# On SomeLk-Finite-Type Euclidean Hypersurfaces

Authors: Akram Mohammadpouri; S. M. B. Kashani
Journal: ISRN Geometry
 (2012)

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## Abstract

We study someLk-finite-type Euclidean hypersurfaces. We classify Lk-1-type Euclidean hypersurfaces and Lk-null-2-type Euclidean hypersurfaces with at most two distinct principal curvatures. We also prove that, under some mild restrictions, there exists no Lk-null-3-type Euclidean hypersurface.


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## Body

## 1. Introduction

The study of submanifolds of finite type began in the late seventies with Chen’s attempts to find the best possible estimate of the total mean curvature of compact submanifolds of Euclidean space and to find a notion of “degree” for submanifolds of Euclidean space (see [1] for details). Since then the subject has had a rapid development and so many mathematicians contribute to it; seeo the excellent survey of Chen [2]. By definition, an isometrically immersed submanifold x:Mn→ℝn+k is said to be of finite type if x has a finite decomposition as x-x0=∑i=1p‍xi, for some positive integer p<+∞ and Δxi=κixi, κi∈ℝ,1≤i≤p where x0 is constant, and xi, 1≤i≤p, are nonconstant smooth maps, and Δ is the Laplace operator of M. If all κi's are mutually different, then Mn is said to be of p-type. If, in particular, one of κi's is zero, then M is said to be of null p-type.As is well known, the Laplace operator of a hypersurfaceM immersed into ℝn+1 is an (intrinsic) second-order linear differential operator which arises naturally as the linearized operator of the first variation of the mean curvature for normal variations of the hypersurface. From this point of view, the Laplace operator Δ can be seen as the first one of a sequence of n operators L0=Δ, L1,…,Ln-1, where Lk stands for the linearized operator of the first variation of the (k+1)th mean curvature arising from normal variations of the hypersurface (see [3, 4]). These operators are given by Lk(f)=tr(Pk∘∇2f) for any f∈C∞(M), where Pk denotes the kth Newton transformation associated to the second fundamental from of the hypersurface and ∇2f is the hessian of f (see the next section for details).In contrast to the operatorΔ, which is elliptic, in general the operators Lk are not elliptic but they still share some nice properties with Laplacian of M. Therefore, from this point of view, it seems natural and interesting to generalize the definition of finite-type hypersurface by replacing Δ by Lk and study the properties of such hypersurfaces. Having this idea, for the first time in [5], the second author inspired by the private communication with Alías, introduced such hypersurfaces and called them Lk-finite-type hypersurfaces. In [5], he began to study generalized cylinders, ruled surfaces, and some revolution hypersurfaces from the point of view of L1-finiteness type.In this paper which is a natural continuation of [5], we study some Lk-finite-type hypersurfaces in the Euclidean space. The structure of the paper is as follows.In Section2, we give preliminaries.In Section3, we classify Lk-1-type Euclidean hypersurfaces and Lk-null-2-type Euclidean hypersurfaces with at most two distinct principal curvatures. We follow the work of Ferrández and Lucas [6], for Euclidean hypersurfaces of null-2-type. They proved that such hypersurfaces with at most two distinct principal curvatures are locally isometric to a generalized cylinder. This is a generalization of Chen's Theorem in [7], stating that null-2-type surfaces are circular cylinders. Here we generalize the result of [6] and prove that any Euclidean hypersurface of Lk-null-2-type, (k≠n-1), with at most two distinct principal curvatures, is locally isometric to a generalized cylinder. One of the important consequences of this theorem is the classification of conformally flat hypersurfaces Mn of Lk-null-2-type for n>3. We also prove that there is no Ln-1-null-2-type Euclidean hypersurface Mn with at most two distinct principal curvatures. This is a generalization of Chen's result stating that there is no null-2-type plane curve [2, corollary of Theorem 7.3].Section4 is about Lk-3-type Euclidean hypersurfaces. Here we prove that any L1-3-type surface has nonconstant Gaussian curvature. The result is an extension of Theorem 1 of [8], stating that any 3-type surface in ℝ3 has nonconstant mean curvature. Also motivated by the work of Ferrández and Lucas [6], stating that there is no Euclidean hypersurface of null-3-type with constant mean curvature and at most two principal curvatures, we prove that there is no Euclidean hypersurface of Lk-null-3-type with constant Hk+1 and at most two distinct principal curvatures.Our main results are the following theorems.Theorem 1.1.
k
-minimal Euclidean hypersurfaces and open parts of hyperspheres are the only Lk-1-type hypersurfaces in ℝn+1.Theorem 1.2.
There is noLn-1-null-2-type hypersurface in the Euclidean space ℝn+1, with at most two distinct principal curvatures.Theorem 1.3.
Letx:Mn→ℝn+1 be an isometrically immersed Euclidean hypersurface with at most two distinct principal curvatures and multiplicities are greater than one. Then Mn is of Lk-null-2-type (k≠n-1), if only if M is isoparametric, so locally isometric to Sm×ℝn-m, m≥k+1.Theorem 1.4.
Letx:Mn→ℝn+1 be an isometrically immersed Euclidean hypersurface with at most two distinct principal curvatures, one of them is simple. Then Mn is of Lk-null-2-type (k≠n-1) if only if M is isoparametric, so locally isometric to ℝ×Sn-1(r) for 0≤k<n-1 or ℝn-1×S1(r) for k=0.In Theorems1.3 and 1.4, if we assume that M is complete, then M is globally isometric to a generalized cylinder.Theorem 1.5.
There is noL1-3-type surface in ℝ3 with constant Gaussian curvature.Theorem 1.6.
There is no hypersurface ofLk-null-3-type in ℝn+1 with constant Hk+1 and at most two distinct principal curvatures.


## 2. Preliminaries

In this section we recall some notions and prerequisites about (Lk-finite type) hypersurfaces of the Euclidian space from [9, 10].Consider an isometrically immersed hypersurfacex:Mn→ℝn+1 in the Euclidean space. We choose a local orthonormal frame {eA}1≤A≤n+1 in ℝn+1, with dual coframe {ωA}1≤A≤n+1, such that, at each point of M, e1,…,en are tangent to M and en+1 is the positively oriented unit normal vector. We will make use of the following convention on the ranges of indices:
(2.1)1≤A,B,C,…,≤n+1;1≤i,j,k,…,≤n.
Then the structure equations of ℝn+1 are given by
(2.2)dωA=∑B=1n+1‍ωAB∧ωB,ωAB+ωBA=0,dωAB=∑C=1n+1‍ωAC∧ωCB.
When restricted to M, we have ωn+1=0 and
(2.3)0=dωn+1=∑i=1n‍ωn+1i∧ωi.
By Cartan's lemma, there exist functions hij such that
(2.4)ωn+1i=∑j=1n‍hijωj,hij=hji.
This gives the second fundamental form of M, as B=∑i,j‍hijωiωjen+1. The mean curvature H is defined by H=(1/n)∑i‍hii. From (2.2)–(2.4) we obtain the structure equations of M, (see [10]):
(2.5)dωi=∑j=1n‍ωij∧ωj,ωij+ωji=0,dωij=∑k=1n‍ωik∧ωkj-12∑k,l=1n‍Rijklωk∧ωl,
and the Gauss equations
(2.6)Rijkl=(hikhjl-hilhjk),
where Rijkl denotes the components of the Riemannian curvature tensor of M.Lethijk denote the covariant derivative of hij. We have
(2.7)∑k‍hijkωk=dhij+∑k‍hkjωki+∑k‍hikωkj.
Thus, by exterior differentiation of (2.4), we obtain the Codazzi equation
(2.8)hijk=hikj.
We choose e1,…,en such that
(2.9)hij=λiδij.
Let Hm be mth mean curvature of M, then we have
(2.10)(nm)Hm=∑1≤i1<i2<⋯<im≤n‍λi1⋯λim.
And Hn=λ1⋯λn is called the Gauss-Kronecker curvature of M. A hypersurface with zero (k+1)th mean curvature in ℝn+1 is called k-minimal. To get more information about k-minimal Euclidean hypersurfaces, the reader is referred to [11, 12].The classical Newton transformationsPk:χ(M)→χ(M) are defined inductively by the shape operator S as
(2.11)P0=I,Pk=(nk)HkI-S∘Pk-1,
for every k=1,…,n, where I denotes the identity transformation in χ(M). Equivalently,
(2.12)Pk=∑j=0k‍(-1)j(nk-j)Hk-jSj.
Note that by the Cayley-Hamilton theorem stating that any operator is annihilated by its characteristic polynomial, we have Pn=0.Since eachPk(p) is also a self-adjoint linear operator on each tangent plane TpM which commutes with S(p). Indeed, S(p) and Pk(p) can be simultaneously diagonalized: if {e1,…,en} are the eigenvectors of S(p) corresponding to the eigenvalues λ1(p),…,λn(p), respectively, then they are also the eigenvectors of Pk(p) with corresponding eigenvalues given by
(2.13)μi,k(p)=∑i1<⋯<ik,ij≠i‍λi1(p)⋯λik(p),
for every 1≤i≤n. We have the following formulae for the Newton transformations, [9]:
(2.14)tr(Pk)=ckHk,(2.15)tr(S∘Pk)=ckHk+1,(2.16)tr(S2∘Pk)=(nk+1)(nH1Hk+1-(n-k-1)Hk+2),
where
(2.17)ck=(n-k)(nk)=(k+1)(nk+1).Associated to each Newton transformationPk, we consider the second-order linear differential operator Lk:C∞(M)→C∞(M) given by
(2.18)Lk(f)=tr(Pk∘∇2f).
Here ∇2f:χ(M)→χ(M) denotes the self-adjoint linear operator metrically equivalent to the Hessian of f and is given by
(2.19)〈∇2f(X),Y〉=〈∇X(∇f),Y〉,
where X,Y∈χ(M), ∇f is the gradient of f, and ∇ is the Levi-Civita connections on M.Now we recall the definition of anLk-finite-type hypersurface from [5], which is the basic notion of the paper. We also quote Proposition 2.2 from [5].Definition 2.1.
An isometrically immersed hypersurfacex:Mn→ℝn+1 is said to be of Lk-finite type if x has a finite decomposition x=∑i=0m‍xi, for some positive integer m satisfying the condition that Lkxi=κixi, κi∈ℝ, 1≤i≤m, where xi:Mn→ℝn+1 are smooth maps, 1≤i≤m, and x0 is constant. If all κi's are mutually different, Mn is said to be of Lk-m-type. An Lk-m-type hypersurface is said to be null if some κi;1≤i≤m, is zero. The polynomial p(t)=∏i=1m‍(t-κi) is called the minimal polynomial of M for Lk.We should mention that similar to Proposition 1 of [13], if M is of Lk-finite type, then p(Lk)(x-x0)=0.Proposition 2.2 (see [5]).
If the isometrically immersed hypersurfaceMn⊂ℝn+1 is a generalized cylinder Sm(r)×ℝn-m, then M is of Lk-null-1-type, if k+1>m, and it is of Lk-null-2-type, if k+1≤m.


## 3.Lk-2-Type Hypersurfaces

In this section we would like to follow [6] and consider Lk-2-type hypersurfaces. In [6], it was proved that if M is a nonminimal Euclidean hypersurface of 1- or 2-type and with constant mean curvature, then either (a) M is an open piece of Sn or (b) M is of null-2-type. In Theorem 3.2 we obtain an extension of this result for Lk-1 and 2-type Euclidean hypersurfaces. In Theorem 3.5, we prove that there is no Ln-1-null-2-type hypersurface in ℝn+1 with at most two distinct principal curvatures. It is a generalization of Chen's result stating that there is no null-2-type plane curve [2, corollary of Theorem 7.3]. In [7], Chen proved that null-2-type surfaces are circular cylinders. Later, Ferrández and Lucas got a generalization of Chen's theorem and showed that Euclidean hypersurfaces of null-2-type with at most two distinct principal curvatures are locally isometric to a generalized cylinder [6, Theorem 3.10]. In Theorems 3.11 and 3.12, we generalize this theorem and prove that any Euclidean hypersurface of Lk-null-2-type (k≠n-1), with at most two distinct principal curvatures is locally isometric to a generalized cylinder.According to Takahashi's theorem [14], we know that minimal Euclidean hypersurfaces and open parts of hyperspheres are the only 1-type Euclidean hypersurfaces. In the next theorem, we show that k-minimal hypersurfaces and open parts of hyperspheres are the only Lk-1-type Euclidean hypersurfaces.Theorem 3.1.
k-minimal Euclidean hypersurfaces and open parts of hyperspheres are the only Lk-1-type hypersurfaces in ℝn+1.Proof.
In Theorem 1 of [9] Alías and Gürbüz have classified hypersurfaces in ℝn+1 satisfying the general condition Lkx=Ax+b, where A∈ℝ(n+1)×(n+1) is a matrix and b∈ℝn+1. Section 3 of [9] and Proposition 2.2 show that k-minimal hypersurfaces and open parts of hyperspheres are the only hypersurfaces satisfying the condition Lkx=cIx+b, for some real constant c.Now we can state the next theorem.Theorem 3.2.
Letx:Mn→ℝn+1 be an isometrically immersed hypersurface in the Euclidean space. If it is of Lk-1-type or Lk-2-type for some 0≤k≤n-1 and the (k+1)th mean curvature of M is a nonzero constant, then one of the following two cases occurs:(a)
Mis an open piece of Sn,(b)
M is of Lk-null-2-type.Proof.
The casek=0 was proved by Chen and Lue [15, Theorem 1] and Ferrández and Lucas [6, Corollary 3.2], so that we may consider the case k≥1. Here, we will follow the techniques introduced by Chen and Lue [15, Theorem 1] for our context. According to Theorem 3.1, k-minimal hypersurfaces and open parts of hyperspheres are the only Lk-1-type hypersurfaces. If M is of Lk-2-type, the position vector x of M in ℝn+1 has the following spectral decomposition:
(3.1)x-c=x1+x2,Lkx1=κ1x1,Lkx2=κ2x2,forsomec∈ℝn+1,
so
(3.2)Lk2x=(κ1+κ2)Lkx-κ1κ2(x-c).
We also have
(3.3)Lkx=ckHk+1N,(3.4)Lk2x=-ck(nk+1)Hk+1(nH1Hk+1-(n-k-1)Hk+2)N.
The formula (3.4) holds since Hk+1 is a nonzero constant, see [9]. Therefore, by using (3.2), (3.3), and (3.4), we obtain
(3.5)-ck(nk+1)Hk+1(nH1Hk+1-(n-k-1)Hk+2)N=ck(κ1+κ2)Hk+1N-κ1κ2(x-c).
From (3.5) we have either κ1κ2=0 or x-c is normal to M at every point of M. If κ1κ2=0, then M is of Lk-null-2-type. If x-c is normal to M, then 〈x-c,x-c〉 is a positive constant. In this case, M is an open piece of Sn centered at c; therefore, M is of Lk-1-type, which is not.As a corollary we get the following result.Corollary 3.3.
IfMn is an Lk-2-type Euclidean hypersurface with constant (k+1)th mean curvature, then Mn is of null Lk-2-type, in particular, if n=k+1 then the mean curvature is a nonzero constant.Proof.
By Theorem3.2, M is null, and we have the following spectral decomposition for the position vector x of M in ℝn+1:
(3.6)x=c+x1+x2,Lkx1=0,Lkx2=κ2x2.
Since n=k+1 and M is null, from formula (3.5) we have
(3.7)-(k+1)H1Hk+12N=κ2Hk+1N.
From (3.7) we obtain κ2=-(k+1)H1Hk+1. Since κ2 and Hk+1 are nonzero and constant, so H1 is also nonzero and constant.To express an extension of Chen's result in [2] stating that there is no null-2-type plane curve, we need the next lemma.Lemma 3.4.
Letx:Mn→ℝn+1 be an isometrically immersed Euclidean hypersurface which is of Ln-1-null-2-type, then the Gauss-Kronecker curvature of M is nonzero and constant.Proof.
Let us consider the open set𝒰={p∈M:∇Hn2(p)≠0}, our objective is to show that 𝒰 is empty. From [9], we have
(3.8)Ln-12x=-cn-1Hn∇Hn-2cn-1(S∘Pn-1)∇Hn-cn-1(nH1Hn2-Ln-1Hn)N.
From this relation and (3.2) we obtain that
(3.9)-cn-1Hn∇Hn-2cn-1(S∘Pn-1)∇Hn-cn-1(nH1Hn2-Ln-1Hn)N=cn-1κ2HnN.
Therefore on 𝒰 we get
(3.10)(S∘Pn-1)∇Hn=-12Hn∇Hn.
But by the Cayley-Hamilton theorem we have Pn=0, so
(3.11)S∘Pn-1=HnI,(S∘Pn-1)∇Hn=Hn∇Hn,
which jointly with (3.10) yields ∇Hn2=0 on 𝒰, which is a contradiction.Now, we state one of the main results of the section as follows.Theorem 3.5.
There is noLn-1-null-2-type hypersurface in ℝn+1, with at most two distinct principal curvatures.Proof.
IfM is an Ln-1-null-2-type hypersurface in ℝn+1 with at most two principal curvatures, by applying Corollary 3.3 and Lemma 3.4 we obtain that the mean and Gauss-Kronecker curvatures are constant. Since M is of Ln-1-null-2-type, we conclude that M has exactly two constant principal curvatures. From [16], M is an open piece of ℝp×Sn-p for some p≥1, so the Gauss-Kronecker curvature of M is zero; therefore, M is of Ln-1-null-1-type, which is a contradiction.In [7], Chen proved that null 2-type surfaces are circular cylinders. In sharp contrast to this result, in the next corollary, we claim that there is not any L1-null-2-type surface at all! It is an obvious consequence of Theorem 3.5.Corollary 3.6.
There is noL1-null-2-type surface in ℝ3.By Theorem3.5 and the following theorems from [17, 18], we can prove that there is no Ln-1-null-2-type compact hypersurface in ℝn+1 with everywhere nonzero sectional curvature.Theorem 3.7 (see [17]).
Letx:Mn→ℝn+1 be an isometrically immersed compact hypersurface in the Euclidean space, then the following conditions are equivalent:(i)
Hr=const≠0 for some r∈{2,…,n} and K≥0,(ii)
H=const and K≥0,(iii)
M has parallel second fundamental form (i.e., M is isoparametric with at most two principle curvatures),
whereK is the sectional curvature of M.The next theorem shows that there is no compact isometrically immersed hypersurfaceMn in ℝn+1 with everywhere nonpositive sectional curvature.Theorem 3.8 (see [18]).
LetM be a compact n-dimensional Riemannian manifold, and let M- be a complete simply connected Riemannian manifold of dimension less than 2. If the sectional curvatures K and K- of M and M- satisfy K≤K-≤0, then M cannot be immersed in M-.Now we can express our result.Proposition 3.9.
There is noLn-1-null-2-type compact hypersurface in the Euclidean space ℝn+1 with everywhere nonzero sectional curvature.Proof.
By Theorem3.8, we assume that K>0. By lemma 4.3, Hn is nonzero and constant. So we get from Theorem 3.7 that M is isoparametric with at most two distinct principle curvatures, and this is a contradiction with Theorem 3.5.By considering two different cases we prove thatLk-null-2-type Euclidean hypersurfaces (k≠n-1) with at most two distinct principal curvatures are circular cylinders. Case  I: The multiplicities are greater than one. Case  II: one of the principal curvatures is simple.We use the following Lemma from [19], for the proof of claim.Lemma 3.10.  (see [19], Theorem 2 and its corollary).
LetM be an n-dimensional hypersurface in the Euclidean space ℝn+1 such that multiplicities of principal curvatures are constant. Then the distribution of the space of principal vectors corresponding to each principal curvature is completely integrable. In particular, if the multiplicity of a principal curvature is greater than one, then this principal curvature is constant on each integral submanifold of the corresponding distribution of the space of principal vectors.Theorem 3.11.
Letx:Mn→ℝn+1 be an isometrically immersed hypersurface with at most two distinct principal curvatures and multiplicities are greater than one. Then Mn is of Lk-null-2-type (k≠n-1) if only if M is isoparametric so locally isometric to Sm×ℝn-m, m≥k+1.Proof.
LetMn is of Lk-null-2-type. If Mn is totally umbilical, then Mn is a piece of ℝn or Sn. By using Theorem 3.1, ℝn and Sn are of Lk-1-type, so M is not totally umbilical. Therefore M has two distinct principal curvatures of multiplicities q and n-q, (q,n-q>1).
Let us consider the open set(3.12)𝒰={p∈M:∇Hk+12(p)≠0}.
Our objective is to show that 𝒰 is empty.
Consider{e1,…,en} to be a local orthonormal frame of principal directions of S on 𝒰 such that Sei=λiei for every i=1,…,n. We assume that
(3.13)λ1=λ2=⋯=λq=λ,λq+1=⋯=λn=μ.
Therefore from (2.13) we have
(3.14)Pk+1ei=μi,k+1ei,
with
(3.15)μi,k+1=∑i1<⋯<ik+1,ij≠i‍λi1⋯λik+1.
So we get(3.16)μ1,k+1=⋯=μq,k+1=∑s‍(q-1s)(n-qk+1-s)λsμk+1-s,μq+1,k+1=⋯=μn,k+1=∑s‍(qs)(n-q-1k+1-s)λsμk+1-s.
We obtain from (2.10) that
(3.17)(nk+1)Hk+1=∑s‍(qs)(n-qk+1-s)λsμk+1-s.
Since M is of Lk-null-2-type, the position vector field x satisfies the following equation for some constant m≠0,
(3.18)Lk2x=mLkx.
So by using the formulae of Lkx and Lk2x from [9], we can write
(3.19)-ck(nk+1)Hk+1∇Hk+1-2ck(S∘Pk)(∇Hk+1)-ck((nk+1)Hk+1(nH1Hk+1-(n-k-1)Hk+2)-LkHk+1)N=mckHk+1N.
From (3.19) we get
(3.20)(S∘Pk)(∇Hk+12)=-12(nk+1)Hk+1∇Hk+12.
Since k≠n-1, it follows from the inductive definition of Pk+1 that (3.20) is equivalent to
(3.21)Pk+1(∇Hk+12)=32(nk+1)Hk+1∇Hk+12on𝒰.
Therefore, writing
(3.22)∇Hk+12=∑i=1n‍〈∇Hk+12,ei〉ei,
we see that (3.21) is equivalent to
(3.23)〈∇Hk+12,ei〉(μi,k+1-32(nk+1)Hk+1)=0on𝒰
for every i=1,…,n. So there is no loss of generality, assuming that
(3.24)μ1,k+1=⋯=μq,k+1=32(nk+1)Hk+1.
Let us denote the integral submanifolds throughx∈𝒰 corresponding to λ and μ by 𝒰1q(x) and 𝒰1n-q(x), respectively. From Lemma 3.10, we know that λ is constant on 𝒰1q(x). Equations (3.16), (3.17), and (3.24) imply that μ is constant on 𝒰1q(x). Again by Lemma 3.10, we get that μ is constant on 𝒰1n-q(x). It now follows from [20, page 182, I] that 𝒰 is locally isometric to the Riemannian product of the maximal integral manifolds 𝒰1q(x) and 𝒰1n-q(x). Therefore, μ is constant on 𝒰. By the same assertion, we know that λ is constant on 𝒰, so Hk+1 is constant on 𝒰, which is a contradiction. Hence Hk+1 is constant and nonzero on M. From (3.19), we obtain that tr(S2∘Pk)=m; therefore, tr(S2∘Pk) is constant. By the fact that M has two principal curvatures and Hk+1, tr(S2∘Pk) are constant, we get that the principal curvatures are constant. So M is isoparametric.
A classical result of Segre [16] states that isoparametric hypersurfaces in ℝn+1 are locally isometric to ℝn, Sn(r) and circular cylinder. On the other hand, since Mn is of Lk-null-2-type, by using Proposition 2.2, we conclude that Mn is locally isometric to Sm×ℝn-m, m≥k+1.Theorem 3.12.
Letx:Mn→ℝn+1 be an isometrically immersed Euclidean hypersurface with at most two distinct principal curvatures, one of them is simple. Then Mn is of Lk-null-2-type, (k≠n-1), if only if M is isoparametric, so locally isometric to ℝ×Sn-1(r)0≤k<n-1 or ℝn-1×S1(r) for k=0.Proof.
LetMn be of Lk-null-2-type (k≠n-1). If Mn is totally umbilic, then Mn is a piece of ℝn or Sn. By using Theorem 3.1, ℝn and Sn are of Lk-1-type, so M cannot be totally umbilic. Therefore suppose M has two distinct principal curvatures of multiplicities 1 and n-1.
Let us consider the open set.(3.25)𝒰={p∈M:∇Hk+12(p)≠0}.
We want to prove that𝒰 is empty. If 𝒰≠∅, then we express Hk+1 as a polynomial in λ (the nonsimple principal curvature of M) with constant coefficients, after that we express λ as a constant multiple of the simple principal curvature of M. By using Otsuki's Lemma (Lemma 3.10), the structure equations of M, and the fact that M is of Lk-null-2-type, we get that λ satisfies a polynomial with constant coefficients. So λ is constant; hence, Hk+1 is constant, a contradiction with 𝒰≠∅. So 𝒰 is empty.
Here is the detailed treatment of the proof.
With the assumption that𝒰≠∅, consider {e1,…,en} to be a local orthonormal frame of principal directions of S on 𝒰 such that Sei=λiei for every i=1,…,n. We assume
(3.26)λ1=λ2=⋯=λn-1=λ,λn=μ.
Therefore from (2.10) we have
(3.27)Pk+1ei=μi,k+1ei,
with
(3.28)μi,k+1=∑i1<⋯<ik+1,ij≠i‍λi1⋯λik+1.
So we get
(3.29)μ1,k+1=⋯=μn-1,k+1=(n-2k+1)λk+1+(n-2k)λkμ,μn,k+1=(n-1k+1)λk+1.
We obtain from (2.13) that
(3.30)(nk+1)Hk+1=(n-1k+1)λk+1+(n-1k)λkμ.
Since M is of Lk-null-2-type, the position vector field x satisfies the following equation for some constant m≠0,
(3.31)Lk2x=mLkx.
So by using the formulae of Lkx and Lk2x from [9], we can write
(3.32)-ck(nk+1)Hk+1∇Hk+1-2ck(S∘Pk)(∇Hk+1)-ck((nk+1)Hk+1(nH1Hk+1-(n-k-1)Hk+2)-LkHk+1)N=mckHk+1N.
From (3.32) we get
(3.33)(S∘Pk)(∇Hk+12)=-12(nk+1)Hk+1∇Hk+12.
Since k≠n-1, it follows from the inductive definition of Pk+1 that (3.33) is equivalent to
(3.34)Pk+1(∇Hk+12)=32(nk+1)Hk+1∇Hk+12on𝒰.
Therefore, by the formula
(3.35)∇Hk+12=∑i=1n‍〈∇Hk+12,ei〉ei,
we see that (3.34) is equivalent to
(3.36)〈∇Hk+12,ei〉(μi,k+1-32(nk+1)Hk+1)=0on𝒰
for every i=1,…,n. Hence, for every i such that 〈∇Hk+12,ei〉≠0 on 𝒰 we get
(3.37)μi,k+1=32(nk+1)Hk+1.
So for the expression ∇Hk+12 in (3.35) we consider two cases.
Case  1. 〈∇Hk+12,en〉≠0, by (4.31), we obtain that
(3.38)Hk+1=23(n-k-1)nλk+1.Case  2. 〈∇Hk+12,en〉=0, so on 𝒰 we have 〈∇Hk+12,ej〉≠0 for some j=1,…,n-1. By (4.31) and using the fact that μi,k are the eigenvalues of Pk and the formula of tr(Pk+1), we obtain that
(3.39)Hk+1=(n-k-1)n(-(1/2)n-k+1/2)λk+1.
Both cases require the same calculation, so we consider just Case I.
By Lemma3.10, let us denote the maximal integral submanifold through x∈𝒰, corresponding to λ by 𝒰1n-1(x). We write
(3.40)dλ=∑i‍λ,iωidμ=∑j‍μ,jωj.
Then Lemma 3.10 implies that λ,1=⋯=λ,n-1=0. We can assume that λ>0 on 𝒰, then (3.30) and (3.38) yield
(3.41)μ=k+1-n3k+3λ.
By means of (2.7) and (2.9), we obtain that
(3.42)∑k‍hijkωk=δijdλj+(λi-λj)ωij.
We adopt the notational convention that 1≤a,b,c,…≤n-1.
From (3.41) and (3.42), we have
(3.43)hijk=0,ifi≠j,haab=0,haan=λ,n,hnna=0,hnnn=μ,n.
Combining this with (2.8) and the formula
(3.44)∑i‍haniωi=dhan+∑i‍hinωia+∑i‍haiωin=(λ-μ)ωan,
we obtain from (3.41)
(3.45)ωan=λ,nλ-μωa=(3k+3)λ,n(2k+2+n)λωa.
Therefore we have
(3.46)dωn=∑a‍ωna∧ωa=0.
Notice that we may considerλ to be locally a function of the parameter s, where s is the arc length of an orthogonal trajectory of the family of the integral submanifolds corresponding to λ. We may put ωn=ds.
Thus, forλ=λ(s), we have
(3.47)dλ=λ,nds,λ,n=λ'(s),
so from(3.45), we get
(3.48)ωan=λ,nλ-μωa=(3k+3)λ'(s)(2k+2+n)λωa.
According to the structure equations of ℝn+1 and (3.48), we may compute
(3.49)dωan=∑b=1n-1‍ωab∧ωbn+ωan+1∧ωn+1n=((3k+3)λ'(2k+2+n)λ)∑b=1n-1‍ωab∧ωb-λμωa∧ds,(3.50)dωan=d{(3k+3)λ'(2k+2+n)λωa}=((3k+3)λ'(2k+2+n)λ)ds∧ωa+((3k+3)λ'(2k+2+n)λ)dωa={((3k+3)λ′(2k+2+n)λ)′+((3k+3)λ'(2k+2+n)λ)2}ωa∧ds+((3k+3)λ'(2k+2+n)λ)∑b=1n-1‍ωab∧ωb.
Then we obtain from the two equalities above that
(3.51)((3k+3)λ′(2k+2+n)λ)′-((3k+3)λ'(2k+2+n)λ)2-λμ=0.
Combining (3.51) with (3.41), we have
(3.52)((3k+3)λ′(2k+2+n)λ)′-((3k+3)λ'(2k+2+n)λ)2-(k+1-n3k+3)λ2=0.
Let us define a function β(s), s∈(-∞,+∞) by β=(1/λ)(3k+3)/(2k+2+n), then (3.52) reduces to
(3.53)β′′=(n-k-13k+3)β(-7k-7-2n)/(3k+3).
Integrating (3.53), we obtain
(3.54)(β′)2=(k+1-n2k+2+n)β(-4k-4-2n)/(3k+3)+C,
where C is the constant of integration.
Equation (3.54) is equivalent to
(3.55)(λ′)2=((n+2k+2)(k+1-n)(3k+3)2)λ(14k+4n+14)/(2k+2+n)+C(2+2k+n3k+3)2λ(10k+10+2n)/(2k+2+n).
Now we use the definition of LkHk+1=tr(Pk∘∇2Hk+1) to compute LkHk+1. So we need to compute ∇ea∇Hk+1,  ∇en∇Hk+1,  Pk(ea), and Pk(en).
From (3.51) we have
(3.56)∇Hk+1=2(k+1)(n-k-1)3nλkλ′en.
By using (3.48) and (3.56) we obtain
(3.57)∇ea∇Hk+1=2(k+1)(n-k-1)3nλkλ'∇eaen=2(k+1)(n-k-1)3nλkλ'∑b‍ωnb(ea)eb=-2(n-k-1)(k+1)2n(2k+2+n)λk-1λ'2ea∇en∇Hk+1=2(k+1)(n-k-1)3n∇enλkλ′en=2k(k+1)(n-k-1)3nλk-1λ′2en+2(k+1)(n-k-1)3nλkλ′′en.
Now we compute Pk(ea) and Pk(en):
(3.58)Pk(ea)=μa,kea=(∑i1<⋯<ik,ij≠a‍λi1⋯λik)ea=(n-2k)2k+33k+3λkea,Pk(en)=(n-1k)λken.
From (3.57) and (3.58), we get
(3.59)LkHk+1=ckHk+1((-2k-3)(k+1)(n-k-1)n(2k+2+n)λk-2λ'2+k(k+1)nλk-2λ'2+k+1nλk-1λ′′).
Since Mn is of Lk-null-2-type, hence from (3.32), we get
(3.60)LkHk+1=ckHk+1(m-tr(S2∘Pk))=ckHk+1(m-(n-1k)2nk+3n-2k-2k23k+3λk+2).
Combining (3.58) and (3.60), we have
(3.61)λλ′′+(k+(-2k-3)(n-k-1)2k+2+n)λ′2+(n-1k)n(2nk+3n-2k-2k2)(k+1)(3k+3)λ4-mnk+1λ2-k=0.
Equation (3.52) is equivalent to
(3.62)λλ′′=5k+5+n2k+2+nλ'2+(2k+2+n)(k+1-n)(3k+3)2λ4.
Thus, putting together (3.61) and (3.62) one has
(3.63)4k2+12k-kn-2n+82k+2+nλ'2+(2k+2+n)(k+1-n)+3(n-1k)n(2nk+3n-2k-2k2)(3k+3)2λ4-mnk+1λ2-k=0.
We deduce, using (3.55), (3.63), and (3.38), that Hk+1 is locally constant on 𝒰, which is a contradiction with the definition of 𝒰. Hence Hk+1 is constant and nonzero on M. From the discussion as in the last part of the proof of the Theorem 3.11, we get the result.An important consequence of the theorem is the classification of conformally flat hypersurfacesMn of Lk-null-2-type for n>3.Definition 3.13 (see [21]).
A Riemannian manifold(Mn,g) is called conformally flat if it is locally conformally equivalent to a Euclidean space ℝn, that is, if every point of M has a neighborhood which is conformal to an open set in the Euclidean space ℝn. A submanifold N of the Euclidean space ℝn is said to be conformally flat if N with the induced metric for ℝn is conformally flat.The dimension of the hypersurface plays an important role in the study of conformally flat Euclidean hypersurfaces. Forn=2, the existence of isothermal coordinates means that any Riemannian surface is conformally flat. For n>3, the result of Cartan-Schouten stats that a conformally flat hypersurface is characterized with two principal curvatures that one multiplicity at least n-1 (see [21] for more details). This significant fact is crucial in our classification of Lk-null-2-type conformally flat Euclidean hypersurfaces Mn for n>3.As a simple consequence of Theorem3.12, we obtain the following nice corollary which is an extension of the result of [22].Corollary 3.14.
LetMn be a conformally flat hypersurface of ℝn+1, n>3. Then Mn is of Lk-null-2-type, if only if M is locally isometric to the cylinder ℝ×Sn-1(r)(0<k<n-1).


## 4.Ln-1-3-Type Hypersurfaces in ℝn+1

In Theorem 1 of [8], Hasanis and Vlachos proved that there is no 3-type surface in ℝ3 with constant mean curvature. Later Ferrández and Lucas proved that there is no Euclidean hypersurface of null 3-type with constant mean curvature with at most two distinct principal curvatures [6, Theorem 3.5]. Here we follow Hasanis and Vlachos's work to consider L1-3-type surfaces. Also we get a generalization of Theorem 3.5 of [6] and prove that there is no Euclidean hypersurface of Lk-null-3-type with constant Hk+1 and at most two distinct principal curvatures.Theorem 4.1.
There is noL1-3-type surface in ℝ3 with constant Gaussian curvature.Proof.
Letx:M2→ℝ3 be the position vector of an L1-3-type surface M, and set |x|=r for the corresponding distance function. Let N be the unit normal vector field of M. We decompose the position vector x of M as follows
(4.1)x=xt+<x,N>N,
where xt is the tangential component, then the gradient of 〈x,x〉=r2 on M is given by ∇r2=2xt. By taking covariant derivative of (4.1) and using the Gauss and Weingarten formula, we have
(4.2)∇X∇r2=∇X2xt=2X+2〈x,N〉SX,
for every tangent vector field X∈χ(M). Therefore by using (2.14) and (2.15) we have
(4.3)L1r2=tr(P1∘∇2r2)=4H1+4H2〈x,N〉.
If M has constant Gaussian curvature H2, it is nonzero. In fact, if H2=0, then according to Theorem 3.1, M must be of L1-1-type. So we assume that M has nonzero constant Gaussian curvature H2 and is of L1-3-type. By computing L1ix, i=1,2,3, one finds that
(4.4)L1x=2H2N,L12x=-4H1H22N,(4.5)L13x=8H22(S∘P1)∇H1-4H22(L1H1-2H12H2)N.
Since M is of L1-3-type, we have
(4.6)L13x=(κ1+κ2+κ3)L12x-(κ1κ2+κ2κ3+κ1κ3)L1x+κ1κ2κ3x,
where x=x1+x2+x3, L1xi=κixi, 1≤i≤3. Taking into account (4.4), and comparing the tangential and normal components of L13x in (4.5), (4.6) we obtain the following useful equations:
(4.7)(S∘P1)∇H1=κ1κ2κ38H22xt,(4.8)L1H1=2H2H12+H1(κ1+κ2+κ3)+12H2(κ1κ2+κ2κ3+κ1κ3)-κ1κ2κ34H22〈x,N〉.
For brevity we set κ=κ1κ2κ3/8H22. We distinguish the following two cases.
Case  I
(
κ
=
0
). That is M is of L1-null-3-type, then, by (4.7), (S∘P1)∇H1=0. By the Cayley-Hamilton theorem, we have P2=0, and from the inductive definition of P2 we get that
(4.9)S∘P1=H2I.
So
(4.10)(S∘P1)∇H1=H2∇H1=0.
Since H2≠0 we conclude that H1 is constant, so M is isoparametric, then by Theorem 3.1, M has exactly two constant principal curvatures, therefore from Lemma 2.B of [17] we know that one of the principal curvatures has to be zero, so H2=0, this is a contradiction.
Case II
(
κ
≠
0
). Then from (4.7) and (4.9) we obtain that
(4.11)H2∇H1=κxt,
so we have ∇(H2H1-(κ/2)r2)=0; hence, H2H1-(κ/2)r2 is constant, therefore
(4.12)L1H1=κ2H2L1r2,
from which, by using (4.3), one finds that
(4.13)L1H1=2κH2H1+2κ〈x,N〉,
from which, by using (4.8) we obtain
(4.14)α∇H1=4κ∇〈x,N〉,
where we set
(4.15)α=(-2κH2H1+4H2H1+(κ1+κ2+κ3)).
Since ∇〈x,N〉=-Sxt, by (4.11), (4.14) we have Sxt=-(α/4H2)xt. If xt is identically zero on an open subset V⊆M, then V should be a sphere, thus of L1-1-type, a contradiction. Hence xt is a principal direction with corresponding principal curvature -α/4H2. Therefore, the other principal curvature is 2H1+α/4H2, thus the Gaussian curvature has to be H2=-(α/4H2)(2H1+α/4H2) or
(4.16)16H23+8H1H2α+α2=0.
Substituting α from (4.15) in (4.16), we get the following polynomial equation for H1:
(4.17)(-48H22)H12+(24κ-12H2(κ1+κ2+κ3))H1+(-4κ2H22+2κH2(κ1+κ2+κ3)-16H23)=0.
This shows that H1 is constant, a contradiction. So H2 is nonconstant.We use the following theorem from [10] to prove of Theorem 4.3.Theorem 4.2 (see [10]).
LetM be an n-dimensional hypersurface in a Euclidean space ℝn+1, with constant mth mean curvature Hm and two distinct principal curvatures. If the multiplicities are greater than one, then M is locally isometric to ℝk×Sn-k(c), 2≤k≤n-2.The next theorem will generalize the Ferrández and Lucas result in [6].Theorem 4.3.
There is no hypersurface ofLk-null-3-type in ℝn+1 with constant Hk+1 and at most two distinct principal curvatures.Proof.
Letx:Mn→ℝn+1 be the position vector of an Lk-null-3-type hypersurface M, and let N be the unit normal vector field of M. If M has constant Hk+1, it is nonzero. In fact, if Hk+1=0, then according to Theorem 3.1, M must be of Lk-1-type. So, we assume that M has nonzero constant Hk+1 and is of Lk-null-3-type.
By computingLkix, i=1,2,3 one finds that
(4.18)Lkx=ckHk+1N,Lk2x=-ck(nk+1)Hk+1(nH1Hk+1-(n-k-1)Hk+2)N,(4.19)Lk3x=2ck(nk+1)Hk+1(S∘Pk)∇(nH1Hk+1-(n-k-1)Hk+2)-ck(nk+1)Hk+1(Lk(nH1Hk+1-(n-k-1)Hk+2)-(nH1Hk+1-(n-k-1)Hk+2)2)N.
Since M is of Lk-null-3-type, we have
(4.20)Lk3x=(κ1+κ2+κ3)Lk2x-(κ1κ2+κ2κ3+κ1κ3)Lkx,
where x=x1+x2+x3, Lkxi=κixi, 1≤i≤3.
We prove the theorem in three steps.
Step I
(
k
=
n
-
1
). From (4.18) and comparing the tangential and normal components of Ln-13x in (4.19), (4.20) and by the Cayley-Hamilton theorem we obtain the following useful equation:
(4.21)(S∘Pn-1)∇H1=Hn∇H1=0.
Since Hn≠0, we conclude that H1 is constant; Therefore, M has at most two constant principal curvatures. From [16] and Theorem 3.1, M is an open piece of ℝp×Sn-p for some p≥1, so Gauss-Kronecker curvature of M is zero, therefore M is of Ln-1-null-1-type, which is a contradiction.
Step II
(
k
≠
n
-
1
). M has at most two distinct principal curvatures with multiplicities greater than one.
The conclusion is directly obtained from Theorem4.2 and Proposition 2.2.
Step III
(
k
≠
n
-
1
). M has two distinct principal curvatures, one of them is simple.
By (4.18) and comparing the tangential and normal components of Lk3x in (4.19), (4.20) and using the definition of Pk+1, we obtain that
(4.22)S∘Pk∇(nH1Hk+1-(n-k-1)Hk+2)=(nk+1)Hk+1∇(nH1Hk+1-(n-k-1)Hk+2)-Pk+1∇(nH1Hk+1-(n-k-1)Hk+2)=0.
It gives
(4.23)Pk+1∇(nH1Hk+1-(n-k-1)Hk+2)=(nk+1)Hk+1∇(nH1Hk+1-(n-k-1)Hk+2).
Consider {e1,…,en} to be a local orthonormal frame of principal directions of S on M such that Sei=λiei for every i=1,…,n. We assume that
(4.24)λ1=λ2=⋯=λn-1=λ,λn=μ.
Therefore from (2.13) we have
(4.25)Pk+1ei=μi,k+1ei,
with
(4.26)μi,k+1=∑i1<⋯<ik+1,ij≠i‍λi1⋯λik+1.
So we get(4.27)μ1,k+1=⋯=μn-1,k+1=(n-2k+1)λk+1+(n-2k)λkμ,μn,k+1=(n-1k+1)λk+1.
We obtain from (2.10) that
(4.28)(nk+1)Hk+1=(n-1k+1)λk+1+(n-1k)λkμ.
We can write
(4.29)∇(nH1Hk+1-(n-k-1)Hk+2)=∑i=1n‍〈∇(nH1Hk+1-(n-k-1)Hk+2),ei〉ei,
we see that (4.29) is equivalent to
(4.30)〈∇(nH1Hk+1-(n-k-1)Hk+2),ei〉(μi,k+1-(nk+1)Hk+1)=0,
for every i=1,…,n.
If(nH1Hk+1-(n-k-1)Hk+2) is constant, by the fact that M has two principal curvature and Hk+1, (nH1Hk+1-(n-k-1)Hk+2) are constant, we conclude that M is isoparametric, by using [16] and Proposition 2.2, this is a contradiction. Therefore, for every i such that 〈∇(nH1Hk+1-(n-k-1)Hk+2),ei〉≠0 we get
(4.31)μi,k+1=(nk+1)Hk+1.
So by using (4.27) and (4.28), we conclude that M is isoparametric and this is a contradiction.From Theorem4.3, we can easily obtain the following corollary.Corollary 4.4.
There is noLk-null-3-type conformally flat Euclidean hypersurface with constant Hk+1.



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*Source: 591296-2012-09-06.xml*