Classification of Hypersurfaces with Parallel Laguerre Second Fundamental Form in Rn

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Differential Geometry and its Applications

www.elsevier.com/locate/difgeo

Classiﬁcation of hypersurfaces with parallel Laguerre second fundamental form in Rn ∗ Tongzhu Li a,b, ,1,HaizhongLic,2, Changping Wang d,3

a Department of Mathematics, Capital Normal University, Beijing, China b Department of Mathematics, Beijing Institute of Technology, Beijing, China c Department of Mathematical Sciences, Tsinghua University, Beijing, China d LMAM, School of Mathematical Sciences, Peking University, Beijing, China

article info abstract

Article history: Let x : M → Rn be an umbilical free hypersurface with non-zero principal curvatures. Two Received 10 November 2008 basic invariants of x under the Laguerre transformation group are the Laguerre metric Received in revised form 21 April 2009 g and the Laguerre second fundamental form B. In this paper, we classify all oriented Available online 2 December 2009 hypersurfaces in Rn with a parallel Laguerre second fundamental form B. Communicated by G. Thorbergsson Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved. MSC: primary 53A40 secondary 53B25

Keywords: Laguerre transformation group Laguerre metric Laguerre second fundamental form

1. Introduction

Let URn be the unit tangent bundle over Rn. An oriented sphere in Rn centered at p with radius r can be regarded as the “oriented sphere” {(x,ξ)| x − p = rξ} in U Rn, where x is the position vector and ξ the unit normal of the sphere. An oriented hyperplane in Rn with constant unit normal ξ and constant real number c can be regarded as the “oriented hyperplane” {(x,ξ)| x · ξ = c} in U Rn. A diffeomorphism φ : U Rn → U Rn which takes oriented spheres to oriented spheres, oriented hyperplanes to oriented hyperplanes, preserving the tangential distance of any two spheres, is called a Laguerre transformation. All Laguerre transformations in U Rn form a group of dimension (n + 1)(n + 2)/2, called Laguerre transformation group. An oriented hypersurface x : M → Rn can be identiﬁed as the submanifold (x,ξ): M → U Rn, where ξ is the unit normal ∗ of x.Twohypersurfacesx, x : M → Rn are called Laguerre equivalent, if there is a Laguerre transformation φ : U Rn → U Rn ∗ ∗ such that (x ,ξ ) = φ ◦ (x,ξ). In Laguerre geometry one studies properties and invariants of hypersurfaces in Rn under the Laguerre transformation group.

* Corresponding author at: Department of Mathematics, Beijing Institute of Technology, Beijing, 100081, China. E-mail addresses: [email protected] (T. Li), [email protected] (H. Li), [email protected] (C. Wang). 1 Partially supported by the grant No. 10726026 of Tianyuan Foundation. 2 Partially supported by the grant No. 10531090 of NSFC. 3 Partially supported by the grant No. 10771005 of NSFC.

0926-2245/$ – see front matter Crown Copyright © 2009 Published by Elsevier B.V. All rights reserved. doi:10.1016/j.difgeo.2009.09.005 T. Li et al. / Differential Geometry and its Applications 28 (2010) 148–157 149

Laguerre geometry of surfaces in R3 has been developed by Blaschke and his school (see [1]). Recently there has been some renewed interest for surface of R3 in Laguerre geometry (see [2,4–7]). In [3] the ﬁrst and third authors gave a complete Laguerre invariant system for hypersurfaces in Rn. They showed that two umbilical free oriented hypersurfaces in Rn (n > 3) with non-zero principal curvatures are Laguerre equivalent if and only if they have the same Laguerre metric g and Laguerre second fundamental form B. Our goal in this paper is to classify all oriented hypersurfaces in Rn such that ∇B = 0, where ∇ is the Levi-Civita connection of g. FirstwegivesomeexamplesofhypersurfacesinRn with ∇B = 0.

− n−k ={ ∈ Rn−k+1 | · − 2 =− } Example 1. For any integer k with 1 k n 1wedenotebyH (v, w) 1 v v w 1, w > 0 the − + − − hyperbolic space embedded in the Minkowski space Rn k 1.Wedeﬁnex : Sk 1 × Hn k → Rn by 1 u v x(u, v, w) = (1 + w), . w w In Section 3, we will show that the Laguerre second fundamental form B of x is parallel.

Rn → Rn Before we give the second example we have to describe in detail the Laguerre embedding τ : U 0 U (for details Rn+1 we refer to [3]). Let 1 be the Minkowski space with the inner product

X, Y =X1Y1 +···+ XnYn − Xn+1Yn+1. (1.1) = Rn+1 ∈ Rn−1 Rn Rn+1 Let ν (1, 0, 1) be the light-like vector in 1 with 0 .Let 0 be the degenerate hyperplane in 1 defined by Rn = ∈ Rn+1 | = 0 X 1 X, ν 0 . (1.2) We define Rn = ∈ Rn+1 × Rn+1 | = = = U 0 (x,ξ) 1 1 x, ν 0, ξ,ξ 0, ξ,ν 1 . (1.3) And we define τ : U Rn → U Rn given by 0

τ (x,ξ)= x ,ξ ∈ U Rn, (1.4)

n−1 n−1 where x = (x1, x0, x1) ∈ R × R × R, ξ = (ξ1 + 1,ξ0,ξ1) ∈ R × R × R and x1 x1 1 ξ0 x = − , x0 − ξ0 ,ξ= 1 + , . (1.5) ξ1 ξ1 ξ1 ξ1 → Rn Rn Rn+1 Let x : M 0 be a space-like oriented hypersurface in the degenerate hyperplane 0.Letξ betheuniquevectorin 1 satisfying ξ,dx=0, ξ,ξ=0, ξ,ν=1.

From τ (x,ξ)= (x ,ξ ) ∈ U Rn,wegetahypersurfacex : M → Rn.

Example 2. For any positive integers m1,...,ms with m1 + m2 +···+ms = n − 1 and any non-zero constants λ1,...,λs,we Rn−1 → Rn Rn deﬁne x : 0 to be a space-like oriented hypersurface in 0 given by 2 2 2 2 2 2 λ1|u1| + λ2|u2| +···+λs|us| λ1|u1| + λ2|u2| +···+λs|us| x = , u1, u2,...,us, , 2 2

m1 m2 ms n−1 2 n−1 n where (u1, u2,...,us) ∈ R × R ×···×R = R and |ui| = ui · ui , i = 1,...,s.Thenτ ◦ (x,ξ)= (x ,ξ ) : R → U R , − and we get the hypersurfaces x : Rn 1 → Rn. In Section 3, we will show that the Laguerre second fundamental form B of x is parallel.

Our purpose in this paper is to prove the following classiﬁcation theorem:

− Theorem. Let x : Mn 1 → Rn be an umbilical free hypersurface with non-zero principal curvatures. If its Laguerre second fundamental form is parallel, then x is Laguerre equivalent to an open part of one of the following hypersurfaces:

− − (i) the oriented hypersurface x : Sk 1 × Hn k → Rn given by Example 1; n−1 → Rn (ii) the image of τ of the oriented hypersurface x : R 0 given by Example 2.

We organize the paper as follows. In Section 2 we give Laguerre invariants for hypersurfaces in Rn. In Section 3 we give examples of hypersurfaces with a parallel Laguerre second fundamental form. In Section 4 we prove our main theorem. 150 T. Li et al. / Differential Geometry and its Applications 28 (2010) 148–157

2. Laguerre geometry of hypersurfaces in Rn

In this section, we review Laguerre invariants and structure equations of hypersurfaces in Rn. For more details we refer to [3]. Rn+3 Rn+3 Let 2 be the space , equipped with the inner product

X, Y =−X1Y1 + X2Y2 +···+ Xn+2Yn+2 − Xn+3Yn+3. n+2 Rn+3 n+2 ={ ∈ Rn+3 | = } G Let C be the light-cone in given by C X 2 X, X 0 .LetL be the subgroup of orthogonal group + O (n + 1, 2) on Rn 3 given by 2 LG = T ∈ O (n + 1, 2) | ℘T = ℘ , = − ∈ Rn Rn+3 where ℘ (1, 1, 0, 0), where 0 , is a light-like vector in 2 . − Let x : M → Rn be an umbilical free hypersurface with non-zero principal curvatures. Let ξ : M → Sn 1 be its unit normal. Let {e1, e2,...,en−1} be the orthonormal basis for TM with respect to dx · dx, consisting of unit principal vectors. We write the structure equations of x : M → Rn by = k + ; =− − e j ei(x) Γijek(x) kiδijξ ei(ξ) kiei(x), 1 i, j,k n 1, k where ki = 0 is the principal curvature corresponding to ei .Let

1 r1 + r2 +···+rn−1 ri = , r = ki n − 1 be the curvature radius and mean curvature radius of x respectively. We deﬁne = · − · : → n+2 ⊂ Rn+3 Y ρ(x ξ, x ξ,ξ,1) M C 2 ,

= − 2 where ρ i(ri r) > 0.

− Theorem 2.1. (See [3].) Let x, x˜ : Mn 1 → Rn be two umbilical free oriented hypersurfaces with non-zero principal curvatures. Then x and x˜ are Laguerre equivalent if and only if there exists T ∈ LG such that Y˜ = YT.

From the theorem we know that g =dY, dY=ρ2dξ · dξ = ρ2III is a Laguerre invariant metric, where III is the third fundamental form of x.Wecallg Laguerre metric. Let the Laplacian operator of g,wedeﬁne 1 1 N = Y + Y , Y Y (2.1) n − 1 2(n − 1)2 and 1 1 η = 1 +|x|2 , 1 −|x|2 , x, 0 + r(x · ξ,−x · ξ,ξ,1). 2 2 The geometrical meaning of η(p), p ∈ M, is the sphere in Rn centered at x(p) + r(p)ξ(p) with radius r(p), which is tangent to the hypersurface at the point x(p).From(2.1) we get Y , Y =N, N=0, Y , N=−1, η, η=0, η,℘=−1.

Let {E1, E2,...,En−1} be an orthonormal basis for g =dY, dY with dual basis {ω1, ω2,...,ωn−1}.Thenwehavethe following orthogonal decomposition Rn+3 = { }⊕ ⊕{ } 2 span Y , N span E1(Y ), E2(Y ),...,En−1(Y ) η,℘ . { } Rn+3 We call Y , N, E1(Y ), E2(Y ),...,En−1(Y ), η,℘ a Laguerre moving frame in 2 of x. By taking derivatives of this frame we obtain the following structure equations: Ei(N) = LijE j(Y ) + Ci℘, (2.2) j = + + k + E j Ei(Y ) LijY δijN Γij Ek(Y ) Bij℘, (2.3) k Ei(η) =−Ci Y + BijE j(Y ). (2.4) j T. Li et al. / Differential Geometry and its Applications 28 (2010) 148–157 151

From these equations we obtain the following basic Laguerre invariants:

(i) The Laguerre metric g =dY, dY; B = ⊗ (ii) The Laguerre second fundamental form ij Bijωi ω j ; L = ⊗ (iii) The symmetric 2-tensor ij Lijωi ω j ; = (iv) The Laguerre form C i Ci ωi .

By taking further derivatives of (2.2), (2.3) and (2.4) we get the following relations between these invariants:

L = L , (2.5) ij,k ik, j Ci, j − C j,i = (Bik Lkj − B jk Lki), (2.6) k

Bij,k − Bik, j = C jδik − Ckδij, (2.7)

Rijkl = L jkδil + Lilδ jk − Likδ jl − L jlδik, (2.8) where {Lij,k}, {Ci, j}, {Bij,k} are covariant derivatives of the tensors {Lij, Ci , Bij} with respect to the Laguerre metric g, and Rijkl is the curvature tensor of g. Moreover, we have the following identities (see [3]): 2 = = = − Bij 1, Bii 0, Bij,i (n 2)C j, (2.9) ij i i 1 Lii =− Y , Y , (2.10) 2(n − 1) i Rik =−(n − 3)Lik − Lii δik, (2.11) i (n − 2) R =−2(n − 2) Lii = Y , Y . (2.12) (n − 1) i

In case n > 3, we know from (2.9) and (2.11) that Ci and Lij are completely determined by the Laguerre invariants {g, B}, thus we get

Theorem 2.2. (See [3].) Two umbilical free oriented hypersurfaces in Rn (n > 3) with non-zero principal curvatures are Laguerre equivalent if and only if they have the same Laguerre metric g and Laguerre second fundamental form B.

In case n = 3, a complete Laguerre invariant system for surfaces in R3 is given by {g, B, L}. ˜ ˜ ˜ ˜ −1 ˜ We deﬁne Ei = riei, 1 i (n − 1),then{E1, E2,...,En−1} is an orthonormal basis for III = dξ · dξ .Then{Ei = ρ Ei | 1 i n − 1} is an orthonormal basis for the Laguerre metric g, and we write {ω1, ω2,...,ωn−1} for its dual basis. By direct calculations, we obtain the following local expressions: 2 2 −1 g = (ri − r) III = ρ III, Bij = ρ (ri − r)δij, (2.13) i − C =−ρ 2 E˜ (r) − E˜ (log ρ)(r − r) , (2.14) i i i i −2 ˜ ˜ 1 2 Lij = ρ Hessij(log ρ) − Ei(log ρ)E j(log ρ) + |∇ log ρ| − 1 δij , (2.15) 2 where Hessij and ∇ are Hessian matrix and the gradient with respect to the third fundamental form III = dξ · dξ of the hypersurface x.

3. Hypersurfaces in Rn with a parallel Laguerre second fundamental form

In this section, we show that Examples 1 and 2, given in the introduction, are hypersurfaces with a parallel Laguerre second fundamental form. − − We now prove that Example 1 has a parallel Laguerre second form. Hypersurface x : Sk 1 × Hn k → Rn,inExample 1,is given by u v x(u, v, w) = (1 + w), . w w 152 T. Li et al. / Differential Geometry and its Applications 28 (2010) 148–157

= u v Clearly x is a hypersurface with the unit normal ﬁeld ξ ( w , w ), and the ﬁrst and the second fundamental forms of x are given by 1 I = dx · dx = (1 + w)2 du · du + dv · dv − dw2 , w2 1 II =−dx · dξ =− (1 + w) du · du + dv · dv − dw2 , w2 − 1 − − − respectively. Therefore x has two principal curvatures w+1 and 1 with multiplicity k 1 and n k.From(2.13) we get the Laguerre metric (k − 1)(n − k) g = du · du + dv · dv − dw2 , n − 1 and the Laguerre second fundamental form B = Bijωi ⊗ ω j , where

n − k Bij =− δij, 1 i, j k − 1, (k − 1)(n − 1)

k − 1 Bij = δij, n − k + 1 i, j n − 1. (n − k)(n − 1)

From (2.14) we get

Ci = 0, 1 i n − 1.

B = ⊗ Clearly the Laguerre second fundamental form of x ij Bijωi ω j is parallel. → Rn Rn Let x : M 0 be a space-like oriented hypersurface in the degenerate hyperplane 0.Letξ be the unique vector in Rn+1 1 satisfying ξ,dx=0, ξ,ξ=0, ξ,ν=1.

We deﬁne the shape operator S : TM→ TM by dξ =−dx ◦ S.SinceS is self-adjoint, all eigenvalues {ki } of S are real. Let {e1, e2,...,en−1} be the orthonormal basis for TM with respect to dx · dx, consisting of eigenvectors of S.Wewritethe → Rn structure equation of x : M 0 by = k + ; =− − e j ei(x) Γijek(x) kiδijν ei(ξ) kiei(x), 1 i, j,k n 1. k

We assume that ki = 0, and we deﬁne ri = 1/ki as the curvature radius of x and r = (r1 + r2 +···+rn−1)/(n − 1) as the mean curvature radius of x.Letτ : U Rn → U Rn be the Laguerre embedding deﬁned by (1.5) and (x ,ξ ) = τ ◦ (x,ξ).Weget 0 ahypersurfacex : M → Rn. By a direct calculation we can show that

Y = ρ x,ξ, −x,ξ,ξ = ρ x · ξ , −x · ξ ,ξ , 1 = Y , 1 1 η = 1 +x, x , 1 −x, x , x + r x,ξ, −x,ξ,ξ 2 2 1 2 1 2 = η = 1 + x , 1 − x , x , 0 + r x · ξ , −x · ξ ,ξ , 1 , 2 2

2 = − 2 2 = − 2 where ρ i(ri r ) and ρ i(ri r) . Thus the Laguerre metric is given by

g = ρ2III = ρ 2III = g , where III and III are the third fundamental forms of x and x , respectively. ˜ ˜ ˜ ˜ −1 ˜ We deﬁne Ei = riei ,1 i (n − 1),then{E1, E2,...,En−1} is an orthonormal basis for III = dξ · dξ .Then{Ei = ρ Ei | 1 i n − 1} is an orthonormal basis for the Laguerre metric g =dY, dY, and we write {ω1, ω2,...,ωn−1} for its dual basis. By direct calculation, the basic Laguerre invariants B, L and C of x : M → Rn have the following local expressions: 0 B = Bijωi ⊗ ω j, L = Lijωi ⊗ ω j, C = Ciωi, ij ij i where − − B = ρ 1(r − r)δ , C =−ρ 2 E˜ (r) − E˜ (log ρ)(r − r) , (3.1) ij i ij i i i i −2 ˜ ˜ 1 2 Lij = ρ Hessij(log ρ) − Ei(log ρ)E j(log ρ) + |∇ log ρ| δij , (3.2) 2 T. Li et al. / Differential Geometry and its Applications 28 (2010) 148–157 153

where Hessij and ∇ are Hessian matrix and the gradient with respect to III = dξ · dξ . Now we prove that Example 2, given in the introduction, has a parallel Laguerre second form. We deﬁne ξ as follows 2 2 2 2 |λ1u1| +···+|λsus| − 1 |λ1u1| +···+|λsus| + 1 ξ = − ,λ1u1,λ2u2,...,λsus, − . 2 2 Clearly ξ satisﬁes the following conditions ξ,dx=0, ξ,ξ=0, ξ,ν=1.

By a direct calculation we know that the eigenvalues of S are λ1,λ2,...,λs with multiplicity m1,m2,...,ms, respectively. Set 1 1 ri = , r = (r1 + r2 +···+rn−1). λi n − 1 From (3.1), (3.2) we get the Laguerre invariants of x given by 2 2 ρ = (ri − r) = constant, Ci = 0, Lij = 0, 1 i, j n − 1, i

Bij = biδij, 1 + m1 +···+mi−1 i, j 1 + m1 +···+mi−1 + mi, − where b = √ ri r . i − 2 i (ri r) Thus from (2.8) we get that x is ﬂat with respect to g, i.e.,

Rijkl = 0, 1 i, j,k,l n − 1, and the Laguerre second fundamental form B = Bijωi ⊗ ω j of x is parallel.

4. Proof of the theorem

n−1 n Let x : M → R be a hypersurface with a parallel Laguerre second fundamental form. Then from (2.7) and Bij,k = 0 we get

δijCk − δikC j = 0, ∀i, j,k. (4.1)

Setting i = j = k in (4.1) we get Ck = 0, thus the Laguerre form C vanishes. From (2.6) and C = 0 we can choose an orthonormal basis {Ei | 1 i n − 1} with respect to Laguerre metric g such that

Bij = biδij, Lij = aiδij, 1 i, j n − 1.

We assume that {b1, b2,...,bn−1} has s distinct numbers b1, b2,...,bs with multiplicity k1,k2,...,ks, respectively. Since Bij,k = 0, it follows that b1, b2,...,bs are constants. From (2.9),weget +···+ = 2 +···+ 2 = k1b1 ksbs 0, k1b1 ksbs 1. (4.2) We write = (Bij) diag(b1,..., b1, b2,..., b2,...,bs,..., bs).

k1 k2 ks From (4.2) we know that s 2. First we consider the case that n = 3, then s = 2. From (4.2) we get √ √ 2 2 b1 = , b2 =− . 2 2 √ = 1 = Since Bij,k 0, it follows that 2ω2 0. Thus the Laguerre metric is ﬂat. We can choose local coordinates (u, v) satisfying ∂ ∂ E1 = , E2 = . ∂u ∂ v From (2.5), (2.6) and (2.12), for these basis we have

L12 = 0, L11 + L22 = 0, L11 = constant.

Setting L11 = a, L22 =−a, then the structure equations of (2.2), (2.3) and (2.4) are given by: 154 T. Li et al. / Differential Geometry and its Applications 28 (2010) 148–157

= =− Nu aYu, Nv √aYv , √ 2 2 Yuu = aY + N + ℘, Yuv = 0, Y vv =−aY + N − ℘, √ 2 √ 2 2 2 ηu = Yu, ηv =− Y v . (4.3) 2 2

Since Yuv = 0, we can write Y = F (u) + G(v),from(4.3) we get

F (u) − 2aF (u) = 0, G (v) + 2aG (v) = 0. Without loss of generality we assume a 0. (i) When a = 0, we can get a unique adapted solution (up to a Laguerre transformation) of (4.3) u2 − v2 v2 − u2 u2 + v2 − 1 u2 + v2 + 1 Y = √ , √ , , u, −v, , 2 2 2 2 2 2 N = (0, 0, 1, 0, 0, 1), u2 + v2 + 2 −u2 − v2 + 2 u2 − v2 u v u2 − v2 η = , , √ , √ , √ , √ . (4.4) 4 4 2 2 2 2 2 2 From (2.1) and (4.4) we get that x is Laguerre equivalent to u2 − v2 u(2v2 + 1) v(2u2 + 1) x˜(u, v) = √ , √ , √ . 2(u2 + v2 + 1) 2(u2 + v2 + 1) 2(u2 + v2 + 1) The unit normal vector ﬁeld of x˜ is u2 + v2 − 1 2u −2v ξ˜ = , , . (u2 + v2 + 1) (u2 + v2 + 1) (u2 + v2 + 1) ˜ ˆ 2 → 3 In fact ξ can be constructed as in Example 2.Letx : R R0 be given by u2 − v2 u2 − v2 u2 + v2 − 1 u2 + v2 + 1 xˆ(u, v) = , u, v, , ξˆ = − , u, −v, − . 2 2 2 2

Then (x˜, ξ)˜ = τ ( √1 xˆ, ξ)ˆ . 2 (ii) When a > 0, we can get a unique adapted solution (up to a Laguerre transformation) of (4.3) −1 −1 √ 1 1 √ 1 √ Y = √ + √ cosh 2au , √ + √ cosh 2au , √ cos 2av, a 2 2a a 2 2a 2a 1 √ 1 √ 1 √ √ sin 2av, √ sinh 2au, √ cosh 2au , 2a 2a 2a √ √ √ √ √ √ − 2a √ 2a √ − 2a √ − 2a √ 2a √ 2a √ N = cosh 2au, cosh 2au, cos 2av, sin 2av, sinh 2au, cosh 2au , 2 2 2 2 2 2 √ √ √ √ √ √ cosh 2au cosh 2au cos 2av sin 2av sinh 2au cosh 2au η = − √ , √ + 1, − √ , − √ , √ , √ + 1 . 2 a 2 a 2 a 2 a 2 a 2 a From (2.1) we get that x is Laguerre equivalent to √ √ √ √ √ √ a + cosh 2au √ a + cosh 2au sinh 2au x˜(u, v) = − cos 2av √ √ , − sin 2av √ √ , √ . a cosh 2au a cosh 2au cosh 2au The unit normal vector ﬁeld of x˜ is √ √ √ cos 2av sin 2av sinh 2av ξ˜ = √ , √ , √ . cosh 2au cosh 2au cosh 2au In fact x˜ can be constructed as in Example 1, x˜ : S1 × H1 → R3, √ √ 1 √ 1 √ x˜ − cos 2av, − sin 2av, √ sinh 2au, √ cosh 2av a a √ √ √ √ 1 + √1 cosh 2au √ 1 + √1 cosh 2au a a sinh 2au = − cos 2av √ , − sin 2av √ , √ . √1 cosh 2au √1 cosh 2au cosh 2au a a T. Li et al. / Differential Geometry and its Applications 28 (2010) 148–157 155

Second we consider the case that n 4, then Theorem 2.2 can be applied. In what follows we assume that n 4, thus s 2.

Case 1. s = 2.

= (Bij) diag(b1,..., b1, b2,..., b2).

k1 k2

Set k1 = k − 1 and k2 = n − k.From(4.2) we get that

(k − 1)b1 + (n − k)b2 = 0, − 2 + − 2 = (k 1)b1 (n k)b2 1. Thus we get

n − k k − 1 b1 =− , b2 = . (k − 1)(n − 1) (n − k)(n − 1)

Then we have the following decomposition n−1 TM = span{E1,...,Ek−1}⊕span{Ek,...,En−1}=V 1 ⊕ V 2. { } { | − } = − i = We denote by ωi the dual basis of Ei 1 i n 1 .SinceBij,k 0, it follows that (bi b j )ω j 0. Therefore, we have i = − − ω j 0, 1 i k 1, k j n 1. (4.5) From (4.5) we have

Rijij = 0, 1 i k − 1, k j n − 1. (4.6)

Setting i = 1, j > k − 1in(4.6),from(2.8) we get ak = ak+1 =···=an−1 =−a1, setting i < k, j = k in (4.6),from(2.8) we get a1 = a2 =···=ak−1 =−ak.Thus = − − diag(a1,a2,...,an−1) diag(a1,..., a1, a1,..., a1). k−1 n−k

Since Lij,k = Lik, j , it follows that a1 is constant. n−1 = × From (4.5) the eigenspaces V 1 and V 2 of the Laguerre shape operator S are integral. We can write M M1 M2 = k−1 2 = n−1 2 locally, moreover we deﬁne g1 i=1 ωi and g2 i=k ωi ,thenwehave n−1 M , g = (M1, g1) × (M2, g2). From (2.8) we get

Rijkl = 2a1(δ jkδil − δikδ jl), 1 i, j,k,l k − 1,

Rijkl =−2a1(δ jkδil − δikδ jl), k i, j,k,l n − 1.

Thus (M1, g1) and (M2, g2) are constant curvature manifolds with opposite curvatures. Setting a1 =−σ , without loss of generality we assume σ 0. We consider the following two cases: (1) σ > 0; (2) σ = 0. − − (1) In the case σ > 0, we deﬁne x˜ : Sk 1 × Hn k → Rn by n − 1 u v x˜(u, v, w) = (1 + w), , 2σ (n − k − 1)(k) w w

k−1 → k n−k → Rn−k+1 where u : S R and (v, w) : H 1 are the canonical embeddings. From Example 1 we get that the Laguerre metric g˜ of x˜ 1 2 g˜ = du · du + dv · dv − dw = g˜ 1 + g˜ 2, 2σ ˜ = 1 · ˜ = 1 · − 2 where g1 2σ (du du) and g2 2σ (dv dv dw ). k−1 n−k Since both (M1, g1) and (S , g˜ 1) have the same constant curvature and both (M2, g2) and (H , g˜ 2) have the same constant curvature we can ﬁnd isometries k−1 n−k φ1 : (M1, g1) → S , g˜ 1 ,φ2 : (M2, g2) → H , g˜ 2 . 156 T. Li et al. / Differential Geometry and its Applications 28 (2010) 148–157

n−1 k−1 n−k Because of Example 1,thusφ = (φ1,φ2) : M = M1 × M2 → S × H preserves the Laguerre metric and the Laguerre − − − second fundamental form. From Theorem 2.2 we know that x : Mn 1 → Rn is Laguerre equivalent to x˜ : Sk 1 × Hn k → Rn. − (2) In the case σ = 0, we define x˜ : Rn 1 → Rn by 0 u · u − v · v u · u − v · v x˜(u, v) = , u, v, , 2 2 − − where u ∈ Rk 1, v ∈ Rn k. = u·u−v·v − 1 − u·u−v·v − 1 Then ξ ( 2 2 , u, v, 2 2 ) satisfies ξ,dx˜=0, ξ,ξ=0, ξ,ν=1. − − From Example 2, we get that the Laguerre metric g˜ of x is flat. Thus we can find isometry φ : (Mn 1, g) → (Rn 1, g˜) which preserves the Laguerre metric and the Laguerre second fundamental form of Example 2.FromTheorem 2.2 we know that x ˜ : n−1 → Rn is Laguerre equivalent to x R 0.

Case 2. s > 2. = diag(b1, b2,...,bn−1) diag(b1,..., b1, b2,..., b2,...,bs,..., bs).

k1 k2 ks

Then we have the following decomposition n−1 = { }⊕···⊕ { }= ⊕ ⊕···⊕ TM span E1,...,Ek1 span En−ks−1,...,En−1 V 1 V 2 V s. Denoted by {ω } the dual basis of {E | 1 i n − 1}.Set i i [i]= k ∈ (1, 2,...,n − 1) | bk = bi . = − i = Since Bij,k 0, it follows that (bi b j)ω j 0. Thus we get i = [ ] =[ ] ω j 0, i j . (4.7) From (4.7) we have

Rijij = 0, [i] =[j]. (4.8)

Setting i = i0, [i0] =[j],from(2.8) and (4.8) we have

ai = a j, i, j ∈[k], ai + al = 0, i ∈[i], l ∈[k], [i] =[k]. Since s 3weget

a1 = a2 =···=an−1 = 0. Therefore the Laguerre metric of x is ﬂat. For any non-zero constants λ1,λ2,...,λs satisfying − = ri r bi , 1 i s, − 2 i(ri r) 1 1 ˜ n−1 n where ri = , i = 1,...,s, r = (k1r1 +k2r2 +···+ksrs),wedeﬁnex : R → R to be a space-like oriented hypersurface λi n−1 0 in the degenerate hyperplane Rn given by 0 2 2 2 2 2 2 λ1|u1| + λ2|u2| +···+λr |ur | λ1|u1| + λ2|u2| +···+λr |ur | x˜ = , u1, u2,...,ur , , 2 2

k1 k2 kr n−1 2 where (u1, u2,...,ur ) ∈ R × R ×···×R = R and |ui| = ui · ui . We define ξ as follows 2 2 2 2 λ1|u1| +···+λr |ur | 1 λ1|u1| +···+λr |ur | 1 ξ = − ,λ1u1,λ2u2,...,λr ur , + . 2 2 2 2 Clearly ξ satisfies the following condition ξ,dx˜=0, ξ,ξ=0, ξ,ν=1. − − From Example 2 we know that the Laguerre metric of x˜ is flat. Thus we can find isometry φ : (Mn 1, g) → (Rn 1, g˜) which preserves the Laguerre metric and the Laguerre second fundamental form of Example 2.FromTheorem 2.2 we know that x ˜ Rn−1 → Rn is Laguerre equivalent to x : 0. Thus we complete the proof of the classification theorem. T. Li et al. / Differential Geometry and its Applications 28 (2010) 148–157 157

Acknowledgements

The authors would like to thank the referee for some helpful suggestions.

References

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