Classification of hypersurfaces with two distinct principal curvatures and closed Möbius form in $mathbb{S}^{m + 1}$

$Page 1: Classification of hypersurfaces with two distinct principal curvatures and closed Möbius form in $mathbb{S}^{m + 1}$$
SCIENCE CHINAMathematics

. ARTICLES . July 2012 Vol. 55 No. 7: 1463–1478

doi: 10.1007/s11425-012-4391-1

c© Science China Press and Springer-Verlag Berlin Heidelberg 2012 math.scichina.com www.springerlink.com

Classification of hypersurfaces with two distinctprincipal curvatures and closed Mobius form in S

m+1

LIN LiMiao1,2 & GUO Zhen2,∗

1Department of Mathematics, East China Normal University, Shanghai 200241, China;2Department of Mathematics, Yunnan Normal University, Kunming 650092, China

Email: [email protected], [email protected]

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 princi-

pal curvatures and closed Mobius form, and then we characterize and classify conformally flat hypersurfaces of

dimension larger than 3.

Keywords moebius geometry, principal curvature, conformally flat, Mobius form

MSC(2010) 54A30, 53C21, 53C40

Citation: Lin L M, Guo Z. Classification of hypersurfaces with two distinct principal curvatures and closed Mobius

form in Sm+1. Sci China Math, 2012, 55(7): 1463–1478, doi: 10.1007/s11425-012-4391-1

1 Introduction

Let x : Mm → 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 hijθiθj be the second fundamental form and H be the mean

curvature of x. Define ρ2 = mm−1 |II− 1

m tr(II)I|2, then positive definite 2-form g = ρ2I is invariant under

Mobius transformation group of Sm+1 and is called Mobius metric of x. Three basic Mobius invariants

of x, Mobius form Φ = ρ∑

iCiθi, Blaschke tensor A = ρ2∑

ij Aijθiθj and Mobius second fundamental

form B = ρ2∑

ij Bijθiθj , are defined by (see [6])

Ci = −ρ−2

(

ei(H) +∑

j

(hij −Hδij)ej(logρ)

)

, (1.1)

Aij = −ρ−2(Hessij(logρ)− ei(logρ)ej(logρ)−Hhij)− 1

2ρ−2(‖∇logρ‖2 − 1 +H2)δij , (1.2)

Bij = ρ−1(hij −Hδij), (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 Mobius principal curvatures of x.

In Mobius geometry, since the eigenspaces of B coincide with II of x, the number of distinct principal

curvatures is a Mobius invariant. A hypersurface in Sm+1 with two distinct principal curvatures is Dupin

hypersurface if and only if its Mobius form vanishes. Pinkall [5] showed that Dupin conditions can be

∗Corresponding author

1464 Lin L M et al. Sci China Math July 2012 Vol. 55 No. 7

naturally formulated in the context of Lie sphere geometry and that they are invariant under the Lie sphere

transformation group. In 1985, Ceceil and Ryan [4] have proved that all Dupin hypersurfaces in Sm+1 are

equivalent by a Lie sphere transformation to Sk(r)×S

m−k(√1− r2) and so they gave the classification of

hypersurfaces in Sm+1 (m � 3) with two distinct principal curvatures and vanishing Mobius form under

the Lie sphere transformation group. Note that the Lie sphere transformation group contains the Mobius

transformation group in Sm+1 as a subgroup. In 2002, Li et al. [11] classified hypersurfaces in S

m+1 with

two distinct principal curvatures and vanishing Mobius form under the Mobius transformation group.

Now we are interested in: How about hypersurfaces in Sm+1 with two distinct principal curvatures and

non-vanishing Mobius form under the Mobius transformation group? In [15,16], the authors classified all

hypersurfaces with isotropic Blaschke tensor and all hypersurfaces with constant Mobius curvature. They

found that all these hypersurfaces have two distinct principal curvatures and closed non-vanishing Mobius

form. Here, we will classify all hypersurfaces in Sm+1 (m � 3) with two distinct principal curvatures and

closed Mobius form.

In order to make our main results intuitional, we use the following notations: Rm+31 denotes the Lorentz

space with the inner product 〈·, ·〉 given by

〈Y, Z〉 = −y0z0 + y1z1 + · · ·+ ym+2zm+2,

where Y = (y0, y1, . . . , ym+2), Z = (z0, z1, . . . , zm+2) ∈ Rm+3. Cm+2

+ and Qm+1 denote the positive light

cone and the quadric in real projection space RPm+2, which are defined as follows:

Cm+2+ = {X ∈ R

m+31 : 〈X,X〉 = 0, x0 > 0}, Q

m+1 = {[Y ] ∈ RPm+2 : 〈Y, Y 〉 = 0}.

Let Hm+1 be the hyperbolic space defined by

Hm+1 = {(y0, y1) ∈ R

+ × Rm+1| − y20 + y1y1 = −1}.

Now let σ : Rm+1 → Sm+1 be the inverse of the stereographic projection given by

σ(u) =

(1− |u|21 + |u|2 ,

2u

1 + |u|2)

, (1.4)

and τ : Hm+1 → Sm+1 be the conformal map defined by

τ(y0, y1) =

(1

y0,y1y0

)

, (y0, y1) ∈ Hm+1. (1.5)

The conformal maps σ and τ assign any hypersurface in Rm+1 or Hm+1 to a hypersurface in S

m+1. We

use map π : Cm+2+ → Qm+1 to denote the natural projection. For a hypersurface x : Mm → S

m+1, we

have map

X := π(1, x) :Mm → Qm+1, (1.6)

which is determined by the immersion x and is called the natural map of x. It is known that two

hypersurfaces x, x : Mm → Sm+1 are Mobius equivalent if and only if whose natural maps X, X :

Mm → Qm+1 are equivalent under the action of Lorentz group O(m + 2, 1). Now, we state the main

theorem as follows:

Theorem 1.1. Let x : Mm → Sm+1 (m � 3) be a hypersurface with two distinct principal curvatures

and closed Mobius form. Then x, up to a Mobius transformation of Sm+1, is locally one of the following

hypersurfaces:

(1) the standard torus Sk(r) × S

m−k(√1− r2);

(2) the image of σ of the standard cylinder Sk(1)× R

m−k ⊂ Rm+1;

(3) the image of τ of the standard Sk(r) ×H

m−k(√1 + r2) in H

m+1;

(4) x(M) = σ(Γ × Rm−1), Γ ⊂ R

2, where Γ is any smooth curve with non-constant curvature in R2;

Lin L M et al. Sci China Math July 2012 Vol. 55 No. 7 1465

(5) for any negative constant a, X(M) = π(Hm−1(1/√−a) × Γ), Γ ⊂ S

2(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 × S

m−1(1/√a)), Γ ⊂ H

2(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 Mobius form and they are Dupin hypersurfaces,

but the hypersurfaces in the last three cases have non-vanishing Mobius 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 Mobius geometry, therefore, we

get Corollary 1.2.

Corollary 1.2. Let x : Mm → Sm+1 (m � 4) be an umbilic-free conformally flat hypersurface with

closed Mobius form. Then x, up to a Mobius transformation of Sm+1, is locally one of the following

hypersurfaces:

(1) the standard torus S1(r) × S

m−1(√1− r2);

(2) the image of σ of the standard cylinder S1(1)× R

m−1 ⊂ Rm+1;

(3) the image of τ of the standard S1(r) ×H

m−1(√1 + r2) in H

m+1;

(4) x(M) = σ(Γ × Rm−1), Γ ⊂ R

2, 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 ⊂ S

2(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 × S

m−1(1/√a)), Γ ⊂ H

2(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 Mobius 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 Mobius curvature K. In fact, it was proved that the two classes

of hypersurfaces are all conformally flat hypersurfaces with closed Mobius 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

ds

1

κ

)2

+ a

(1

κ

)2

− 1

mlog

(1

κ

)2

+1

m2= 0,

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

(d

ds

1

κ

)2

+ a

(1

κ

)2

= −K,

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 Mobius

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 Mobius form. We get the local

express of their basic Mobius invariants (Proposition 3.1, Lemma 3.3 and Lemma 3.4) and get Theo-

rem 3.2. In Section 4 we obtain the differential equations characterizing the hypersurfaces with two

distinct principal curvatures and closed non-vanishing Mobius 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 Mobius geometry and then get

Corollary 1.2.


2 Mobius invariants for a hypersurface in Sm+1

In this section, we define Mobius invariants and recall the structure equations for hypersurfaces in Sm+1.

For details we refer to [6].

Let x : Mm → Sm+1 ⊂ R

m+2 be an umbilic-free hypersurface immersed in Sm+1. We define the

Mobius position vector Y :Mm → Rm+31 of x by

Y = ρ(1, x) :Mm → Rm+31 , ρ2 =

m

m− 1(‖II‖2 −mH2) > 0.

Then we have the following:

Theorem 2.1 (See [6]). Two hypersurfaces x, x : Mm → Sm+1 are Mobius equivalent if and only if

there exists T in the Lorentz group O(m+ 2, 1) acting on Rm+31 such that Y = Y T .

Since the Mobius transformation group in Sm+1 is isomorphic to the subgroup O+(m + 2, 1), which

preserves the positive part of the light cone in Rm+31 , from Theorem 2.1 we know 2-form

g = 〈dY, dY 〉 = ρ2dx · dx (2.1)

is a Mobius invariant and is called Mobius metric or Mobius first fundamental form induced by x (cf.

[6, 18, 19]). Let be the Laplacian with respect to (M,g). We define

N = − 1

mY − 1

2m2〈Y, Y 〉Y. (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+31 . Then we have the orthogonal

decomposition

Rm+31 = span{Y,N} ⊕ span{Y1, . . . , Ym} ⊕ V. (2.3)

Let E be a unit vector field of V , then {Y,N, Y1, . . . , Ym, E} forms a moving frame in Rm+31 along M .

We will use the following range of indices in this section: 1 � i, j, k, . . . � m. The structure equations

are given by

dY =∑

i

ωiYi, (2.4)

dN =∑

ij

AijωjYi +∑

i

CiωiE, (2.5)

dYi = −∑

j

AijωjY − ωiN +∑

j

ωijYj +∑

j

BijωjE, (2.6)

dE = −∑

i

CiωiY −∑

ij

BijωjYi, (2.7)

where ωij is the connection form of the Mobius metric g, Aij and Bij are symmetric with respect to (ij).

It is clear that A =∑

ij Aijωi ⊗ ωj, B =∑

ij Bijωi ⊗ ωj , Φ =∑

i Ciωi are Mobius invariants and are

called the Blaschke tensor, Mobius second fundamental form and Mobius form of x, respectively.

Remark 2.2. The relations between A,B,Φ and Euclidean invariants of x are given by (1.1), (1.2)

and (1.3).


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 = BikCj −BijCk, (2.8)

Ci,j − Cj,i =∑

k

BikAkj −∑

k

BjkAki, (2.9)

Bij,k − Bik,j = δijCk − δikCj , (2.10)

Rijkl = BikBjl −BilBjk + δikAjl + δjlAik − δilAjk − δjkAil, (2.11)

Rij = −∑

k

BikBkj + tr(A)δij + (m− 2)Aij , (2.12)

tr(A) =1

2m

(

1 +m

m− 1R

)

,∑

i

Bii = 0,∑

ij

(Bij)2 =

m− 1

m, (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∑

i

Bij,i = −(m− 1)Cj . (2.14)

Remark 2.3. Some recent results about the Mobius geometry of submanifolds can be found in [2,7–16].

3 Mobius invariants of a hypersurface with two distinct principal curvatures

and closed Mobius form in Sm+1

Let x : Mm → Sm+1 ⊂ R

m+2 (m � 3) be an umbilic-free hypersurface with two distinct principal

curvatures and closed Mobius form. The goal of this section is to determine its basic Mobius invariants

{g,Φ,B,A}. It was showed by Li et al. (see [11]) that x has two distinct constant Mobius principal

curvatures, which are given by

λ =1

m

√(m− 1)(m− k)

k, μ = − 1

m

√(m− 1)k

m− k,

where k is the multiplicity of λ.

Proposition 3.1. Let x : Mm → Sm+1 ⊂ R

m+2 be a hypersurface with two distinct principal curva-

tures, then there are only two cases:

(1) Mobius form Φ of M vanishes;

(2) Mobius form Φ of M does not vanish but M has two distinct Mobius principal curvatures m−1m and

− 1m 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 =⋃

p∈M

V1(p), V2 =⋃

p∈M

V2(p), (3.1)

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 TM = 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 .


From (2,10) and (2.14), we get

Bab,j = 0, Bαβ,j = 0, Baα,b = −δabCα, Baα,β = −δαβCa, (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 .

Without loss of generality, we can assume k = 1 and then λ = m−1m , μ = − 1

m .

The hypersurfaces of Case (1) in Proposition 3.1 have been classified by Li et al.

Theorem 3.2 (See [11]). Let x : Mm → Sm+1 (m � 3) be a hypersurface with two distinct principal

curvatures and vanishing Mobius form. Then x is Mobius equivalent to an open part of one of the following

hypersurfaces in Sm+1:

(1) the standard torus Sk(r) × S

m−k(√1− r2);

(2) the image of σ of the standard cylinder Sk(1)× R

m−k ⊂ Rm+1;

(3) the image of τ of the standard Sk(r) ×H

m−k(√1 + r2) in H

m+1.

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,

B1j =m− 1

mδ1j , Bαj = − 1

mδαj , Aij = Aiiδij , (3.2)

B11,j = 0, Bαβ,j = 0, B1α,j = −C1δαj , Cα = 0, ω1α = −C1ωα. (3.3)

Hence dω1 = 0 which implies V2 is integrable according to Frobenius theorem. Denote the integral

manifold of V2 by Nm−1. Then, locally, we have

Mm = L×Nm−1,

where L is an interval in R1. For each u ∈ L, g induces a positive definite metric g = g|{u}×Nm−1 on

{u} ×Nm−1. Since dΦ = 0 and Cα = 0, there exists locally a smooth function f = f(u) such that

Φ = df, C1 = f ′.

Let Rαβγδ, Rαβ and R denote the components of curvature tensor, components of Ricci curvature tensor

and scalar curvature of ({u} ×Nm−1, g), respectively.

Lemma 3.3.

Rαβγλ = Rαβγλ − C21δαβγλ, R1α1β =

(∂C1

∂u− C2

1

)

δαβ, R1αβγ = 0, (3.4)

where δαβγλ = δαγδβλ − δαλδβγ.

Proof. Choose E1 = ∂∂u , then {ω1 = du, ωα} is the dual frame of {E1, Eα}. So g = du2 +

∑α ω

2α

and g =∑

α ω2α where ωα = ωα|{u}×Nm−1 . Let ωij be the connection form of (M,g) and ωαβ be the

connection form of ({u}×Nm−1, g). Denote dN as the exterior differential operator on T ∗Nm−1. Because

the exterior differential operator on T ∗(L×Nm−1) is d = du ∧ ∂∂u + dN , then

du ∧(∂ωα

∂u+ C1ωα

)

=∑

β

ωβ ∧ (ωβα − ωβα), (3.5)

which implies that there exists function aαβ such that

ωαβ = ωαβ + aαβdu, aαβ = ωαβ

(∂

∂u

)

.


So

− 1

2

∑

γλ

Rαβγλωγ ∧ ωλ = dN ωαβ −∑

γ

ωαγ ∧ ωγβ,

− 1

2

∑

ij

Rαβijωi ∧ ωj = dN ωαβ −∑

γ

ωαγ ∧ ωγβ + C21ωα ∧ ωβ

+ du ∧(∂ωαβ

∂u− dNaαβ −

∑

γ

aαγ ωγβ +∑

γ

aγβωαγ

)

− 1

2

∑

ij

R1αijωi ∧ ωj

= −(∂C1

∂u− C2

1

)

ω1 ∧ ωα,

then we get (3.4) immediately.

Lemma 3.4. For fixed u, ({u} ×Nm−1, g) has constant curvature.

Proof. For the chosen basis in Lemma 3.3, from (3.4) and Gauss equation (2.11) we have

∂C1

∂u− C2

1 = −m− 1

m2+A11 +Aαα, 2 � α � m, (3.6)

which implies all Aαα equal (2 � α � m). Denote A11 = r1, Aαα = r2, then

Rαβαβ =1

m2+ 2r2, Rαβαβ = 2

(1

2m2+ r2 +

1

2C2

1

)

, α = β.

After calculating the covariant derivations of Aαβ , we get

Aαα,γ = Eγ(r2), Aαγ,α = 0, α = γ.

Because Cγ = 0, then Aαα,γ = Aαγ,α = 0. So Eγ(Rαβαβ) = 2Eγ(r2) = 0 which means ({u} ×Nm−1, g)

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

g = du2 + e2θ(u,v2,...,vm)(dv22 + · · ·+ dv2m), (3.7)

where e(·) denotes exponential function exp, θ is a smooth function on L×N . Then

E1 =∂

∂u, Eα = e−θ ∂

∂vα, ω1 = du, ωα = eθdvα, ω1α = −C1e

θdvα. (3.8)

Lemma 3.5. 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

∂vαdvβ − ∂h

∂vβdvα, (3.10)

where h is a smooth function on N .

Proof. The details can be found in [15].

Lemma 3.6.

∂2h

∂v2α+∂2h

∂v2β+

m∑

γ=2

(∂h

∂vγ

)2

−(∂h

∂vα

)2

−(∂h

∂vβ

)2

= −ae2h, α = β, (3.11)

Aαα = r2 =1

2

(

ae2f − (f ′)2 − 1

m2

)

, A11 = r1 = f ′′ − 1

2(f ′)2 +

2m− 1

2m2− 1

2ae2f , (3.12)

where a is a constant.


Proof. Using (3.8) and (3.10) to calculate the curvature tensor, we get

e2θRαβαβ = −∂2h

∂v2α− ∂2h

∂v2β− (f ′)2e2θ −

∑

γ

(∂h

∂vγ

)2

+

(∂h

∂vα

)2

+

(∂h

∂vβ

)2

, α = β.

Let

aαβ(v) := e−2h

[

− ∂2h

∂v2α− ∂2h

∂v2β−∑

γ

(∂h

∂vγ

)2

+

(∂h

∂vα

)2

+

(∂h

∂vβ

)2]

,

then

Rαβαβ = e2faαβ(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

Aαα = r2 =1

2

(

e2fa(v)− (f ′)2 − 1

m2

)

, (3.13)

A11 = r1 = f ′′ − 1

2(f ′)2 +

2m− 1

2m2− 1

2e2fa(v), (3.14)

Rαβαβ = a(v)e2f , α = β. (3.15)

Lemma 3.4 implies that Rαβαβ is independent of v and then a(v) is constant from (3.15). Hence there

hold (3.11), (3.12).

Now view quadratic form g = e2h(dv22 + · · · + dv2m) as a metric on manifold Nm−1. According to

Lemma 3.6, g has constant curvature a showing that there exists a local coordinate, still denoted by

(v2, . . . , vm), such that

g =dv22 + · · ·+ dv2m(1 + a

4‖v‖2)2,

where ‖v‖2 =∑m

α=2 v2α.

As summary, we come to the following conclusion:

Theorem 3.7. For an m (� 3)-dimensional hypersurface Mm with two distinct principal curvatures

and closed non-vanishing Mobius form in Sm+1, under a suitable local coordination (u, v2, . . . , vm), the

Mobius invariants of Mm can be expressed as follows:

g = du2 + e−2f(u)

(

e2h(v)∑

α

dv2α

)

, (3.16)

B =m− 1

mdu2 − 1

me−2f(u)

(

e2h(v)∑

α

dv2α

)

, (3.17)

Φ = f ′du, A = r1du2 + r2e

−2f(u)

(

e2h(v)∑

α

dv2α

)

, (3.18)

where f = f(u) is a smooth function, r1, r2 are given by (3.12) and

h(v) = − log

(

1 +a

4‖v‖2

)

. (3.19)

Theorem 3.7 shows that all the Mobius invariants in structure equations are determined by the functions

f and h, and so we can get the hypersurfaceMm by integrating the structure equations. We will complete

the procedure in the next section.


4 Differential equations of a hypersurface with two principal curvatures and

closed non-vanishing Mobius form in Sm+1

From the structure equations (2.4)–(2.7), by using (3.16)–(3.18) and Eα = e−θ ∂∂vα

, we get

Yuu = −(

r1Y +N − m− 1

mE

)

; (4.1)

Nu = r1Yu + f ′E,∂N

∂vα= r2

∂Y

∂vα; (4.2)

Eu = −(

f ′Y +m− 1

mYu

)

,∂E

∂vα=

1

m

∂Y

∂vα; (4.3)

∂2Y

∂u∂vα+ f ′ ∂Y

∂vα= 0; (4.4)

(

r2Y +N +1

mE − f ′Yu

)

δαβ = −efFαβ , (4.5)

where

Fαβ = ef−2h

(∂2Y

∂vβ∂vα−(∂h

∂vβ

∂Y

∂vα+

∂h

∂vα

∂Y

∂vβ

)

+∑

γ

∂h

∂vγ

∂Y

∂vγδαβ

)

.

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 − efFαα, (4.6)

N = − (m− 1)r2 + r1m

Y +m− 1

mf ′Yu − 1

mYuu − m− 1

mefFαα. (4.7)

From (4.3) and (4.6), noting r1 − r2 = −ae2f + f ′′ + 1m , we know that Y satisfies

Yuuu + f ′Yuu +

(

f ′′ + (r1 − r2) +m− 1

m

)

Yu + ((r1 − r2)′ + f ′)Y = (Fααe

f )u. (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 = ehζ, then

Fαβ = e−h

((∂2h

∂vα∂vβ− ∂h

∂vα

∂h

∂vβ+∑

γ

(∂h

∂vγ

)2

δαβ

)

ζ +∂2ζ

∂vα∂vβ+∑

γ

∂h

∂vγ

∂ζ

∂vγδαβ

)

. (4.10)

Because Fαβ satisfies

Fαα = Fββ, Fαβ = 0, α = β.

Thus, from (4.10) we have

∂2ζ

∂vα∂vβ= −

(∂2h

∂vα∂vβ− ∂h

∂vα

∂h

∂vβ

)

ζ, α = β, (4.11)

∂2ζ

∂v2α+

(∂2h

∂v2α− ∂h

∂vα

∂h

∂vα

)

ζ =∂2ζ

∂v2β+

(∂2h

∂v2β− ∂h

∂vβ

∂h

∂vβ

)

ζ. (4.12)

Because of the function h given by (3.19), we get

∂2ζ

∂vα∂vβ= 0,

∂2ζ

∂v2α=∂2ζ

∂v2β, α = β, (4.13)


which shows

ζ = ‖v‖2�a+m∑

α=2

vα�bα + �c, (4.14)

where �a, �bα and �c are constant vectors. With (4.10) and (3.19), we have

Fαα + aη = 2�a+a

2�c. (4.15)

Now consider the unknown function ξ. By substituting (4.9) into (4.8), we have

ξ′′′ − 2f ′ξ′′ + (−ae2f + (f ′)2 − f ′′ + 1)ξ′ − af ′e2fξ = f ′e2f(

2�a+a

2�c

)

. (4.16)

From (4.14)–(4.16) we have the following inclusion:

Theorem 4.1. Let Y be the Mobius position vector of the immersion x : Mm → Sm+1 (m � 3) with

two distinct principal curvatures and non-vanishing closed Mobius 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

η(v) = eh(v)(

‖v‖2�a+m∑

α=2

vα�bα + �c

)

. (4.17)

Remark 4.2. The conditions satisfied by constant vectors �a, �bα, �c will be given in the next section.

5 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, (5.1)⟨∂Y

∂u,∂Y

∂u

⟩

= 1,

⟨∂Y

∂u,∂Y

∂vα

⟩

= 0,

⟨∂Y

∂vα,∂Y

∂vβ

⟩

= e−2f+2hδαβ . (5.2)

Lemma 5.1. Let �a, �bα, �c and ξ be the quantities given in Theorem 4.1. Then

〈ξ + �c, ξ + �c〉 = 0, 〈ξ′, ξ′〉 = e2f ; (5.3)

〈ξ + �c,�bα〉 = 0,

⟨

ξ + �c, 2�a− a

2�c

⟩

+ 1 = 0; (5.4)

〈�bα,�bβ〉 = δαβ,

⟨

2�a− a

2�c,�bα

⟩

= 0,

⟨

2�a− a

2�c, 2�a− a

2�c

⟩

= a. (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

∂η

∂vα

∣∣∣∣v=0

= �bα,∂2η

∂v2α

∣∣∣∣v=0

= 2�a− a

2�c. (5.6)

(5.1) and (5.6) imply

⟨

ξ + η,∂η

∂vα

⟩

= 0,

⟨∂η

∂vα,∂η

∂vα

⟩

+

⟨

ξ + η,∂2η

∂v2α

⟩

= 0. (5.7)


So

〈ξ + �c,�bα〉 = 0,

⟨

ξ + �c, 2�a− a

2�c

⟩

+ 〈�bα,�bα〉 = 0. (5.8)

The third equation of (5.2) implies

⟨

− a

2vαη + 2vα�a+�bα,−a

2vβη + 2vβ�a+�bβ

⟩

= δαβ,

then 〈�bα,�bβ〉 = δαβ and

⟨

− a

2vαη + 2vα�a+�bα,−a

2η − a

2vα

∂η

∂vα+ 2�a

⟩

= 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 cosu, z2 = ef sinu, ψ(u) = z1

∫

z1du+ z2

∫

z2du, (5.11)

y0 =

∫

ψdu, y1 =

∫

z1du, y2 =

∫

z2du. (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

ξ = 2�ay0 + y1 �A+ y2 �B, (5.13)

where �A and �B are constant vectors. From Lemma 5.1, there yields

〈�a,�a〉 = 0, 〈�a,�bα〉 = 0, 〈�bα,�bβ〉 = δαβ , (5.14)

〈�bα, �A〉y1 + 〈�bα, �B〉y2 + 〈�bα,�c〉 = 0, (5.15)

〈�a, �A〉y1 + 〈�a, �B〉y2 + 〈�c,�a〉 = −1

2, (5.16)

〈 �A, �A〉z21 + 〈 �B, �B〉z22 + 4〈�a, �A〉ψz1 + 4〈�a, �B〉ψz2 + 2〈 �A, �B〉z1z2 = e2f , (5.17)

y21〈 �A, �A〉+ y22〈 �B, �B〉+ 4〈�a, �A〉y0y1 + 4〈�a, �B〉y0y2 + 2〈 �A, �B〉y1y2+ 4〈�a,�c〉y0 + 2〈 �A,�c〉y1 + 2〈 �B,�c〉y2 + 〈�c,�c〉 = 0. (5.18)

Since y1, y2, 1 are linear independent, from (5.15) and (5.16),

〈�bα, �A〉 = 〈�bα, �B〉 = 〈�bα,�c〉 = 0, 〈�a, �A〉 = 〈�a, �B〉 = 0, 〈�c,�a〉 = −1

2, (5.19)

and

(〈 �A, �A〉 − 1)z21 + (〈 �B, �B〉 − 1)z22 + 2〈 �A, �B〉z1z2 = 0.

Since z21 , z22 and z1z2 are linear independent,

〈 �A, �A〉 = 〈 �B, �B〉 = 1, 〈 �A, �B〉 = 0,


and

y21 + y22 − 2y0 + 2〈 �A,�c〉y1 + 2〈 �B,�c〉y2 + 〈�c,�c〉 = 0. (5.20)

By differentiating (5.20) and noting that y′0 = y′1y1 + y′2y2, we have

y′1〈 �A,�c〉+ y′2〈 �B,�c〉 = 0,

which shows

〈 �A,�c〉 = 〈 �B,�c〉 = 0, y21 + y22 + 〈�c,�c〉 = 2y0.

Hence we can take {�a,�bα,�c, �A, �B}, up to a Lorentz transformation in Rm+31 , as follows:

�a = (1,−1, 0, 0, . . . , 0), �c = (c1, c2, 0, 0, . . . , 0),

�bα = (0, . . . , 0︸︷︷︸

α+2

, 1, 0, . . . , 0), 2 � α � m,

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

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 + ‖v‖2,−2y0 + c2 − ‖v‖2, y1, y2, v2, . . . , vm)

=

(

y21 + y22 + ‖v‖2 + 1

4,1

4− (y21 + y22 + ‖v‖2), y1, y2, v

)

.

Noting ρ(1, x) = e−f (ξ + η), we have

x =

(1− 4(y21 + y22 + ‖v‖2)4(y21 + y22 + ‖v‖2) + 1

, 22(y1, y2, v)

4(y21 + y22 + ‖v‖2) + 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

ξ = ξ + �c+1

a

(

2�a− a

2�c

)

,

then we can write (4.16) as

ξ′′′ − 2f ′ξ′′ + (−ae2f + (f ′)2 − f ′′ + 1)ξ′ − af ′e2f ξ = 0. (5.22)

From Lemma 5.1 we have

〈ξ, ξ〉 = −1

a,

⟨

ξ, 2�a− a

2�c

⟩

= 0, 〈ξ,�bα〉 = 0, 〈ξ′, ξ′〉 = e2f . (5.23)

Subcase II-1. a < 0.

In this case, (5.5) shows that 2�a − a2�c is a time-like vector. We can take it and �bα, up to a Lorentz

transformation in Rm+31 , as follows:

2�a− a

2�c = (

√−a, 0, . . . , 0), �bα = (0, . . . , 0︸︷︷︸

α−1

, 1, . . . , 0), 2 � α � m.


Let�A = (0, . . . , 0, 1, 0, 0), �B = (0, . . . , 0, 0, 1, 0), �C = (0, . . . , 0, 0, 1).

Then {2�a− a2�c,

�A, �B, �C,�bα} is a Lorentz orthonormal basis in Rm+31 . From (5.23) we see that

ξ(u) ∈ S2(1/

√−a) ⊂ span{ �A, �B, �C} ∼= R3,

and

η(v)− �c = eh(v)(1

2

(

2�a− a

2�c

)

‖v‖2 +∑

α

�bαvα

)

. (5.24)

(5.23) and (5.24) yield

ξ(u) + η(v) =

(1− a

4‖v‖2√−a(1 + a4‖v‖2)

,v

1 + a4‖v‖2

, ξ(u)

)

. (5.25)

Locally, we can take Nm−1 = {v : v ∈ Rm−1, ‖v‖ < 2/

√−a}. Let φ : Nm−1 → Hm−1(1/

√−a) denote

the inverse stereographic projection which is defined by

φ(v) =

(1− a

4‖v‖2√−a(1 + a4‖v‖2)

,v

1 + a4‖v‖2

)

.

Then (5.25) implies

(ξ + η)(Mm) = φ(Nm−1)× ξ(L) = Hm−1(1/

√−a)× Γ2.

Hence

X(Mm) = π((ξ + η)(Mm)) = 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 2�a − a2�c is a space-like vector. We can take it and �bα, up to a Lorentz

transformation in Rm+31 , as follows:

2�a− a

2�c = (0, 0, 0,−√

a, 0, . . . , 0), �bα = (0, . . . , 0︸︷︷︸

α+2

, 1, . . . , 0), 2 � α � m.

Let�A = (1, 0, 0, 0, . . . , 0), �B = (0, 1, 0, 0, . . . , 0), �C = (0, 0, 1, 0, . . . , 0).

Then {2�a− a2�c,

�A, �B, �C,�bα} is a Lorentz orthonormal basis in Rm+31 . From (5.23) we see that

ξ(u) ∈ H2(1/

√a) ⊂ span{ �A, �B, �C} ∼= R

31,

and

ξ(u) + η(v) =

(

ξ(u),1− a

4‖v‖2√a(1 + a

4‖v‖2),

v

1 + a4‖v‖2

)

. (5.26)

Set Nm−1 = Rm−1 ∪ {∞} and let ψ : Rm−1 ∪ {∞} → S

m−1(1/√a) denote the inverse stereographic

projection, which is defined by

ψ(v) =

(1− a

4‖v‖2√a(1 + a

4‖v‖2),

v

1 + a4‖v‖2

)

.

Then (5.26) implies

(ξ + η)(Mm) = ξ(L)× ψ(Nm−1) = Γ3 × Sm−1(1/

√a).


Hence

X(Mm) = π((ξ + η)(Mm)) = π(Γ3 × Sm−1(1/

√a)).

We will show the meaning of parameter u. The curvature sphere corresponding to the Mobius principal

curvature − 1m is

P = E − 1

mY :Mm → S

m+21 , (5.27)

where Sm+21 = {Z ∈ R

m+31 : 〈Z,Z〉 = 1}, called (m+2)-dimensional de Sitter space. The second equation

of (4.3) shows ⟨∂P∂u

,∂P∂u

⟩

= 1,∂P∂vα

= 0, 2 � α � m, (5.28)

which means P degenerates into a curve in Sm+21 and u is its arc-length parameter.

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 − efFαα. (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−fξ′, (5.31)

P + P ′′ = −ef[

a(ξ + �c) + 2�a− a

2�c

]

, (5.32)

P = −ef[

a(ξ + �c) + 2�a− a

2�c

]

+ (e−fξ′)′. (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+ 1a (2�a− a

2�c), the relations between P and ξ are as follows:

P = −aef ξ + (e−f ξ′)′, (5.35)

P ′ = −e−f ξ′, (5.36)

ξ = −1

ae−f (P + P ′′). (5.37)

Next we show that the curves Γ2 and Γ3 are determined by function f . Let s denote the arc parameter

of Γi (i = 2, 3). From (5.23) we see that dsdu = ef . Set ˙ξ = dξ

ds and q = ef , then (5.22), (5.23) imply ξ

satisfies the following equation system:

q2...ξ + qq ¨ξ + (1 + (m− 2)q2 − (m− 3)aq2) ˙ξ − aqqξ = 0, (5.38)

〈ξ, ξ〉 = −1

a, 〈 ˙ξ, ˙ξ〉 = 1. (5.39)

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:

¨ξ = aξ +1

qP , P = −1

q˙ξ.


Thus, by using (5.38) we see that {√|a|ξ, ˙ξ,P} is an orthonormal frame field along the curve and

κ = 〈 ¨ξ,P〉 = 1

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 e2ϕg 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 Mobius invariants.

Proposition 5.2. An m-dimensional umbilic-free hypersurface x : Mm → Sm+1 (m � 4) is confor-

mally flat if and only if its Mobius second fundamental form B has two distinct principal curvatures m−1m

and − 1m of multiplicities 1 and m− 1.

Proof. Let W =∑

ijklWijklωi ⊗ ωj ⊗ ωk ⊗ ωl, S =∑

ij Sijωi ⊗ ωj be the Weyl curvature tensor and

Schouten tensor of x given by

Sij = Rij − 1

2(m− 1)Rδij , Wijkl = Rijkl − 1

m− 2Aijkl , (5.41)

where Aijkl = Sikδjl + Sjlδik − Silδjk − Sjkδil. From (5.41), (2.12) and (2.13) we get

Wijkl = BikBjl −BilBjk − 1

m− 2

[

−∑

h

BihBhkδjl −∑

h

BjhBhlδik

+∑

h

BihBhlδjk +∑

h

BjhBhkδil +1

m(δikδjl − δilδjk)

]

. (5.42)

At a fixed point p ∈M we can choose a tangent orthonormal frame {Ei} of (M,g) such that Bij = Biiδij ,

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

BiiBjj +1

m− 2(B2

ii +B2jj)−

1

m(m− 2)= 0, 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, B22 = · · · = Bmm = μ.

Using (2.13), we have

λ =m− 1

m, μ = − 1

m. (5.45)

Conversely, if B11 = λ, B22 = · · · = Bmm = μ, it is easy to see that (5.43) holds and then M is

conformally flat.


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)(Cjδik − Ckδij). (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.�

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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Classification of hypersurfaces with two distinct principal curvatures and closed Möbius form in $mathbb{S}^{m + 1}$