold M ˆ (Y might be degenerate.) (3) x is Möbius equivalent to a minimal Wintgen ideal submanifold in space form Sm+ p (if Fˆ > 0) or Hm+ p (if Fˆ < 0) or Rm+ p (if Fˆ = 0). . Remark 7.3 For a submanifold x : M m → Q m+ p (c) in a real space form of constant sectional curvature c, it is well known that if the dimension of its relative nullity subspace is constant, then the leaves of the integrable relative nullity distribution are totally geodesic submanifolds in the ambient Q m+ p (c) [9,19]. In particular, if x is a minimal Wintgen ideal submanifold, from (1.2) we see that the relative nullity subspace of x is D⊥ = Span{E 3 , E 4 , . . . , E m }, which is orthogonal with D. It is obvious from (5.19) and (5.20) that D⊥ is integrable. Hence x M m is foliated by totally geodesic submanifolds with dimension m − 2 in Q m+ p (c).

8 The generic irreducible Wintgen ideal submanifolds Definition 8.1 Let x : M m → Sm+ p be an irreducible Wintgen ideal submanifold with dimension m ≥ 3. If x is not equivalent to any minimal submanifold in a space form, then we call x a generic Wintgen ideal submanifold. (See Corollary 4.10.) Equivalently, the jet M+ p+2 bundle V of its conformal Gauss map has the full dimension as the total space R1 . According to Theorem 5.5, a Wintgen ideal submanifold x is generic if and only if the plane-bundle Span{Y, Yˆ } on M m is not flat, which is an open condition. This implicitly implies that there are many generic Wintgen ideal submanifolds. In the following we will show that such submanifolds are abundant by constructing a large family of codimension two examples in Sm+2 . Let Cm+4 be the complex linear space of dimension m + 4 with the bilinear extension 1 of the real Lorentz inner product. Endow the complex projective space CP m+3 with the pseudo-Hermitian Fubini-Study metric in the usual way. Let Qm+2 := {[v] ∈ CP m+3 |v ∈ Cm+4 , v, v = 0} 1 . be the complex quadratic hypersurface corresponding to isotropic lines in Cm+4 1 According to our previous work in [13], a codimension two Wintgen ideal submanifold M m → Sm+2 always comes from a 1-isotropic holomorphic curve (with local complex coordinate z): ξ : M → Cm+4 = Rm+4 ⊗ C, s.t. ξ, ξ = 0, ξz , ξz = 0. 1 1

(8.1)

Given ξ : M → Cm+4 , M m → Sm+2 can be recovered as a sphere bundle over the Riemann 1 surface M. At each point of M, the fiber is the (m − 2)-dimensional sphere determined by the 4-dimensional spacelike subspace (see Remark 2.2) Span{Re(ξ ), Im(ξ ), Re(ξz ), Im(ξz )} ⊂ Rm+4 . 1

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In a more geometric language, M m is the enveloping submanifold of the two-parameter family of codim-4 spheres in Sm+2 defined by ξ [13]. According to our result in [13], such a submanifold M m is always Wintgen ideal provided that it is immersed with first normal subspace spanned by Re(ξ ), Im(ξ ). Requiring M m to be generic is equivalent to that ξ : D ⊂ C → Cm+4 is fully immersed, i.e., all the derivatives 1

m+4 ∂ . So our task is to find fully immersed, { ∂z k ξ |k = 0, 1, 2 . . .} must span the full space C1 1-isotropic holomorphic curves, which can be done as follows. Let g(z) = (g1 (z), g2 (z), . . . , gm (z)) be a holomorphic vector-valued function defined on a domain U of C containing 0, z ∈ U . which describes a holomorphic curve in Cm . We assume that it is full, i.e., it is not contained in any low-dimensional subspace of Cm . Define

 z 1 − g, g 1 + g, g f = g, , . (8.2) 2 2i 0 k

ξ=

1 −  f, f 1 +  f, f , f, . 2 2

(8.3)

ξ is clearly holomorphic. Recalling our convention Y, Z := −Y0 Z 0 + Y1 Z 1 + · · · + Ym+ p+1 Z m+ p+1 , it is straightforward to verify ξ, ξ = 0, ξz , ξz =  f z , f z = 0. It is easy to see that for a general g, our ξ would give a full map into Cm+4 . 1

References 1. Burstall, F., Ferus, D., Leschke, K., Pedit, F., Pinkall, U.: Conformal geometry of surfaces in the 4-sphere and quaternions Lecture Notes in Mathematics, vol. 1772. Springer (2002) 2. Chen, B.Y.: Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures. Ann. Glob. Anal. Geom. 38, 145–160 (2010) 3. Choi, T., Lu, Z.: On the DDVV conjecture and the comass in calibrated geometry (I). Math. Z. 260, 409–429 (2008) 4. Dajczer, M., Tojeiro, R.: A class of austere submanifolds. Ill. J. Math. 45(3), 735–755 (2001) 5. Dajczer, M., Tojeiro, R.: All superconformal surfaces in R 4 in terms of minimal surfaces. Math. Z. 261(4), 869–890 (2009) 6. Dajczer, M., Tojeiro, R.: Submanifolds of codimension two attaining equality in an extrinsic inequality. Math. Proc. Camb. Philos. Soc. 146(2), 461–474 (2009) 7. De Smet, P.J., Dillen, F., Verstraelen, L., Vrancken, L.: A pointwise inequality in submanifold theory. Arch. Math. (Brno) 35, 115–128 (1999) 8. Dillen, F., Fastenakels, J., Van Der Veken, J.: Remarks on an inequality involving the normal scalar curvature. In: Proceedings of the International Congress on Pure and Applied Differential GeometryPADGE Brussels, pp. 83–92. Shaker Verlag, Aachen (2007) 9. Ferus, D.: On the completeness of nullity foliations. Mich. Math. J. 18, 61–64 (1971) 10. Ge, J., Tang, Z.: A proof of the DDVV conjecture and its equality case. Pac. J. Math. 237, 87–95 (2008) 11. Guadalupe, I., Rodríguez, L.: Normal curvature of surfaces in space forms. Pac. J. Math. 106, 95–103 (1983) 12. Li, T., Ma, X., Wang, C., Xie, Z.: New examples and classification of Möbius homogeneous Wintgen ideal submanifolds. in preparation 13. Li, T., Ma, X., Wang, C., Xie, Z.: Wintgen ideal submanifolds of codimension two, complex curves, and Möbius geometry. Tohoku Math. J. 68(4), 621–638 (2016) 14. Li, T., Ma, X., Wang, C.: Deformation of hypersurfaces preserving the möbius metric and a reduction theorem. Adv. Math. 256, 156–205 (2014) 15. Li, T., Ma, X., Wang, C.: Wintgen ideal submanifolds with a low-dimensional integrable distribution. Front. Math. China 10(1), 111–136 (2015) 16. Liu, H., Wang, C., Zhao, G.: Möbius isotropic submanifolds in S n . Tohoku Math. J. 53, 553–569 (2001)

123

Ann Glob Anal Geom 17. Lu, Z.: Normal scalar curvature conjecture and its applications. J. Funct. Anal. 261, 1284–1308 (2011) 18. Ma, X., Xie, Z.X.: The Möbius geometry of Wintgen ideal submanifolds. In: ICM 2014 Satellite Conference on Real and Complex Submanifolds, Springer Proceedings in Mathematics & Statistics, vol. 106, pp. 411–425 (2014) 19. Maltz, R.: Isometric immersions into spaces of constant curvature. Ill. J. Math. 15, 490–502 (1971) 20. Petrovié-torga˘sev, M., Verstraelen, L.: On Deszcz symmetries of Wintgen ideal submanifolds. Arch. Math. 44, 57–67 (2008) 21. Van der Veken, J.: Ideal Lagrangian submanifolds. to appear in Recent Advances in the Geometry of Submanifolds, Contemporary Mathematics, American Mathematical Society 22. Wang, C.: Möbius geometry of submanifolds in S n . Manuscr. Math. 96, 517–534 (1998) 23. Wintgen, P.: Sur l’inégalité de Chen-Willmore. C. R. Acad. Sci. Paris 288, 993–995 (1979) 24. Xie, Z.X.: Wintgen ideal submanifolds with vanishing Möbius form. Ann. Glob. Anal. Geom. 48, 331–343 (2015) 25. Xie, Z.X., Li, T.Z., Ma, X., Wang, C.P.: Möbius geometry of three dimensional Wintgen ideal submanifolds In S5 . Sci. China Math. 57, 1203–1220 (2014)

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