LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n

Bull Korean Math Soc 50 (013), No 6, pp 1781 1797 http://dxdoiorg/104134/bkms0135061781 LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n Shichang Shu and Yanyan Li Abstract Let x : M R n be an n 1-dimensional hypersurface in R n, L be the Laguerre Blaschke tensor, B be the Laguerre second fundamental form and D = L + λb be the Laguerre para-blaschke tensor of the immersion x, where λ is a constant The aim of this article is to study Laguerre Blaschke isoparametric hypersurfaces and Laguerre para- Blaschke isoparametric hypersurfaces in R n with three distinct Laguerre principal curvatures one of which is simple We obtain some classification results of such isoparametric hypersurfaces 1 Introduction In Laguerre differential geometry, T Li and C Wang[5] studied invariants of hypersurfaces in Euclidean space R n under the Laguerre transformation group The Laguerre transformations are the Lie sphere transformations which take oriented hyperplanes in R n to oriented hyperplanes and preserve the tangential distance Let UR n be the unit tangent bundle over R n An oriented sphere in R n centered at p with radius r can be regarded as the oriented sphere {(x,ξ) x p = rξ} in UR n, where x is the position vector and ξ the unit normal vector of the sphere An oriented hyperplane in R n with constant unit normal vector ξ and constant real number c can be regarded as the oriented hyperplane {(x,ξ) x ξ = c} in UR n A diffeomorphism φ : UR n UR n 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 UR n form a group of dimension (n+1)(n+), called Laguerre transformation group An oriented hypersurface x : M R n can be identified as the submanifold (x,ξ) : M UR n, where ξ is the unit normal of x Two hypersurfaces x,x : M R n are called Received May, 011; Revised November 8, 011 010 Mathematics Subject Classification 53C4, 53C0 Key words and phrases Laguerre characterization, Laguerre form, Laguerre Blaschke tensor, Laguerre second fundamental form Project supported by NSF of Shaanxi Province (SJ08A31) and NSF of Shaanxi Educational Committee(11JK0479, 010JK893) 1781 c 013 The Korean Mathematical Society

178 SHICHANG SHU AND YANYAN LI Laguerre equivalent, if there is a Laguerre transformation φ : UR n UR n such that (x,ξ ) = φ (x,ξ) (see [4]) In [5], T Li and C Wang gave a complete Laguerre invariant system for hypersurfaces in R n They proved that two umbilical free oriented hypersurfaces in R n 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 We should notices that the Laguerre geometry of surfaces in R 3 has been studied by Blaschke in [1] and other authors in [3, 6, 7] Let R n+3 be the space R n+3 equipped with the inner product X,Y = X 1 Y 1 +X Y + +X n+ Y n+ X n+3 Y n+3 Let C n+ be the light-cone in R n+3 given by C n+ = {X R n+3 X,X = 0} Let LG be the subgroup of orthogonalgroupo(n+1,)onr n+3 givenbylg = {T O(n+1,) ζt = ζ}, where ζ = (1, 1,0,0) and 0 R n is a light-like vector in R n+3 Let x : M R n be an umbilic free hypersurface with non-zero principal curvatures, ξ : M S n 1 be its unit normal vector Let {e 1,e,,e n 1 } be the orthonormal basis for TM with respect to dx dx, consisting of unit principal vectors The structure equations of x : M R n are (see [4]) (11) e j (e i (x)) = k Γ k ij e k(x)+k i δ ij ξ, e i (ξ) = k i e i (x), i,j,k = 1,,n 1, where k i 0 is the principal curvature corresponding to e i Let (1) r i = 1, r = r 1 +r + +r n 1, k i n 1 be the curvature radius and mean curvature radius of x respectively We define Y = ρ(x ξ, x ξ,ξ,1) : M C n+ R n+3, where ρ = i (r i r) > 0 From[5], weknowthatthelaguerre metric g oftheimmersionxcanbedefined by g = dy,dy Let {E 1,E,,E n 1 } be an orthonormal basis for g with dual basis {ω 1,ω,,ω n 1 } The Laguerre form C, Laguerre Blaschke tensor L and Laguerre second fundamental form B of the immersion x are defined by n 1 (13) C = C i ω i, L = i=1 n 1 i,j=1 L ij ω i ω j, B = n 1 i,j=1 B ij ω i ω j, respectively, where C i, L ij and B ij are defined by formulas (10) (1) in Section We should notices that g, C, L and B are Laguerre invariants (see [5]) By making use of the two important Laguerre invariants, the Laguerre Blaschke tensor L and the Laguerre second fundamental form B of the immersion x, we define a symmetric (0,) tensor D = L+λB which is so called the Laguerre para-blaschke tensor of x, where λ is a constant An eigenvalue of the Laguerre Blaschke tensor is called a Laguerre Blaschke eigenvalue of x, an eigenvalue of the Laguerre second fundamental form is called a Laguerre principal curvature of x and an eigenvalue of the Laguerre para-blaschke tensor is called a Laguerre para-blaschke eigenvalue of x An umbilic free hypersurface

LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n 1783 x : M R n is called a Laguerre isoparametric hypersurface if C 0 and the Laguerre principal curvatures of the immersion x are constant, an umbilic free hypersurface x : M R n is called a Laguerre Blaschke isoparametric hypersurface if C 0 and the Laguerre Blaschke eigenvalues of the immersion x are constant, and an umbilic free hypersurface x : M R n is called a Laguerre para-blaschke isoparametric hypersurface if C 0 and the Laguerre para-blaschke eigenvalues of the immersion x are constant An umbilic free hypersurface x : M R n is called a Laguerre para-isotropic hypersurface, if there are two functions λ and µ on x such that L+λB+µg = 0 and C 0 If λ = 0, we call x a Laguerre isotropic hypersurface It should be noted that if x is a Laguerre para-isotropic hypersurface, or a Laguerre isotropic hypersurface, then the Laguerre para-blaschke eigenvalues, or the Laguerre Blaschke eigenvalues of x are all equal We define the Laguerre embedding τ : UR n 0 UR n (see [5]) Let R n+1 1 be the Minkowski space with the inner product X,Y = X 1 Y 1 + + X n Y n X n+1 Y n+1 Let ν = (1,0,1) be the light-like vector in R n+1 1, 0 R n 1 Let R n 0 be the degenerate hyperplane in R n+1 1 defined by R n 0 = {X Rn+1 1 X,ν = 0} We define (14) UR n 0 = {(x,ξ) Rn+1 1 R n+1 1 x,ν = 0, ξ,ξ = 0, ξ,ν = 1} The Laguerre embedding τ : UR n 0 URn is defined by (15) τ(x,ξ) = (x,ξ ) UR n, where x = (x 1,x 0,x 1 ) R R n 1 R, ξ = (ξ 1 +1,ξ 0,ξ 1 ) R R n 1 R and ( (16) x = x 1,x 0 x ) ( 1 ξ 0, ξ = 1+ 1, ξ ) 0 ξ 1 ξ 1 ξ 1 ξ 1 Let x : M R n 0 be a space-like oriented hypersurface in the degenerate hyperplane R n 0 Let ξ be the unique vector in Rn+1 1 satisfying ξ,dx = 0, ξ,ξ = 0, ξ,ν = 1 From τ(x,ξ) = (x,ξ ) UR n, we may obtain a hypersurface x : M R n We should notice that it is one of the important aims to characterize hypersurfaces in terms of Laguerre invariants Concerning this topic, recently, T Li, H Li and C Wang [4] studied the Laguerre geometry of hypersurfaces with parallel Laguerre second fundamental form in R n and obtained the following result: Theorem 11 ([4]) Let x : M R n be an umbilic free hypersurface with non-zero principal curvatures If the Laguerre second fundamental form of x is parallel, then x is Laguerre equivalent to an open part of one of the following hypersurfaces: (1) the oriented hypersurface x : S k 1 H n k R n given by Example 1; or () the image of τ of the oriented hypersurface x : R n 1 R n 0 given by Example

1784 SHICHANG SHU AND YANYAN LI The aim of this article is to continue this topic, we shall study Laguerre Blaschke isoparametric hypersurfaces and Laguerre para-blaschke isoparametric hypersurfaces in R n with three distinct Laguerre principal curvatures one of which is simple We obtain the following results: Theorem 1 Let x : M R n be an n 1-dimensional Laguerre Blaschke isoparametric hypersurface in R n (n 5) with three distinct Laguerre principal curvatures one of which is simple Then x is a Laguerre isoparametric hypersurface with non-parallel Laguerre second fundament form or a Laguerre isotropic hypersurface Theorem 13 Let x : M R n be an n 1-dimensional Laguerre para- Blaschke isoparametric hypersurface in R n (n 5) and D = L + λb, λ 0, be the Laguerre para-blaschke tensor of x If x has three distinct Laguerre principal curvatures one of which is simple, then (i) x is a Laguerre isoparametric hypersurface with non-parallel Laguerre second fundament form, or (ii) x is a Laguerre para-isotropic hypersurface, or (iii) x is Laguerre equivalent to an open part of the image of τ of the oriented hypersurface x : R n 1 R n 0 given by Example From Theorem 1 and Theorem 13, we easily see that: Corollary 14 Let x : M R n be an n 1-dimensional Laguerre para- Blaschke isoparametric hypersurface in R n (n 5) and D = L + λb be the Laguerre para-blaschke tensor of x If x has three distinct Laguerre principal curvatures one of which is simple, then (i) x is a Laguerre isoparametric hypersurface with non-parallel Laguerre second fundament form, or (ii) x is a Laguerre isotropic hypersurface for λ = 0, or (iii) x is a Laguerre para-isotropic hypersurface or x is Laguerre equivalent to an open part of the image of τ of the oriented hypersurface x : R n 1 R n 0 given by Example for λ 0 Laguerre invariants and fundamental formulas In this section, we review the Laguerre invariants and fundamental formulas on Laguerre geometry of hypersurfaces in R n, for more details, see [5] Let x : M R n be an n 1-dimensional umbilical free hypersurface with vanishing Laguerre form in R n Let {E 1,,E n 1 } denote a local orthonormal frame for Laguerre metric g = dy,dy with dual frame {ω 1,,ω n 1 } Putting Y i = E i (Y), then we have (1) N = 1 n 1 Y + 1 (n 1) Y, Y Y, () Y,Y = N,N = 0, Y,N = 1,

LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n 1785 and the following orthogonal decomposition: (3) R n+3 = Span{Y,N} Span{Y 1,,Y n 1 } V, where {Y,N,Y 1,,Y n 1,η, } forms a moving frame in R n+3 and V = {η, } is called Laguerre normal bundle of x We use the following range of indices throughout this paper: 1 i,j,k,l,m n 1 The structure equations on x with respect to the Laguerre metric g can be written as (4) dy = i ω i Y i, (5) dn = i ψ i Y i +ϕη, (6) dy i = ψ i Y ω i N + j ω ij Y j +ω in+1 η, (7) d = ϕy i ω in+1 Y i, where {ψ i,ω ij,ω in+1,ϕ} are 1-forms on x with (8) ω ij +ω ji = 0, dω i = j ω ij ω j, and (9) ψ i = j L ij ω j, L ij = L ji, ω in+1 = j B ij ω j, B ij = B ji, ϕ = i C i ω i We define Ẽi = r i e i, 1 i n 1, then {Ẽ1,,Ẽn 1} is an orthonormal basis for III = dξ dξ and {E i = ρ 1 Ẽ i } is an orthonormal basis for the Laguerre metric g with dual frame {ω 1,,ω n 1 } L ij, B ij and C i are locally defined functions and satisfy (10) L ij = ρ { Hess ij (logρ) Ẽi(logρ)Ẽj(logρ)+ 1 ( logρ 1 ) δ ij }, (11) (1) B ij = ρ 1 (r i r)δ ij, C i = ρ { } Ẽ i (r) Ẽi(logρ)(r i r), where g = i (r i r) III = ρ III, r i and r are defined by (1), Hess ij and are the Hessian matrix and the gradient with respect to the third fundamental form III = dξ dξ of x (see [5])

1786 SHICHANG SHU AND YANYAN LI Defining the covariant derivative of C i,l ij,b ij by (13) C i,j ω j = dc i + C j ω ji, j j (14) L ij,k ω k = dl ij + L ik ω kj + L kj ω ki, k k k (15) B ij,k ω k = db ij + B ik ω kj + B kj ω ki k k k We have from [5] that (16) dω ij = ω ik ω kj 1 R ijkl ω k ω l, R ijkl = R jikl, k k,l (17) B ii = 0, i Bij = 1, B ij,i = (n )C j, trl = R (n ) i,j i (18) L ij,k = L ik,j, (19) C i,j C j,i = k (B ik L kj B jk L ki ), (0) B ij,k B ik,j = C j δ ik C k δ ij, (1) R ijkl = L jk δ il +L il δ jk L ik δ jl L jl δ ik, where R ijkl and R denote the curvature tensor and the scalar curvature with respect to the Laguerre metric g on x Since the Laguerre form C 0, we have for all indices i,j,k () B ij,k = B ik,j, B ik L kj = B kj L ki k k Denote by D = i,j D ijω i ω j the (0,) Laguerre para-blaschke tensor, then (3) D ij = L ij +λb ij, 1 i,j n, where λ is a constant The covariant derivative of D ij is defined by (4) D ij,k ω k = dd ij + D ik ω kj + D kj ω ki k k k From (18) and (), we have for all indices i,j,k that (5) D ij,k = D ik,j We recall the following examples of hypersurfaces in R n with parallel Laguerre second fundamental form (see [4]):

LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n 1787 Example 1 ([4]) For any integer k, 1 k n 1, we define a hypersurface x : S k 1 H n k R n by ( u x(u,v,w) = w w) (1+w), v, where H n k = {(v,w) R1 n k+1 v v w = 1,w > 0} denotes the hyperbolic space embedded in the Minkowski space R1 n k+1 From [4], we know that x has two distinct Laguerre principal curvatures n k B 1 = (k 1)(n 1), B k 1 = (n k)(n 1), the Laguerre form is zero and the Laguerre second fundamental form of x is parallel Example ([4]) For any positive integers m 1,,m s with m 1 + +m s = n 1 and any non-zero constants λ 1,,λ s, we define x : R n 1 R n 0 to be a spacelike oriented hypersurface in R n 0 given by { λ1 u 1 + +λ s u s x =,u 1,u,,u s, λ 1 u 1 + +λ s u s }, where (u 1,,u s ) R m1 R ms = R n 1 and u i = u i u i,i = 1,,s Then τ (x,ξ) = (x,ξ ) : R n 1 UR n, and we obtain the hypersurfaces x : R n 1 R n From [4], we know that x has s(s 3) distinct Laguerre principal curvatures: r i r B i = i (r 1 i s, i r), where r i = 1, r = k 1r 1 +k r + +k s r s, k i n 1 and k i 0 is the principal curvature corresponding to e i We also know that the Laguerre form is zero, L ij = 0 for 1 i,j n 1 and the Laguerre second fundamental form of x is parallel 3 Propositions and lemmas Throughout this section, we shall make the following convention on the ranges of indices: 1 a,b m 1, m 1 +1 p,q m 1 +m, m 1 +m +1 α,β m 1 +m +m 3 = n 1, 1 i,j,k n 1 Let L, B and D denote the n n-symmetric matrices (L ij ), (B ij ) and (D ij ), respectively, where L ij, B ij and D ij are defined by (10), (11) and (3) From () and (3), we know that BL = LB, DL = LD and BD = DB Thus, we may choose a local orthonormal basis {E 1,E,,E n } such that L ij = L i δ ij, B ij = B i δ ij, D ij = D i δ ij,

1788 SHICHANG SHU AND YANYAN LI where L i, B i and D i are the Laguerre Blaschke eigenvalues, the Laguerre principal curvatures and the Laguerre para-blaschke eigenvalues of the immersion x In the proof of the following propositions and theorems, we agree on the fact that a local orthonormal basis {E 1,E,,E n } may be always chosen such that L ij = L i δ ij,b ij = B i δ ij,d ij = D i δ ij Proposition 31 Let x : M R n be an n 1-dimensional hypersurface with vanishing Laguerre form in R n If the multiplicity of a Laguerre principal curvature is constant and greater than 1, then this Laguerre principal curvature is constant along its leaf Proof Let B i, i = 1,,n 1, be the Laguerre principal curvatures of x with constant multiplicities We choose a local orthonormal frame {E 1,,E n 1 } such that E i is a unit principal vector with respect to B i From (15), we have (31) B ij,k = E k (B i )δ ij +Γ j ik (B i B j ), where Γ j ik is the Levi-Civita connection for the Laguerre metric g given by (3) ω ij = k Γ j ik ω k, Γ j ik = Γi jk From (), we know that B ii,j = B ij,i Thus from (31), we get (33) E j (B i ) = Γ j ii (B i B j ) for i j Without loss of generality, we may assume that B 1 is the Laguerre principal curvature of x with constant multiplicity m 1 and m 1, that is, for 1 a m 1 we have B a = B 1 From (33), we have E a (B 1 ) = Γ a 11(B 1 B a ) = 0 for a 1, and E 1 (B 1 ) = E 1 (B a ) = Γ 1 aa(b a B 1 ) = 0 for a 1 Thus E a (B 1 ) = 0 for any a This implies that B 1 is constant along its leaf We complete the proof of Proposition 31 We may prove the following proposition by reasoning as in [] Proposition 3 Let x : M R n be an n 1-dimensional hypersurface in R n (n 5) with vanishing Laguerre form and three distinct Laguerre principal curvatures B 1, B, B 3 one of which is simple Then either B 1, B, B 3 are constants or B ap,n 1 = 0 for every a,p Proof From (16), (8) and (3), the curvature tensor of x may be given by (see []) (34) R ijkl =E k (Γ j il ) E l(γ j ik )+ m Γ j im Γm kl

LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n 1789 m Γ j im Γm lk + m Γ m ilγ j mk m Γ m ikγ j ml Since x has three distinct Laguerre principal curvatures B 1, B, B 3 one of which is simple and n 5, without loss of generality, we may assume that m 3 = 1, m 1 1 and m From (17), we have Thus (35) m 1 db 1 +m db +m 3 db 3 = 0, m 1 B 1 db 1 +m B db +m 3 B 3 db 3 = 0 m 1 db 1 B 3 B = m db B 1 B 3 = m 3dB 3 B B 1 From Proposition 31 and (35), we have (36) E p (B ) = E p (B 1 ) = E p (B 3 ) = 0, and from (31), we have (37) (38) (39) Γ p ab = Γα ab = 0,a b, Γα pq = 0,p q, Γp aa = Γp bb, Γα aa = Γα bb, Γ p aα = B ap,α B 1 B, Γ a αp = B αa,p B 3 B 1, Γ α pa = B pα,a B B 3 (i) If m 1, from Proposition 31 and (35), we have E a (B 1 ) = E a (B ) = E a (B 3 ) = 0 From (31), (33), (36) and (39), we have (310) (311) From (38), we have (31) Γ p ab = Γa pq = 0, Γa n 1n 1 = Γp n 1n 1 = 0, Γ n 1 aa = E n 1(B 1 ) B 1 B 3, Γ n 1 pp = E n 1(B ) B B 3 Γ p an 1 = B ap,n 1, Γ p n 1b B 1 B = B bp,n 1, Γ n 1 bq = B bq,n 1, B 3 B B 1 B 3 Γ n 1 qb = B bq,n 1 B B 3 Thus, from (34), (37) and (310) (31), we have R apbq = E b (Γ p aq ) E q(γ p ab )+ m Γ p am Γm bq (313) m Γ p am Γm qb + m Γ m aq Γp mb m Γ m ab Γp mq = Γ p an 1 Γn 1 bq Γ p an 1 Γn 1 qb +Γaq n 1 Γ p n 1b Γn 1 ab Γpq n 1 = B ap,n 1B bq,n 1 +B aq,n 1 B bp,n 1 +E n 1 (B 1 )E n 1 (B )δ ab δ pq (B 1 B 3 )(B 3 B )

1790 SHICHANG SHU AND YANYAN LI On the other hand, from (1), we have (314) R apbq = (L a +L p )δ ab δ pq (313) and (314) imply that Putting B ap,n 1 B bq,n 1 +B aq,n 1 B bp,n 1 = { (L a +L p )(B 1 B 3 )(B 3 B ) E n 1 (B 1 )E n 1 (B )}δ ab δ pq (315) a,p = (L a +L p )(B 1 B 3 )(B 3 B ) E n 1 (B 1 )E n 1 (B ), we get If a = b, we have B ap,n 1 B bq,n 1 +B aq,n 1 B bp,n 1 = a,p δ ab δ pq (316) B ap,n 1 B aq,n 1 = a,p δ pq From (315), we know that a,p depends on a,p If L 1 = L = = L n 1, from (315), we see that for any a,p, all a,p are equal If there is p 0, such that B ap0,n 1 0, 1 a m 1 By (316), we have B ap,n 1 = 0 for other p(p p 0 ) By (316) again, if p = q, then Bap,n 1 = a,p for any p Since for any a,p, all a,p are equal, we have Bap 0,n 1 = a,p 0 = a,p = Bap,n 1 = 0 for p 0,p(p p 0 ) Thus B ap0,n 1 = 0, this is a contradiction Therefore we know that B ap,n 1 = 0 for any a,p If at least two of L 1,L,,L n 1 are not equal, since m and m 1, we may prove that there exists at most one p, such that a,p 0 for any a, 1 a m 1 and there exists at most one a, such that a,p 0 for any p, m 1 +1 p m 1 +m In fact, if there exists more than one p, for example p 1, p, (p 1 p ) such that a,p1 0, a,p 0 By (316), we have Bap,n 1 = a,p for any p Thus Bap 1,n 1 = a,p 1 0, Bap,n 1 = a,p 0 By (316) again, we see that B ap1,n 1B ap,n 1 = 0, this is a contradiction Thus, we know that there exists at most one p, such that a,p 0 for any a, 1 a m 1 By the same proof as above, we also know that there exists at most one a, such that a,p 0 for any p, m 1 +1 p m 1 +m If there exists at most one p, such that a,p 0 for any a, possibly, say a,p0 0 for any a From (315), we have (317) L a +L p = E n 1(B 1 )E n 1 (B ) (B 1 B 3 )(B 3 B ), p p 0 By (317), we know that L a = L b for any a,b On the other hand, since we know that there exists at most one a, such that a,p 0 for any p, possibly, say a0,p 0 for any p By (315), we also have L a +L p = E n 1(B 1 )E n 1 (B ) (B 1 B 3 )(B 3 B ), a a 0,

LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n 1791 and we know that L p = L q for any p,q Thus, from (315) we see that only a,p = 0 for any a,p holds exactly Thus, by (316), we have B ap,n 1 = 0 for any p and a (ii) If m 1 = 1, from (33) and (37), we have (318) (319) Γ 1 pq = Γn 1 pq = 0, Γ p n 1n 1 = Γp 11 = 0, Γ 1 n 1n 1 = E 1(B 3 ) B 3 B 1, Γ 1 pp = E 1(B ) B B 1, Γ n 1 11 = E n 1(B 1 ) B 1 B 3, Γ n 1 pp = E n 1(B ) B B 3 From (34), (37), (38), (318) and (319), by a similar calculation as in (i), we have (30) B 1p,n 1 B 1q,n 1 = υ p δ pq for any p,q, where (31) υ p = (B 1 B )(B 1 B 3 ) { E 1 (B )E 1 (B 3 ) (L p +L n 1 )+ (B 1 B )(B 1 B 3 ) + [E n 1(B ) E n 1 (B 3 )]E n 1 (B ) (B B 3 ) E n 1(E n 1 (B )) B B 3 + E n 1(B ) (B B 3 ) From (31), we know that υ p depends on p If L 1 = L = = L n 1, from (31), we see that for any p, all υ p are equal By the same proof as in (i), we know that B 1p,n 1 = 0 for any p If at least two of L 1,L,,L n 1 are not equal, since m, by the same proof as in (i), we easily know that there exists at most one p, such that υ p 0 If for any p, υ p = 0, by (30), we have B 1p,n 1 = 0 If there is p 0, such that υ p0 0 and υ p = 0, for other p(p p 0 ), we have (3) υ p0 = υ p0 υ p = (B 1 B )(B 1 B 3 )(L p0 L p ) On the other hand, since m 1 = 1, m 3 = 1 and B ij,k is symmetric for all indices i,j,k, interchanging 1 and n in the above equations, we also have, (33) B n 1p,1 B n 1q,1 = ω p δ pq, where (34) { ω p = (B 3 B )(B 3 B 1 ) (L p +L 1 )+ E n 1(B )E n 1 (B 1 ) (B 3 B )(B 3 B 1 ) + [E 1(B ) E 1 (B 1 )]E 1 (B ) (B B 1 ) E 1(E 1 (B )) B B 1 + E 1(B ) (B B 1 ) From (34), we know that ω p depends on p By the same assertion as above, we know that there exists at most one p, such that ω p 0 If for any p, ω p = 0, by (33), we have B 1p,n 1 = 0 Otherwise, we may prove that ω p0 0 for the above p 0 in (3) In fact, by (30), we have } }

179 SHICHANG SHU AND YANYAN LI B1p 0,n 1 = υp 0 0 On the other hand, by (33), we have Bn 1p 0,1 = ωp 0 Since B 1p0,n 1 = B n 1p0,1, we have ω p0 = υ p0 0 Since there exists at most one p, such that ω p 0, we know that for other p(p p 0 ), ω p = 0 By (34), we also have (35) υ p0 = ω p0 = ω p0 ω p = (B 3 B )(B 3 B 1 )(L p0 L p ) Thus, from (3) and (35), we have and (B 1 B 3 )(B 1 B +B 3 )(L p0 L p ) = 0 If L p0 = L p, by (3), we have υ p0 = 0, this contradicts with υ p0 0 Thus (36) B 1 B +B 3 = 0, db 1 db +db 3 = 0 From (35) and (36), we easily know that db 1 = db = db 3 = 0, that is, B 1,B,B 3 are constants We complete the proof of Proposition 3 4 Proof of theorems Proof of Theorem 1 Let B 1, B, B 3 be the three distinct Laguerre principal curvatures of multiplicities m 1, m, m 3 and one of which is simple Since n 5, without loss of generality, we may assume that m 3 = 1, m 1 1 and m By Proposition 3, we know that either B 1,B,B 3 are constants or B ap,n 1 = 0 for any a,p In the first case, we know that x is a Laguerre isoparametric hypersurface with non-parallel Laguerre second fundament form In the second case, if B ap,n 1 = 0 for any a,p, we may consider two cases: (i) If m 1, since B ap,n 1 = 0 for every a,p, setting p = q in (316), we have a,p = 0 for any a,p From (315), we get for any 1 a m 1 and m 1 +1 p m 1 +m, (41) L a +L p = E n 1(B 1 )E n 1 (B ) (B 1 B 3 )(B 3 B ) Thus, we know that L a = L b for any a,b and L p = L q for any p,q This implies that x has at most three distinct Laguerre Blaschke eigenvalues L a,l p,l n 1 with multiplicities m 1, m, 1 and m 1,m (ii) If m 1 = 1, since B 1p,n 1 = 0 for any p, setting p = q in (30), we have υ p = 0 for any p, m 1 +1 p m 1 +m By (31), (4) L p = L n 1 + E 1 (B )E 1 (B 3 ) (B 1 B )(B 1 B 3 ) + [E n 1(B ) E n 1 (B 3 )]E n 1 (B ) (B B 3 ) E n 1(E n 1 (B )) B B 3 + E n 1(B ) (B B 3 ) Thus, we know that x has at most three distinct Laguerre Blaschke eigenvalues L 1,L p,l n 1 with multiplicities 1, m, 1 and m

LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n 1793 Next, we may prove that x : M R n is only a Laguerre isotropic hypersurface, that is, the number of distinct Laguerre Blaschke eigenvalues is only 1 In fact, if L a,l p,l n 1 are the three constant Laguerre Blaschke eigenvalues with multiplicities m 1, m, 1 and the number of distinct Laguerre Blaschke eigenvalues is or 3 From (14), we have (43) L ij,k = E k (L i )δ ij +Γ j ik (L i L j ), where Γ j ik is the Levi-Civita connection for the Laguerre metric g From (18), we know that L ii,j = L ij,i Thus (44) E j (L i ) = Γ j ii (L i L j ) for i j If the number of distinct Laguerre Blaschke eigenvalues is 3, that is, L a L p L n 1, 1 a m 1, m 1 +1 p m 1 +m and m, from (44), we have (45) 0 = E a (L p ) = Γ a pp (L p L a ), 0 = E n 1 (L p ) = Γ n 1 pp (L p L n 1 ) Thus Γ a pp = Γ n 1 pp = 0 On the other hand, from (15), we have (46) B ij,k = E k (B i )δ ij +Γ j ik (B i B j ), where Γ j ik is the Levi-Civita connection for the Laguerre metric g From (), we know that B ii,j = B ij,i Thus (47) E j (B i ) = Γ j ii (B i B j ) for i j We get E a (B p ) = Γ a pp(b p B a ) = 0, E n 1 (B p ) = Γ n 1 pp (B p B n 1 ) = 0 That is, E a (B ) = 0, E n 1 (B ) = 0 Since we assume that m, from Proposition 31, we have E p (B ) = 0 Thus, B is constant Therefore, from (35), we know that B 1 and B 3 are constants If the number of distinct Laguerre Blaschke eigenvalues is, when m 1, without loss of the generality, we may assume that L a = L p L n 1 By (44), we have 0 = E n 1 (L a ) = Γ n 1 aa (L a L n 1 ) Thus, Γ n 1 aa = 0 On the other hand, by (47) From (35), we have (48) E n 1 (B 1 ) = Γ n 1 aa (B 1 B 3 ) = 0 m 1 E i (B 1 ) B 3 B = m E i (B ) B 1 B 3 = m 3E i (B 3 ) B B 1 By Proposition 31, we have E a (B 1 ) = 0, 1 a m 1 and E p (B ) = 0, m 1 +1 p m 1 +m By (48), we have E a (B 3 ) = E p (B 3 ) = E n 1 (B 3 ) = 0, that is, B 3 is constant By (35) again, we know that B 1 and B are constants

1794 SHICHANG SHU AND YANYAN LI When m 1 = 1, without loss of the generality, we may assume that L 1 = L n 1 L p By (44), we have 0 = E 1 (L p ) = Γ 1 pp(l p L 1 ), 0 = E n 1 (L p ) = Γ n 1 pp (L p L n 1 ) Thus Γ 1 pp = Γn 1 pp = 0 On the other hand, by (47) E 1 (B ) = Γ 1 pp(b B 1 ) = 0, E n 1 (B ) = Γ n 1 pp (B B 3 ) = 0 From Proposition 31, we have E p (B ) = 0, p 1 + m Thus B is constant By (35) again, we know that B 1 and B 3 are constants To sum up, we know that if the number of distinct Laguerre Blaschke eigenvalues is or 3, then B 1, B and B 3 are constants From (46), we have B ab,k = B pq,k = B αβ,k = 0 for any a,b,p,q,α,β,k Since we know that B ap,n 1 = 0 for every a,p, we get B ij,k = 0 for any i,j,k, that is, the Laguerre second fundamental form of x is parallel From (15), it follows that (49) 0 = db i δ ij +(B i B j )ω ij, 1 i,j n 1 If B i B j, we have ω ij = 0 If for some k such that ω ik 0 and ω kj 0, by (49) we have B i = B k = B j, this is in contradiction with B i B j Thus, from (16), we have R ijij = 0 for B i B j From (1), it follows that (410) L i +L j = 0 for B i B j Let B 1 = B a, B = B p, B 3 = B α be the three distinct Laguerre principal curvatures with multiplicities m 1, m, m 3 one of which is simple, where 1 a m 1, m 1 +1 p m 1 +m, m 1 +m +1 α n 1 Since B a B p, B a B α and B p B α, from (410), we have L a +L p = 0, L a +L α = 0 and L p + L α = 0 This implies that L a = 0, L p = 0 and L α = 0 for any a,p,α That is L i = 0 for any i This is a contradiction with the assumption that the number of distinct Laguerre Blaschke eigenvalues is or 3 Therefore, we know that the number of distinct Laguerre Blaschke eigenvalues is only 1, that is, x is only a Laguerre isotropic hypersurface This completes the proof of Theorem 1 Proof of Theorem 13 By the same assertion as in the proof of Theorem 1, we know that either B 1,B,B 3 are constants and x is a Laguerre isoparametric hypersurface with non-parallel Laguerre second fundament form, or B ap,n 1 = 0 for any a,p If B ap,n 1 = 0 for any a,p, we may consider two cases: (i) If m 1, since B ap,n 1 = 0 for every a,p, setting p = q in (316), we have a,p = 0 for any a,p From (315) and (3), we get for any 1 a m 1 and m 1 +1 p m 1 +m, (411) D a +D p = λ(b 1 +B ) E n 1(B 1 )E n 1 (B ) (B 1 B 3 )(B 3 B )

LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n 1795 From (411), we know that for any a, all D a are the same and for any p, all D p are the same Thus, we know that x has at most three distinct Laguerre para- Blaschke eigenvalues D a,d p,d n 1 with multiplicities m 1, m, 1 and m 1,m (ii) If m 1 = 1, since B 1p,n 1 = 0 for any p, setting p = q in (30), we have υ p = 0 for any p, m 1 +1 p m 1 +m By (31) and (3), (41) D p = D n 1 +λ(b +B 3 )+ E 1 (B )E 1 (B 3 ) (B 1 B )(B 1 B 3 ) + [E n 1(B ) E n 1 (B 3 )]E n 1 (B ) (B B 3 ) E n 1(E n 1 (B )) B B 3 + E n 1(B ) (B B 3 ) From (41), we know that for any p, all D p are the same Thus, x has at most three distinct para-blaschke eigenvalues D 1,D p,d n 1 with multiplicities 1, m, 1 and m If the number of the distinct Laguerre para-blaschke eigenvalues of D a, D p, D n 1 is 1, then x is a Laguerre para-isotropic hypersurface If the number of the distinct Laguerre para-blaschke eigenvalues of D a, D p, D n 1 is, we may prove that this case does not occur In fact, if m 1, without loss of the generality, we may assume that D a = D p D n 1 From (4), we have D ij,k = E k (D i )δ ij +Γ j ik (D i D j ), where Γ j ik is the Levi-Civita connection for the Laguerre metric g From (5), we know that D ii,j = D ij,i Thus (413) E j (D i ) = Γ j ii (D i D j ) for i j By (413), we have Thus, Γ n 1 aa = 0 From (47), we get 0 = E n 1 (D a ) = Γ n 1 aa (D a D n 1 ) E n 1 (B 1 ) = Γ n 1 aa (B 1 B 3 ) = 0 Combining with E a (B 1 ) = 0, 1 a m 1, E p (B ) = 0, m 1 +1 p m 1 +m, (48) and (35), we easily see that B 1, B and B 3 are constants If m 1 = 1, without loss of the generality, we may assume that D 1 = D n 1 D p By (413), we have 0 = E 1 (D p ) = Γ 1 pp (D p D 1 ), 0 = E n 1 (D p ) = Γ n 1 pp (D p D n 1 ) Thus Γ 1 pp = Γn 1 pp = 0 On the other hand, by (47) E 1 (B ) = Γ 1 pp (B B 1 ) = 0, E n 1 (B ) = Γ n 1 pp (B B 3 ) = 0 Combining with E p (B ) = 0, p 1+m and (35), we easily see that B 1, B and B 3 are constants

1796 SHICHANG SHU AND YANYAN LI By the same assertion as in the proof of Theorem 1, we know that the Laguerre second fundamental form of x is parallel and L i = 0 for any i Since λ 0, it follows that x has three distinct Laguerre para-blaschke eigenvalues λb 1,λB,λB 3 This is a contradiction If the number of the distinct Laguerre para-blaschke eigenvalues of D a, D p, D n 1 is 3, that is, D a D p D n 1, 1 a m 1, m 1 +1 p m 1 +m and m, from (413), we have (414) 0 = E a (D p ) = Γ a pp (D p D a ), 0 = E n 1 (D p ) = Γ n 1 pp (D p D n 1 ) Thus Γ a pp = Γ n 1 pp = 0 By (47), we get E a (B p ) = Γ a pp (B p B a ) = 0, E n 1 (B p ) = Γ n 1 pp (B p B n 1 ) = 0 That is, E a (B ) = 0, E n 1 (B ) = 0 Combining with E p (B ) = 0 and (35), we know that B 1, B and B 3 are constants By the same assertion as in the proof of Theorem 1, we know that the Laguerre second fundamental form of x is parallel and L i = 0 for any i From the result of Theorem 11 and Example 1 Example, we know that x is Laguerre equivalent to an open part of the image of τ of the oriented hypersurface x : R n 1 R n 0 given by Example This completes the proof of Theorem 13 Acknowledgment The authors would like to thank the referee for his/her many valuable remarks and suggestions that really improve the paper References [1] W Blaschke, Vorlesungen über Differential geometrie, Springer, Berlin, Heidelberg, New York, Vol 3, 199 [] G H Li, Möbius hypersurfaces in S n+1 with three distinct principal curvatures, J Geom 80 (004), no 1-, 154 165 [3] T Z Li, Laguerre geometry of surfaces in R 3, Acta Math Sin (Engl Ser) 1 (005), no 6, 155 1534 [4] T Z Li, H Z Li, and C P Wang, Classification of hypersurfaces with parallel Laguerre second fundamental form in R n, Differential Geom Appl 8 (010), no, 148 157 [5] T Z Li and C P Wang, Laguerre geometry of hypersurfaces in R n, Manuscripta Math 1 (007), no 1, 73 95 [6] E Musso and L Nicolodi, A variational problem for surfaces in Laguerre geometry, Trans Amer Math Soc 348 (1996), no 11, 431 4337 [7], Laguerre geometry of surfaces with plane lines of curvature, Abh Math Sem Univ Hamburg 69 (1999), 13 138 Shichang Shu School of Mathematics and Information Science Xianyang Normal University Xianyang 71000 Shaanxi, P R China E-mail address: shushichang@16com

LAGUERRE CHARACTERIZATIONS OF HYPERSURFACES IN R n 1797 Yanyan Li School of Mathematics and Information Science Xianyang Normal University Xianyang 71000 Shaanxi, P R China E-mail address: muzilyy@163com