Classification of hypersurfaces with two distinct ...

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SCIENCE CHINA Mathematics

. ARTICLES .

July 2012 Vol. 55 No. 7: 1463–1478 doi: 10.1007/s11425-012-4391-1

Classification of hypersurfaces with two distinct principal curvatures and closed M¨ obius form in Sm+1 LIN LiMiao1,2 & GUO Zhen2,∗ 1Department

of Mathematics, East China Normal University, Shanghai 200241, China; of Mathematics, Yunnan Normal University, Kunming 650092, China Email: [email protected], [email protected]

2Department

Received February 16, 2011; accepted December 26, 2011; published online March 15, 2012

Abstract Let x be an m-dimensional umbilic-free hypersurface in an (m + 1)-dimensional unit sphere Sm+1 (m  3). In this paper, we classify and explicitly express the hypersurfaces with two distinct principal curvatures and closed M¨ obius form, and then we characterize and classify conformally flat hypersurfaces of dimension larger than 3. Keywords MSC(2010)

moebius geometry, principal curvature, conformally flat, M¨ obius form 54A30, 53C21, 53C40

Citation: Lin L M, Guo Z. Classification of hypersurfaces with two distinct principal curvatures and closed M¨ obius form in Sm+1 . Sci China Math, 2012, 55(7): 1463–1478, doi: 10.1007/s11425-012-4391-1

1

Introduction

Let x : M m → Sm+1 be an m-dimensional umbilic-free hypersurface in an (m + 1)-dimensional unit sphere Sm+1 and {ei } be a local orthonormal tangent frame field of x for the induced metric I = dx · dx  with dual frame field {θi }. Let II = ij hij θi θj be the second fundamental form and H be the mean m 1 curvature of x. Define ρ2 = m−1 |II − m tr(II)I|2 , then positive definite 2-form g = ρ2 I is invariant under M¨ obius transformation group of Sm+1 and is called M¨obius metric of x. Three basic M¨obius invariants   of x, M¨obius form Φ = ρ i Ci θi , Blaschke tensor A = ρ2 ij Aij θi θj and M¨ obius second fundamental  form B = ρ2 ij Bij θi θj , are defined by (see [6])    (hij − Hδij )ej (logρ) , (1.1) Ci = −ρ−2 ei (H) + j

1 Aij = −ρ−2 (Hessij (logρ) − ei (logρ)ej (logρ) − Hhij ) − ρ−2 (∇logρ2 − 1 + H 2 )δij , 2 Bij = ρ−1 (hij − Hδij ),

(1.2) (1.3)

where Hessij and ∇ are the Hessian matrix and the gradient with respect to the induced metric I = dx·dx. We call the eigenvalues of B the M¨obius principal curvatures of x. In M¨ obius geometry, since the eigenspaces of B coincide with II of x, the number of distinct principal curvatures is a M¨ obius invariant. A hypersurface in Sm+1 with two distinct principal curvatures is Dupin hypersurface if and only if its M¨obius form vanishes. Pinkall [5] showed that Dupin conditions can be ∗ Corresponding

author

c Science China Press and Springer-Verlag Berlin Heidelberg 2012 

math.scichina.com

www.springerlink.com

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naturally formulated in the context of Lie sphere geometry and that they are invariant under the Lie sphere m+1 are transformation group. In 1985, Ceceil and Ryan [4] have proved √ that all Dupin hypersurfaces in S k m−k 2 equivalent by a Lie sphere transformation to S (r) × S ( 1 − r ) and so they gave the classification of hypersurfaces in Sm+1 (m  3) with two distinct principal curvatures and vanishing M¨ obius form under the Lie sphere transformation group. Note that the Lie sphere transformation group contains the M¨obius transformation group in Sm+1 as a subgroup. In 2002, Li et al. [11] classified hypersurfaces in Sm+1 with two distinct principal curvatures and vanishing M¨obius form under the M¨obius transformation group. Now we are interested in: How about hypersurfaces in Sm+1 with two distinct principal curvatures and non-vanishing M¨obius form under the M¨obius transformation group? In [15,16], the authors classified all hypersurfaces with isotropic Blaschke tensor and all hypersurfaces with constant M¨obius curvature. They found that all these hypersurfaces have two distinct principal curvatures and closed non-vanishing M¨obius form. Here, we will classify all hypersurfaces in Sm+1 (m  3) with two distinct principal curvatures and closed M¨obius form. denotes the Lorentz In order to make our main results intuitional, we use the following notations: Rm+3 1 space with the inner product ·, · given by Y, Z = −y0 z0 + y1 z1 + · · · + ym+2 zm+2 , m+2 where Y = (y0 , y1 , . . . , ym+2 ), Z = (z0 , z1 , . . . , zm+2 ) ∈ Rm+3 . C+ and Qm+1 denote the positive light m+2 cone and the quadric in real projection space RP , which are defined as follows: m+2 = {X ∈ Rm+3 : X, X = 0, x0 > 0}, C+ 1

Qm+1 = {[Y ] ∈ RP m+2 : Y, Y  = 0}.

Let Hm+1 be the hyperbolic space defined by Hm+1 = {(y0 , y1 ) ∈ R+ × Rm+1 | − y02 + y1 y1 = −1}. Now let σ : Rm+1 → Sm+1 be the inverse of the stereographic projection given by   1 − |u|2 2u σ(u) = , , 1 + |u|2 1 + |u|2 and τ : Hm+1 → Sm+1 be the conformal map defined by   1 y1 , τ (y0 , y1 ) = , (y0 , y1 ) ∈ Hm+1 . y0 y0

(1.4)

(1.5)

The conformal maps σ and τ assign any hypersurface in Rm+1 or Hm+1 to a hypersurface in Sm+1 . We m+2 use map π : C+ → Qm+1 to denote the natural projection. For a hypersurface x : M m → Sm+1 , we have map (1.6) X := π(1, x) : M m → Qm+1 , which is determined by the immersion x and is called the natural map of x. It is known that two  : obius equivalent if and only if whose natural maps X, X hypersurfaces x, x  : M m → Sm+1 are M¨ m m+1 are equivalent under the action of Lorentz group O(m + 2, 1). Now, we state the main M →Q theorem as follows: Theorem 1.1. Let x : M m → Sm+1 (m  3) be a hypersurface with two distinct principal curvatures and closed M¨ obius form. Then x, up to a M¨ obius transformation of Sm+1 , is locally one of the following hypersurfaces: √ (1) the standard torus Sk (r) × Sm−k ( 1 − r2 ); m−k (2) the image of σ of the standard cylinder Sk (1) × ⊂ Rm+1 ; √R k m−k 2 (3) the image of τ of the standard S (r) × H ( 1 + r ) in Hm+1 ; m−1 2 (4) x(M ) = σ(Γ × R ), Γ ⊂ R , where Γ is any smooth curve with non-constant curvature in R2 ;

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√ √ (5) for any negative constant a, X(M ) = π(Hm−1 (1/ −a) × Γ), Γ ⊂ S2 (1/ −a), where Γ is any √ smooth curve with non-constant geodesic curvature in S2 (1/ −a); √ √ (6) for any positive constant a, X(M ) = π(Γ × Sm−1 (1/ a)), Γ ⊂ H2 (1/ a), where Γ is any smooth √ curve with non-constant geodesic curvature in H2 (1/ a). The hypersurfaces in the first three cases have vanishing M¨obius form and they are Dupin hypersurfaces, but the hypersurfaces in the last three cases have non-vanishing M¨obius form and then they are not Dupin hypersurfaces. Because an umbilic-free conformally flat hypersurface in Sm+1 (m  4) can be characterized as a hypersurface with two distinct principal curvatures in M¨obius geometry, therefore, we get Corollary 1.2. Corollary 1.2. Let x : M m → Sm+1 (m  4) be an umbilic-free conformally flat hypersurface with closed M¨ obius form. Then x, up to a M¨ obius transformation of Sm+1 , is locally one of the following hypersurfaces: √ (1) the standard torus S1 (r) × Sm−1 ( 1 − r2 ); (2) the image of σ of the standard cylinder S1 (1) × Rm−1 ⊂ Rm+1 ; √ (3) the image of τ of the standard S1 (r) × Hm−1 ( 1 + r2 ) in Hm+1 ; (4) x(M ) = σ(Γ × Rm−1 ), Γ ⊂ R2 , where Γ is any smooth curve with non-constant curvature in R2 ; √ √ (5) for any negative constant a, X(M ) = π(Hm−1 (1/ −a) × Γ), Γ ⊂ S2 (1/ −a), where Γ is any √ smooth curve with non-constant geodesic curvature in S2 (1/ −a); √ √ (6) for any positive constant a, X(M ) = π(Γ × Sm−1 (1/ a)), Γ ⊂ H2 (1/ a), where Γ is any smooth √ curve with non-constant geodesic curvature in H2 (1/ a). Remark 1.3. Cartan proved that any conformally flat hypersurface with dimension m  4 is (piece of) a branched channel hypersurface (envelope of a 1-parameter family of spheres). Cartan’s result was re-proven several times (see [17]). Our Corollary 1.2 further more explicitly expresses conformally flat hypersurfaces with closed M¨obius form. So Corollary 1.2 provides a new method of constructing channel hypersurfaces. Remark 1.4. There are two important subclasses of the class of hypersurfaces described in Corollary 1.2: one is the class of the hypersurfaces with isotropic Blaschke tensor and another is the class of the hypersurfaces with constant M¨obius curvature K. In fact, it was proved that the two classes of hypersurfaces are all conformally flat hypersurfaces with closed M¨obius form. So, the classification results in papers [15] and [16] can be seen as special cases of Corollary 1.2, where, for the first subclass, in each case the geodesic curvature κ(s) of curve Γ satisfies O.D.E.: 

d 1 ds κ

2

 2  2 1 1 1 1 +a − log + 2 = 0, κ m κ m

for the second subclass, in each case the geodesic curvature κ(s) of curve Γ satisfies O.D.E.: 

d 1 ds κ

2

 2 1 +a = −K, κ

where s is the arc-parameter of the curve. The above O.D.E.s are obtained by using Theorem 3.2. We organize the paper in five sections. In Section 2 we review the structure equations and M¨obius invariants of a hypersurface in Sm+1 . In Section 3 we study hypersurfaces with two distinct principal curvatures and focus on these hypersurfaces with closed non-vanishing M¨ obius form. We get the local express of their basic M¨obius invariants (Proposition 3.1, Lemma 3.3 and Lemma 3.4) and get Theorem 3.2. In Section 4 we obtain the differential equations characterizing the hypersurfaces with two distinct principal curvatures and closed non-vanishing M¨ obius form (Theorem 4.1). In Section 5 we finish the proof of Theorem 1.1. We also give a new proof of Cartan’s result in M¨ obius geometry and then get Corollary 1.2.

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M¨ obius invariants for a hypersurface in Sm+1

In this section, we define M¨obius invariants and recall the structure equations for hypersurfaces in Sm+1 . For details we refer to [6]. Let x : M m → Sm+1 ⊂ Rm+2 be an umbilic-free hypersurface immersed in Sm+1 . We define the of x by M¨obius position vector Y : M m → Rm+3 1 Y = ρ(1, x) : M m → Rm+3 , 1

ρ2 =

m (II2 − mH 2 ) > 0. m−1

Then we have the following: Theorem 2.1 (See [6]). Two hypersurfaces x, x  : M m → Sm+1 are M¨ obius equivalent if and only if m+3 there exists T in the Lorentz group O(m + 2, 1) acting on R1 such that Y = Y T . Since the M¨obius transformation group in Sm+1 is isomorphic to the subgroup O+ (m + 2, 1), which , from Theorem 2.1 we know 2-form preserves the positive part of the light cone in Rm+3 1 g = dY, dY  = ρ2 dx · dx

(2.1)

is a M¨obius invariant and is called M¨obius metric or M¨obius first fundamental form induced by x (cf. [6, 18, 19]). Let be the Laplacian with respect to (M, g). We define N =−

1 1  Y, Y Y.

Y − m 2m2

(2.2)

Let {E1 , . . . , Em } be a local orthonormal basis for (M, g) with dual basis {ω1 , . . . , ωm } and write Ei (Y ) = Yi , then Y, Y  = N, N  = 0,

Y, N  = 1,

Yi , Yj  = δij ,

Yi , Y  = Yi , N  = 0,

1  i, j  m.

Let V be the orthogonal complement of span{Y, N, Y1 , . . . , Ym } in Rm+3 . Then we have the orthogonal 1 decomposition = span{Y, N } ⊕ span{Y1 , . . . , Ym } ⊕ V. Rm+3 1

(2.3)

Let E be a unit vector field of V , then {Y, N, Y1 , . . . , Ym , E} forms a moving frame in Rm+3 along M . 1 We will use the following range of indices in this section: 1  i, j, k, . . .  m. The structure equations are given by dY =



ωi Yi ,

i

dN =

 ij

dYi = −

Aij ωj Yi +



 i



Ci ωi E,

i

Aij ωj Y − ωi N +

j

dE = −

(2.4)

Ci ωi Y −



 j

Bij ωj Yi ,

(2.5) ωij Yj +



Bij ωj E,

(2.6)

j

(2.7)

ij

obius metric g, Aij and Bij are symmetric with respect to (ij). where ωij is the connection form of the M¨    It is clear that A = ij Aij ωi ⊗ ωj , B = ij Bij ωi ⊗ ωj , Φ = i Ci ωi are M¨obius invariants and are called the Blaschke tensor, M¨obius second fundamental form and M¨obius form of x, respectively. Remark 2.2. and (1.3).

The relations between A, B, Φ and Euclidean invariants of x are given by (1.1), (1.2)

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Let Rijkl , Rij and R be components of the curvature tensor, components of Ricci curvature tensor and scalar curvature of g, respectively. By exterior differentiation of structure equations, we can get the integrability conditions for structure equations as follows (see [6]): Aij,k − Aik,j = Bik Cj − Bij Ck ,   Ci,j − Cj,i = Bik Akj − Bjk Aki , k

(2.8) (2.9)

k

Bij,k − Bik,j = δij Ck − δik Cj ,

(2.10)

Rijkl = Bik Bjl − Bil Bjk + δik Ajl + δjl Aik − δil Ajk − δjk Ail ,  Rij = − Bik Bkj + tr(A)δij + (m − 2)Aij ,

(2.11)

k

tr(A) =

  1 m R , 1+ 2m m−1



Bii = 0,



i

ij

(Bij )2 =

m−1 , m

(2.12) (2.13)

where {Aij,k },{Bij,k } and {Ci,j } are covariant derivatives of A, B and Φ with respect to the connection induced by g. From (2.10) and (2.13) we have  Bij,i = −(m − 1)Cj . (2.14) i

Remark 2.3.

3

Some recent results about the M¨obius geometry of submanifolds can be found in [2,7–16].

M¨ obius invariants of a hypersurface with two distinct principal curvatures and closed M¨ obius form in Sm+1

Let x : M m → Sm+1 ⊂ Rm+2 (m  3) be an umbilic-free hypersurface with two distinct principal curvatures and closed M¨ obius form. The goal of this section is to determine its basic M¨obius invariants {g, Φ, B, A}. It was showed by Li et al. (see [11]) that x has two distinct constant M¨obius principal curvatures, which are given by   1 (m − 1)(m − k) 1 (m − 1)k λ= , μ=− , m k m m−k where k is the multiplicity of λ. Proposition 3.1. Let x : M m → Sm+1 ⊂ Rm+2 be a hypersurface with two distinct principal curvatures, then there are only two cases: (1) M¨ obius form Φ of M vanishes; (2) M¨ obius form Φ of M does not vanish but M has two distinct M¨ obius principal curvatures m−1 m and 1 −m of multiplicities 1 and m − 1. Proof. Since B has eigenvalues λ and μ of multiplicities k and m − k, we can define two distributions V1 and V2 as follows:   V1 = V1 (p), V2 = V2 (p), (3.1) p∈M

p∈M

where V1 (p) and V2 (p) are the eigenspaces corresponding to λ and μ at a point p ∈ M , with dim(V1 (p)) = k and dim(V2 (p)) = m − k. Thus T M = V1 ⊕ V2 . For convenience, we will make 1  i, j, . . .  m;

1  a, b, . . .  k;

k + 1  α, β, . . .  m.

Choose a local orthonormal tangent frame field {E1 , . . . , Em } of T (M ) in a neighborhood of p, such that V1 = span{E1 , . . . , Ek } and V2 = span{Ek+1 , . . . , Em }. Then, with respect to this basis, Baj = λδaj ,

Bαj = μδαj .

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From (2,10) and (2.14), we get Bab,j = 0,

Bαβ,j = 0,

Baα,b = −δab Cα ,

Baα,β = −δαβ Ca ,

(k − 1)Ca = 0,

(m − 1 − k)Cα = 0.

If Φ = 0, there exists a point p ∈ M such that Φ(p) = 0. Then k = 1 or k = m − 1 at p and also on M . 1 Without loss of generality, we can assume k = 1 and then λ = m−1 m , μ = −m. The hypersurfaces of Case (1) in Proposition 3.1 have been classified by Li et al. Theorem 3.2 (See [11]). Let x : M m → Sm+1 (m  3) be a hypersurface with two distinct principal curvatures and vanishing M¨ obius form. Then x is M¨ obius equivalent to an open part of one of the following hypersurfaces in Sm+1 : √ (1) the standard torus Sk (r) × Sm−k ( 1 − r2 ); m−k ⊂ Rm+1 ; (2) the image of σ of the standard cylinder Sk (1) × √R k m−k ( 1 + r2 ) in Hm+1 . (3) the image of τ of the standard S (r) × H In following we discuss Case (2) in Proposition 3.1. For convenience, we will make 1  i, j, . . .  m; 2  α, β, . . .  m. Because dΦ = 0, we can choose a local orthonormal tangent frame field {E1 , E2 , . . . , Em } of T (M ) in a neighborhood of a point p ∈ M such that A and B are diagonalized simultaneously, V1 = span{E1 } and V2 = span{E2 , . . . , Em }. Then, with respect to this basis, m−1 1 δ1j , Bαj = − δαj , Aij = Aii δij , m m = 0, Bαβ,j = 0, B1α,j = −C1 δαj , Cα = 0,

B1j = B11,j

(3.2) ω1α = −C1 ωα .

(3.3)

Hence dω1 = 0 which implies V2 is integrable according to Frobenius theorem. Denote the integral manifold of V2 by N m−1 . Then, locally, we have M m = L × N m−1 , g = g|{u}×N m−1 on where L is an interval in R1 . For each u ∈ L, g induces a positive definite metric  {u} × N m−1 . Since dΦ = 0 and Cα = 0, there exists locally a smooth function f = f (u) such that Φ = df,

C1 = f  .

αβ and R  denote the components of curvature tensor, components of Ricci curvature tensor αβγδ , R Let R and scalar curvature of ({u} × N m−1 ,  g), respectively. Lemma 3.3. αβγλ − C12 δαβγλ , Rαβγλ = R

 R1α1β =

 ∂C1 − C12 δαβ , ∂u

where δαβγλ = δαγ δβλ − δαλ δβγ .

R1αβγ = 0,

(3.4)

 ∂ Proof. Choose E = ∂u , then {ω1 = du, ωα } is the dual frame of {E1 , Eα }. So g = du2 + α ωα2  2 1 and g = α ω α where ω α = ωα |{u}×N m−1 . Let ωij be the connection form of (M, g) and ω αβ be the connection form of ({u}×N m−1 ,  g). Denote dN as the exterior differential operator on T ∗ N m−1 . Because ∂ + dN , then the exterior differential operator on T ∗ (L × N m−1 ) is d = du ∧ ∂u    ∂ωα du ∧ + C1 ωα = ωβ ∧ (ωβα − ω βα ), (3.5) ∂u β

which implies that there exists function aαβ such that  αβ + aαβ du, ωαβ = ω

aαβ = ωαβ

 ∂ . ∂u

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So −

 1 Rαβγλ ω γ ∧ ω λ = dN ω αβ − ω αγ ∧ ω γβ , 2 γ γλ

−

1

2

Rαβij ωi ∧ ωj = dN ω αβ −

ij





ω αγ ∧ ω γβ + C12 ωα ∧ ωβ

γ

  ∂ω αβ − dN aαβ − aαγ ω γβ + aγβ ω αγ ∂u γ γ   ∂C1 2 − C1 ω1 ∧ ωα , =− ∂u + du ∧

 −

1 R1αij ωi ∧ ωj 2 ij

then we get (3.4) immediately. Lemma 3.4. Proof.

For fixed u, ({u} × N m−1 , g) has constant curvature.

For the chosen basis in Lemma 3.3, from (3.4) and Gauss equation (2.11) we have m−1 ∂C1 − C12 = − + A11 + Aαα , ∂u m2

2  α  m,

(3.6)

which implies all Aαα equal (2  α  m). Denote A11 = r1 , Aαα = r2 , then   1 1 1 2 αβαβ = 2 C + r + , α = β. Rαβαβ = 2 + 2r2 , R 2 m 2m2 2 1 After calculating the covariant derivations of Aαβ , we get Aαα,γ = Eγ (r2 ),

Aαγ,α = 0,

α = γ.

˜ αβαβ ) = 2Eγ (r2 ) = 0 which means ({u} × N m−1 ,  g) Because Cγ = 0, then Aαα,γ = Aαγ,α = 0. So Eγ (R has constant curvature for a fixed u. From Lemma 3.4 and ω1α = −f  ωα , there exists a local coordinate (u, v2 , . . . , vm ) of M such that g is expressed as 2 ), (3.7) g = du2 + e2θ(u,v2 ,...,vm ) (dv22 + · · · + dvm where e(·) denotes exponential function exp, θ is a smooth function on L × N . Then E1 = Lemma 3.5.

∂ , ∂u

Eα = e−θ

∂ , ∂vα

ω1 = du,

ωα = eθ dvα ,

ω1α = −C1 eθ dvα .

(3.8)

The function θ has the expression θ(u, v2 , . . . , vm ) = −f (u) + h(v2 , . . . , vm ),

(3.9)

and the connection forms, except ω1α , are given by ωαβ =

∂h ∂h dvβ − dvα , ∂vα ∂vβ

(3.10)

where h is a smooth function on N . Proof.

The details can be found in [15].

Lemma 3.6.

2  2  2 m  ∂h ∂h ∂ 2 h ∂ 2 h  ∂h + 2 + − − = −ae2h , α = β, ∂vα2 ∂vβ γ=2 ∂vγ ∂vα ∂vβ   1 1 1 2m − 1 1 2f 2f  2 − ae , Aαα = r2 = ae − (f ) − 2 , A11 = r1 = f  − (f  )2 + 2 m 2 2m2 2

where a is a constant.

(3.11) (3.12)

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Using (3.8) and (3.10) to calculate the curvature tensor, we get e2θ Rαβαβ = −

Let

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 ∂2h ∂2h − 2 − (f  )2 e2θ − 2 ∂vα ∂vβ γ



∂h ∂vγ



2 +

∂h ∂vα



2 +

∂h ∂vβ

2 ,

α = β.

 2  2  2

∂h ∂h ∂ 2 h ∂ 2 h  ∂h + + aαβ (v) := e−2h − 2 − 2 − , ∂vα ∂vβ ∂vγ ∂vα ∂vβ γ

then Rαβαβ = e2f aαβ (v) − (f  )2 ,

α = β.

On the other hand, from Lemma 3.4, aαβ (v) is independent of the choice of α and β. Denote aαβ (v) by a(v), then from (3.6), (3.7) and (3.12) there hold   1 2f 1 e a(v) − (f  )2 − 2 , 2 m 1 2m − 1 1 A11 = r1 = f  − (f  )2 + − e2f a(v), 2 2 2m 2 αβαβ = a(v)e2f , α = β. R Aαα = r2 =

(3.13) (3.14) (3.15)

αβαβ is independent of v and then a(v) is constant from (3.15). Hence there Lemma 3.4 implies that R hold (3.11), (3.12). 2 Now view quadratic form gˆ = e2h (dv22 + · · · + dvm ) as a metric on manifold N m−1 . According to Lemma 3.6, gˆ has constant curvature a showing that there exists a local coordinate, still denoted by (v2 , . . . , vm ), such that 2 dv 2 + · · · + dvm gˆ = 2 a , (1 + 4 v2 )2  2 where v2 = m α=2 vα . As summary, we come to the following conclusion:

Theorem 3.7. For an m ( 3)-dimensional hypersurface M m with two distinct principal curvatures and closed non-vanishing M¨ obius form in Sm+1 , under a suitable local coordination (u, v2 , . . . , vm ), the m M¨ obius invariants of M can be expressed as follows:    dvα2 , (3.16) g = du2 + e−2f (u) e2h(v) α

  1 −2f (u) 2h(v)  2 m−1 2 du − e dvα , B= e m m α     2 −2f (u) 2h(v) 2 dvα , e Φ = f du, A = r1 du + r2 e

(3.17) (3.18)

α

where f = f (u) is a smooth function, r1 , r2 are given by (3.12) and   a h(v) = − log 1 + v2 . 4

(3.19)

Theorem 3.7 shows that all the M¨ obius invariants in structure equations are determined by the functions f and h, and so we can get the hypersurface M m by integrating the structure equations. We will complete the procedure in the next section.

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Differential equations of a hypersurface with two principal curvatures and closed non-vanishing M¨ obius form in Sm+1

From the structure equations (2.4)–(2.7), by using (3.16)–(3.18) and Eα = e−θ ∂v∂α , we get   m−1 Yuu = − r1 Y + N − E ; m ∂Y ∂N Nu = r1 Yu + f  E, = r2 ; ∂vα ∂vα   m−1 1 ∂Y ∂E Eu = − f  Y + Yu , = ; m ∂vα m ∂vα ∂2Y ∂Y + f = 0; ∂u∂vα ∂vα   1  r2 Y + N + E − f Yu δαβ = −ef Fαβ , m where Fαβ = ef −2h



∂2Y − ∂vβ ∂vα



∂h ∂Y ∂h ∂Y + ∂vβ ∂vα ∂vα ∂vβ

 +

(4.1) (4.2) (4.3) (4.4) (4.5)

  ∂h ∂Y δαβ . ∂vγ ∂vγ γ

In the following we concentrate on the solution of the partial differential equations (4.1)–(4.5). From (4.1) and (4.5) we have E = (r1 − r2 )Y + Yuu + f  Yu − ef Fαα , N =−

(4.6)

(m − 1)r2 + r1 m−1  1 m−1 f Y + f Yu − Yuu − e Fαα . m m m m

1 From (4.3) and (4.6), noting r1 − r2 = −ae2f + f  + m , we know that Y satisfies   m−1 Yuuu + f  Yuu + f  + (r1 − r2 ) + Yu + ((r1 − r2 ) + f  )Y = (Fαα ef )u . m

(4.7)

(4.8)

The equation (4.4) implies there exists vector functions ξ = ξ(u) and η = η(v) such that Y = e−f (ξ(u) + η(v)).

(4.9)

Let η = eh ζ, then      ∂h 2  ∂h ∂ζ ∂2h ∂h ∂h ∂ 2ζ −h Fαβ = e − + δαβ ζ + + δαβ . ∂vα ∂vβ ∂vα ∂vβ ∂vγ ∂vα ∂vβ ∂vγ ∂vγ γ γ Because Fαβ satisfies Fαα = Fββ ,

Fαβ = 0,

(4.10)

α = β.

Thus, from (4.10) we have   ∂2h ∂2ζ ∂h ∂h =− − ζ, α = β, ∂vα ∂vβ ∂vα ∂vβ ∂vα ∂vβ  2  2   ∂ h ∂ h ∂h ∂h ∂h ∂h ∂ 2ζ ∂2ζ + − + − ζ = ζ. ∂vα2 ∂vα2 ∂vα ∂vα ∂vβ2 ∂vβ2 ∂vβ ∂vβ

(4.11) (4.12)

Because of the function h given by (3.19), we get ∂2ζ = 0, ∂vα ∂vβ

∂2ζ ∂2ζ = 2, 2 ∂vα ∂vβ

α = β,

(4.13)

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which shows ζ = v a + 2

m 

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vαbα + c,

(4.14)

α=2

where a, bα and c are constant vectors. With (4.10) and (3.19), we have a Fαα + aη = 2a + c. 2

(4.15)

Now consider the unknown function ξ. By substituting (4.9) into (4.8), we have   a ξ  − 2f  ξ  + (−ae2f + (f  )2 − f  + 1)ξ  − af  e2f ξ = f  e2f 2a + c . 2

(4.16)

From (4.14)–(4.16) we have the following inclusion: Theorem 4.1. Let Y be the M¨ obius position vector of the immersion x : M m → Sm+1 (m  3) with two distinct principal curvatures and non-vanishing closed M¨ obius form. Then Y = e−f (u) (ξ(u) + η(v)), where vector function ξ satisfies (4.16), function h(v) is given by (3.19) and vector function η is given by   m  h(v) 2  vα bα + c . (4.17) v a + η(v) = e α=2

Remark 4.2.

5

The conditions satisfied by constant vectors a, bα , c will be given in the next section.

Proof of Theorem 1.1

From Proposition 3.1, Theorems 3.2 and 3.7, we see that it is necessary to determine the constant vectors a, bα , c in Theorem 4.1. From Theorems 3.7 and 4.1, Y satisfies the following conditions: Y = e−f (ξ(u) + η(v)), Y, Y  = 0,   ∂Y ∂Y ∂Y ∂Y , = 1, , = 0, ∂u ∂u ∂u ∂vα Lemma 5.1.



∂Y ∂Y , ∂vα ∂vβ



(5.1) = e−2f +2h δαβ .

(5.2)

Let a, bα , c and ξ be the quantities given in Theorem 4.1. Then ξ  , ξ   = e2f ;  a ξ + c, 2a − c + 1 = 0; 2   a a a 2a − c, bα = 0, 2a − c, 2a − c = a. 2 2 2

ξ + c, ξ + c = 0,

(5.3)

ξ + c, bα  = 0,

(5.4)

bα , bβ  = δαβ ,

(5.5)

Proof. (5.1) implies ξ + η, ξ + η = 0, for all (u, v). In particular, h(0) = 0, η(0) = c, ξ +c, ξ +c = 0, and then ξ + c, ξ   = 0. The first equation of (5.2) implies e−2f (−2f  ξ + c, ξ   + ξ  , ξ  ) = 1. Hence ξ  , ξ   = e2f . From the expressions of h and η we have

2 a ∂η

bα , ∂ η = = 2a − c. ∂vα v=0 ∂vα2 v=0 2

(5.6)

(5.1) and (5.6) imply  ∂η ξ + η, = 0, ∂vα



∂η ∂η , ∂vα ∂vα



 ∂2η + ξ + η, 2 = 0. ∂vα

(5.7)

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So ξ + c, bα  = 0,

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 a ξ + c, 2a − c + bα , bα  = 0. 2

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(5.8)

The third equation of (5.2) implies  a a   − vα η + 2vαa + bα , − vβ η + 2vβa + bβ = δαβ , 2 2 then bα , bβ  = δαβ and 

a ∂η a a + 2a − vα η + 2vαa + bα , − η − vα 2 2 2 ∂vα

= 0.

(5.9)

Then by differentiating the two sides of (5.9) by ∂/∂vα and taking v = 0, we get the second equation and the third equation of (5.5). We need to discuss the cases a = 0, a < 0 and a > 0. Case I. a = 0. Let y be a component of the unknown vector value function ξ in the equation (4.16). We consider the ordinary equation y  − 2f  y  + ((f  )2 − f  + 1)y  = f  e2f . (5.10) Let z1 = ef cos u, z2 = ef sin u,   y0 = ψdu, y1 = z1 du,

  ψ(u) = z1 z1 du + z2 z2 du,  y2 = z2 du.

(5.11) (5.12)

Then the general solution of (5.10) can be expressed by y = y0 + Ay1 + By2 , where A and B are constant. Thus the general solution of (4.14) can be expressed as  + y2 B,  ξ = 2ay0 + y1 A

(5.13)

 and B  are constant vectors. From Lemma 5.1, there yields where A a, bα  = 0,

bα , bβ  = δαβ ,

(5.14)

 1 + bα , By  2 + bα , c = 0, bα , Ay  1 + a, By  2 + c, a = − 1 , a, Ay 2 2 2 2f         A, Az1 + B, Bz2 + 4a, Aψz1 + 4a, Bψz 2 + 2A, Bz1 z2 = e ,

(5.15)

a, a = 0,

 A  y12 A,

+

 B  y22 B,

(5.16) (5.17)

 0 y1 + 4a, By  0 y2 + 2A,  By  1 y2 + 4a, Ay

 cy1 + 2B,  cy2 + c, c = 0. + 4a, cy0 + 2A,

(5.18)

Since y1 , y2 , 1 are linear independent, from (5.15) and (5.16),  = bα , B  = bα , c = 0, bα , A

 = a, B  = 0, a, A

1 c, a = − , 2

and  A  − 1)z 2 + (B,  B  − 1)z 2 + 2A,  Bz  1 z2 = 0. (A, 1 2 Since z12 , z22 and z1 z2 are linear independent,  A  = B,  B  = 1, A,

 B  = 0, A,

(5.19)

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and  cy1 + 2B,  cy2 + c, c = 0. y12 + y22 − 2y0 + 2A,

(5.20)

By differentiating (5.20) and noting that y0 = y1 y1 + y2 y2 , we have  c + y2 B,  c = 0, y1 A, which shows  c = B,  c = 0, A,

y12 + y22 + c, c = 2y0 .

 B},  up to a Lorentz transformation in Rm+3 , as follows: Hence we can take {a, bα , c, A, 1 a = (1, −1, 0, 0, . . . , 0),

c = (c1 , c2 , 0, 0, . . . , 0),

bα = (0, . . . , 0, 1, 0, . . . , 0),   

2  α  m,

α+2

 = (0, 0, 1, 0, . . . , 0), A

 = (0, 0, 0, 1, . . . , 0). B

Noting a, c = − 12 (see (5.19)), we have c1 + c2 =

1 , 2

c1 + c, c =

1 , 4

c2 − c, c =

1 . 4

Hence ξ + η = (2y0 + c1 + v2 , −2y0 + c2 − v2 , y1 , y2 , v2 , . . . , vm )   1 1 = y12 + y22 + v2 + , − (y12 + y22 + v2 ), y1 , y2 , v . 4 4 Noting ρ(1, x) = e−f (ξ + η), we have   1 − 4(y12 + y22 + v2 ) 2(y1 , y2 , v) , 2 x= . 4(y12 + y22 + v2 ) + 1 4(y12 + y22 + v2 ) + 1

(5.21)

Without lose of generality, we can use v instead of 2v and c1 , c0 instead of 2c1 , 2c0 . Set (u, v) = (y1 , y2 , v), which is an (m − 1)-dimensional cylinder surface in Rm+1 . Denote the inverse stereographic projection by σ which is defined by (1.4). Then, from (5.21), x = σ ◦ . Case II. a = 0. Let   1 a ξ¯ = ξ + c + 2a − c , a 2 then we can write (4.16) as ξ¯ − 2f  ξ¯ + (−ae2f + (f  )2 − f  + 1)ξ¯ − af  e2f ξ¯ = 0. From Lemma 5.1 we have ¯ ξ ¯ = −1, ξ, a

 ¯ 2a − a c = 0, ξ, 2

¯ bα  = 0, ξ,

ξ¯ , ξ¯  = e2f .

(5.22)

(5.23)

Subcase II-1. a < 0. In this case, (5.5) shows that 2a − a2 c is a time-like vector. We can take it and bα , up to a Lorentz transformation in Rm+3 , as follows: 1 √ a 2a − c = ( −a, 0, . . . , 0), bα = (0, . . . , 0, 1, . . . , 0), 2  α  m.    2 α−1

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Let  = (0, . . . , 0, 1, 0, 0), A

 = (0, . . . , 0, 0, 1, 0), B

 = (0, . . . , 0, 0, 1). C

 B,  C,  bα } is a Lorentz orthonormal basis in Rm+3 . From (5.23) we see that Then {2a − a2 c, A, 1 √ ¯  B,  C}  ∼ ξ(u) ∈ S2 (1/ −a) ⊂ span{A, = R3 ,

     1 a bα vα . 2a − c v2 + 2 2 α

(5.24)

 1 − a4 v2 v ¯ √ , ξ(u) . , −a(1 + a4 v2 ) 1 + a4 v2

(5.25)

and η(v) − c = eh(v) (5.23) and (5.24) yield  ξ(u) + η(v) =

√ √ Locally, we can take N m−1 = {v : v ∈ Rm−1 , v < 2/ −a}. Let φ : N m−1 → Hm−1 (1/ −a) denote the inverse stereographic projection which is defined by   1 − a4 v2 v φ(v) = √ , . −a(1 + a4 v2 ) 1 + a4 v2 Then (5.25) implies √ ¯ = Hm−1 (1/ −a) × Γ2 . (ξ + η)(M m ) = φ(N m−1 ) × ξ(L) Hence

√ X(M m ) = π((ξ + η)(M m )) = π(Hm−1 (1/ −a) × Γ2 ),

where X and π are the maps defined in §1. Subcase II-2. a > 0. In this case, (5.5) shows that 2a − a2 c is a space-like vector. We can take it and bα , up to a Lorentz , as follows: transformation in Rm+3 1 √ a 2a − c = (0, 0, 0, − a, 0, . . . , 0), 2

bα = (0, . . . , 0, 1, . . . , 0),   

2  α  m.

α+2

Let  = (1, 0, 0, 0, . . . , 0), A

 = (0, 1, 0, 0, . . . , 0), B

 = (0, 0, 1, 0, . . . , 0). C

 B,  C,  bα } is a Lorentz orthonormal basis in Rm+3 . From (5.23) we see that Then {2a − a2 c, A, 1 √ ¯  B,  C}  ∼ ξ(u) ∈ H2 (1/ a) ⊂ span{A, = R31 , and

  1 − a4 v2 v ¯ √ , ξ(u), . (5.26) a(1 + a4 v2 ) 1 + a4 v2 √ Set N m−1 = Rm−1 ∪ {∞} and let ψ : Rm−1 ∪ {∞} → Sm−1 (1/ a) denote the inverse stereographic projection, which is defined by   1 − a4 v2 v , . ψ(v) = √ a(1 + a4 v2 ) 1 + a4 v2 ξ(u) + η(v) =

Then (5.26) implies

√ (ξ + η)(M m ) = ξ(L) × ψ(N m−1 ) = Γ3 × Sm−1 (1/ a).

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√ X(M m ) = π((ξ + η)(M m )) = π(Γ3 × Sm−1 (1/ a)).

We will show the meaning of parameter u. The curvature sphere corresponding to the M¨ obius principal 1 curvature − m is 1 , (5.27) P = E − Y : M m → Sm+2 1 m = {Z ∈ Rm+3 : Z, Z = 1}, called (m + 2)-dimensional de Sitter space. The second equation where Sm+2 1 1 of (4.3) shows  ∂P ∂P ∂P , = 0, 2  α  m, (5.28) = 1, ∂u ∂u ∂vα and u is its arc-length parameter. which means P degenerates into a curve in Sm+2 1 In following we will give the relationship between curves P and ξ to explain the meaning of the constrain condition (5.22). On the one hand, (4.6) implies P = Yuu + f  Yu + (−ae2f + f  )Y − ef Fαα .

(5.29)

On the other hand, the first equation of (4.3) implies P  = −f  Y − Yu .

(5.30)

By differentiating (5.30) and taking v → 0 we have P  = −e−f ξ  ,

a  f P + P = −e a(ξ + c) + 2a − c , 2

a P = −ef a(ξ + c) + 2a − c + (e−f ξ  ) . 2

(5.31) (5.32) (5.33)

Hence, for case a = 0, P is a solution of the following equation: P  + P = −2ef a.

(5.34)

For case a = 0, by setting ξ¯ = ξ + c + a1 (2a − a2 c), the relations between P and ξ¯ are as follows: P = −aef ξ¯ + (e−f ξ¯ ) , P  = −e−f ξ¯ , 1 ξ¯ = − e−f (P + P  ). a

(5.35) (5.36) (5.37)

Next we show that the curves Γ2 and Γ3 are determined by function f . Let s denote the arc parameter ¯ ds of Γi (i = 2, 3). From (5.23) we see that du = ef . Set ξ¯˙ = ddsξ and q = ef , then (5.22), (5.23) imply ξ¯ satisfies the following equation system: ... ¯ + (1 + (m − 2)q˙2 − (m − 3)aq 2 )ξ¯˙ − aq q˙ξ¯ = 0, q 2 ξ¯ + q q˙ξ¨ (5.38) ¯˙ ξ ¯˙ = 1. ¯ ξ ¯ = − 1 , ξ, (5.39) ξ, a √ √ It is well known that a curve in S2 (1/ −a)(a < 0) or H2 (1/ a)(a > 0) is determined by the geodesic curvature, up to a transformation in O(3) or O(1, 2). We can calculate the geodesic curvature κ of Γi by using (5.38) and (5.39). In fact, (5.38) is equivalent to the following: ¯ = aξ¯ + 1 P, ξ¨ q

1 ¯˙ P˙ = − ξ. q

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Thus, by using (5.38) we see that {

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 ¯ ξ, ¯˙ P} is an orthonormal frame field along the curve and |a|ξ, ¨¯ P = 1 . κ = ξ, q

(5.40)

Conversely, it is easy to check that for any smooth function f , the curve with geodesic curvature κ = e−f √ √ in S2 (1/ −a)(a < 0) or H2 (1/ a)(a > 0) must satisfy the equation system (5.38), (5.39). By concluding above all and Theorem 3.1, we complete the proof of Theorem 1.1. A Riemannian manifold (M, g) is called conformally flat if to each point p ∈ M , there exists a function ϕ defined on some neighborhood of this point such that the metric e2ϕ g on the neighborhood is flat. It is known that M with dimension greater than 3 is conformally flat if and only if its Weyl curvature tensor W vanishes. And M with dimension 3 is conformally flat if and only if its Schouten tensor S is a Codazzi tensor. To see that Theorem 1.1 implies Corollary 1.2, we need the following classical result. Here we present a reproof by using M¨obius invariants. Proposition 5.2. An m-dimensional umbilic-free hypersurface x : M m → Sm+1 (m  4) is conformally flat if and only if its M¨ obius second fundamental form B has two distinct principal curvatures m−1 m 1 and − m of multiplicities 1 and m − 1.   Proof. Let W = ijkl Wijkl ωi ⊗ ωj ⊗ ωk ⊗ ωl , S = ij Sij ωi ⊗ ωj be the Weyl curvature tensor and Schouten tensor of x given by Sij = Rij −

1 Rδij , 2(m − 1)

Wijkl = Rijkl −

1 Aijkl , m−2

where Aijkl = Sik δjl + Sjl δik − Sil δjk − Sjk δil . From (5.41), (2.12) and (2.13) we get   1 Wijkl = Bik Bjl − Bil Bjk − Bih Bhk δjl − Bjh Bhl δik − m−2 h h

  1 + Bih Bhl δjk + Bjh Bhk δil + (δik δjl − δil δjk ) . m h

(5.41)

(5.42)

h

At a fixed point p ∈ M we can choose a tangent orthonormal frame {Ei } of (M, g) such that Bij = Bii δij , 1  i, j  m. Since the Weyl curvature tensor is curvature tensor, it is determined by {Wijij }. So M (dimM > 3) is conformally flat if and only if Bii Bjj +

1 1 2 2 (Bii = 0, + Bjj )− m−2 m(m − 2)

i = j.

(5.43)

If M (dim > 3) is conformally flat, from (5.43), by taking i, j, k, l such that they are distinct each other, we can get (Bii − Bjj )(Bkk − Bll ) = 0, (5.44) which means at most two of Bii , Bjj , Bkk , Bll are distinct. This and the assumption of umbilic-free imply that there are two of Bii , Bjj , Bkk , Bll which are distinct. Without lose of generality, we suppose Bii = Bjj = Bkk = Bll . Since i, j, k, l are distinct with each other and range from 1 to m, we know that Bii has multiplicity 1 and Bjj has multiplicity m − 1. For convenience, we arrange E1 , . . . , Em such that B11 = λ, Using (2.13), we have

1 m−1 , μ=− . (5.45) m m = · · · = Bmm = μ, it is easy to see that (5.43) holds and then M is λ=

Conversely, if B11 = λ, B22 conformally flat.

B22 = · · · = Bmm = μ.

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Remark 5.3. For M with dimension 3, after calculating the covariant derivations of Sij from (5.41) and (2.12), we can see that S is a Codazzi tensor if and only if Bik,j (Bii + 2Bjj ) = (2Bii + Bjj )(Cj δik − Ck δij ).

(5.46)

So, from (3.2), (3.3) and m = 3, we see that M must be conformally flat if it has two distinct principal curvatures. But if M is conformally flat, it may have three distinct principal curvatures. Proof of Corollary 1.2. It is quite evident that Proposition 5.2 and Theorem 1.1 imply Corollary 1.2.2 Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos. 10561010, 10861013). The authors would like to thank the referees for very helpful suggestions.

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Mathematics CONTENTS

Vol. 55 No. 7 July 2012

Articles Weak crossed biproducts and weak projections ....................................................................................................................................... FERNÁNDEZ VILABOA Jose Manuel, GONZÁLEZ RODRÍGUEZ Ramon & RODRÍGUEZ RAPOSO Ana Belen

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Maximal subalgebras of the general linear Lie algebra containing Cartan subalgebras ........................................................................... WANG DengYin, GE Hui & LI XiaoWei

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Projection pressure and Bowen’s equation for a class of self-similar fractals with overlap structure ...................................................... WANG ChenWei & CHEN ErCai

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Invariance of shift-invariant spaces .......................................................................................................................................................... ZHANG QingYue & SUN WenChang

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Hardy spaces HLp(n) associated with operators satisfying k-Davies-Gaffney estimates ......................................................................... CAO Jun & YANG DaChun

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K0-group and similarity of operator weighted shifts ................................................................................................................................. LI JueXian

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A new proof of Wu’s theorem on vortex sheets ....................................................................................................................................... WANG Chao & ZHANG ZhiFei

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Hierarchy paths in the Hatcher-Thurston complex ................................................................................................................................... MA JiMing

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A multilevel preconditioner for the C-R FEM for elliptic problems with discontinuous coefficients ................................................... WANG Feng, CHEN JinRu & HUANG PeiQi

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