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ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 591296, 23 pages
http://dx.doi.org/10.5402/2012/591296
Research Article

On Some πΏπ‘˜-Finite-Type Euclidean Hypersurfaces

Department of Pure Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran

Received 24 May 2012; Accepted 26 July 2012

Academic Editors: I.Β Biswas, A.Β Ferrandez, and G.Β Martin

Copyright Β© 2012 Akram Mohammadpouri and S. M. B. Kashani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

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 β€œdegree” 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 π‘₯βˆΆπ‘€π‘›β†’β„π‘›+π‘˜ is said to be of finite type if π‘₯ has a finite decomposition as π‘₯βˆ’π‘₯0=βˆ‘π‘π‘–=1π‘₯𝑖, for some positive integer 𝑝<+∞ and Ξ”π‘₯𝑖=πœ…π‘–π‘₯𝑖, πœ…π‘–βˆˆβ„,1≀𝑖≀𝑝 where π‘₯0 is constant, and π‘₯𝑖, 1≀𝑖≀𝑝, are nonconstant smooth maps, and Ξ” is the Laplace operator of 𝑀. If all πœ…π‘–'s are mutually different, then 𝑀𝑛 is said to be of 𝑝-type. If, in particular, one of πœ…π‘–'s is zero, then 𝑀 is said to be of null 𝑝-type.

As is well known, the Laplace operator of a hypersurface 𝑀 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 Ξ” can be seen as the first one of a sequence of 𝑛 operators 𝐿0=Ξ”, 𝐿1,…,πΏπ‘›βˆ’1, where πΏπ‘˜ stands for the linearized operator of the first variation of the (π‘˜+1)th mean curvature arising from normal variations of the hypersurface (see [3, 4]). These operators are given by πΏπ‘˜(𝑓)=tr(π‘ƒπ‘˜βˆ˜βˆ‡2𝑓) for any π‘“βˆˆπΆβˆž(𝑀), where π‘ƒπ‘˜ denotes the π‘˜th Newton transformation associated to the second fundamental from of the hypersurface and βˆ‡2𝑓 is the hessian of 𝑓 (see the next section for details).

In contrast to the operator Ξ”, which is elliptic, in general the operators πΏπ‘˜ 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 Ξ” by πΏπ‘˜ 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 AlΓ­as, introduced such hypersurfaces and called them πΏπ‘˜-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 πΏπ‘˜-finite-type hypersurfaces in the Euclidean space. The structure of the paper is as follows.

In Section 2, we give preliminaries.

In Section 3, we classify πΏπ‘˜-1-type Euclidean hypersurfaces and πΏπ‘˜-null-2-type Euclidean hypersurfaces with at most two distinct principal curvatures. We follow the work of FerrΓ‘ndez 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 πΏπ‘˜-null-2-type, (π‘˜β‰ π‘›βˆ’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 πΏπ‘˜-null-2-type for 𝑛>3. We also prove that there is no πΏπ‘›βˆ’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].

Section 4 is about πΏπ‘˜-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 FerrΓ‘ndez 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 πΏπ‘˜-null-3-type with constant π»π‘˜+1 and at most two distinct principal curvatures.

Our main results are the following theorems.

Theorem 1.1. π‘˜βˆ’minimal Euclidean hypersurfaces and open parts of hyperspheres are the only πΏπ‘˜-1-type hypersurfaces in ℝ𝑛+1.

Theorem 1.2. There is no πΏπ‘›βˆ’1-null-2-type hypersurface in the Euclidean space ℝ𝑛+1, with at most two distinct principal curvatures.

Theorem 1.3. Let π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 be an isometrically immersed Euclidean hypersurface with at most two distinct principal curvatures and multiplicities are greater than one. Then 𝑀𝑛 is of πΏπ‘˜-null-2-type (π‘˜β‰ π‘›βˆ’1), if only if 𝑀 is isoparametric, so locally isometric to π‘†π‘šΓ—β„π‘›βˆ’π‘š, π‘šβ‰₯π‘˜+1.

Theorem 1.4. Let π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 be an isometrically immersed Euclidean hypersurface with at most two distinct principal curvatures, one of them is simple. Then 𝑀𝑛 is of πΏπ‘˜-null-2-type (π‘˜β‰ π‘›βˆ’1) if only if 𝑀 is isoparametric, so locally isometric to β„Γ—π‘†π‘›βˆ’1(π‘Ÿ) for 0β‰€π‘˜<π‘›βˆ’1 or β„π‘›βˆ’1×𝑆1(π‘Ÿ) for π‘˜=0.

In Theorems 1.3 and 1.4, if we assume that 𝑀 is complete, then 𝑀 is globally isometric to a generalized cylinder.

Theorem 1.5. There is no 𝐿1-3-type surface in ℝ3 with constant Gaussian curvature.

Theorem 1.6. There is no hypersurface of πΏπ‘˜-null-3-type in ℝ𝑛+1 with constant π»π‘˜+1 and at most two distinct principal curvatures.

2. Preliminaries

In this section we recall some notions and prerequisites about (πΏπ‘˜-finite type) hypersurfaces of the Euclidian space from [9, 10].

Consider an isometrically immersed hypersurface π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 in the Euclidean space. We choose a local orthonormal frame {𝑒𝐴}1≀𝐴≀𝑛+1 in ℝ𝑛+1, with dual coframe {πœ”π΄}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: 1≀𝐴,𝐡,𝐢,…,≀𝑛+1;1≀𝑖,𝑗,π‘˜,…,≀𝑛.(2.1) Then the structure equations of ℝ𝑛+1 are given by π‘‘πœ”π΄=𝑛+1𝐡=1πœ”π΄π΅βˆ§πœ”π΅,πœ”π΄π΅+πœ”π΅π΄=0,π‘‘πœ”π΄π΅=𝑛+1𝐢=1πœ”π΄πΆβˆ§πœ”πΆπ΅.(2.2) When restricted to 𝑀, we have πœ”π‘›+1=0 and 0=π‘‘πœ”π‘›+1=𝑛𝑖=1πœ”π‘›+1π‘–βˆ§πœ”π‘–.(2.3) By Cartan's lemma, there exist functions β„Žπ‘–π‘— such that πœ”π‘›+1𝑖=𝑛𝑗=1β„Žπ‘–π‘—πœ”π‘—,β„Žπ‘–π‘—=β„Žπ‘—π‘–.(2.4) This gives the second fundamental form of 𝑀, as βˆ‘π΅=𝑖,π‘—β„Žπ‘–π‘—πœ”π‘–πœ”π‘—π‘’π‘›+1. The mean curvature 𝐻 is defined by βˆ‘π»=(1/𝑛)π‘–β„Žπ‘–π‘–. From (2.2)–(2.4) we obtain the structure equations of 𝑀, (see [10]): π‘‘πœ”π‘–=𝑛𝑗=1πœ”π‘–π‘—βˆ§πœ”π‘—,πœ”π‘–π‘—+πœ”π‘—π‘–=0,π‘‘πœ”π‘–π‘—=π‘›ξ“π‘˜=1πœ”π‘–π‘˜βˆ§πœ”π‘˜π‘—βˆ’12π‘›ξ“π‘˜,𝑙=1π‘…π‘–π‘—π‘˜π‘™πœ”π‘˜βˆ§πœ”π‘™,(2.5) and the Gauss equations π‘…π‘–π‘—π‘˜π‘™=ξ€·β„Žπ‘–π‘˜β„Žπ‘—π‘™βˆ’β„Žπ‘–π‘™β„Žπ‘—π‘˜ξ€Έ,(2.6) where π‘…π‘–π‘—π‘˜π‘™ denotes the components of the Riemannian curvature tensor of 𝑀.

Let β„Žπ‘–π‘—π‘˜ denote the covariant derivative of β„Žπ‘–π‘—. We have ξ“π‘˜β„Žπ‘–π‘—π‘˜πœ”π‘˜=π‘‘β„Žπ‘–π‘—+ξ“π‘˜β„Žπ‘˜π‘—πœ”π‘˜π‘–+ξ“π‘˜β„Žπ‘–π‘˜πœ”π‘˜π‘—.(2.7) Thus, by exterior differentiation of (2.4), we obtain the Codazzi equation β„Žπ‘–π‘—π‘˜=β„Žπ‘–π‘˜π‘—.(2.8) We choose 𝑒1,…,𝑒𝑛 such that β„Žπ‘–π‘—=πœ†π‘–π›Ώπ‘–π‘—.(2.9) Let π»π‘š be π‘šth mean curvature of 𝑀, then we have βŽ›βŽœβŽœβŽπ‘›π‘šβŽžβŽŸβŽŸβŽ π»π‘š=1≀𝑖1<𝑖2<β‹―<π‘–π‘šβ‰€π‘›πœ†π‘–1β‹―πœ†π‘–π‘š.(2.10) And 𝐻𝑛=πœ†1β‹―πœ†π‘› is called the Gauss-Kronecker curvature of 𝑀. A hypersurface with zero (π‘˜+1)th mean curvature in ℝ𝑛+1 is called π‘˜-minimal. To get more information about π‘˜-minimal Euclidean hypersurfaces, the reader is referred to [11, 12].

The classical Newton transformations π‘ƒπ‘˜βˆΆπœ’(𝑀)β†’πœ’(𝑀) are defined inductively by the shape operator 𝑆 as 𝑃0=𝐼,π‘ƒπ‘˜=βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π»π‘˜πΌβˆ’π‘†βˆ˜π‘ƒπ‘˜βˆ’1,(2.11) for every π‘˜=1,…,𝑛, where 𝐼 denotes the identity transformation in πœ’(𝑀). Equivalently, π‘ƒπ‘˜=π‘˜ξ“π‘—=0(βˆ’1)π‘—βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜βˆ’π‘—π‘˜βˆ’π‘—π‘†π‘—.(2.12) Note that by the Cayley-Hamilton theorem stating that any operator is annihilated by its characteristic polynomial, we have 𝑃𝑛=0.

Since each π‘ƒπ‘˜(𝑝) is also a self-adjoint linear operator on each tangent plane 𝑇𝑝𝑀 which commutes with 𝑆(𝑝). Indeed, 𝑆(𝑝) and π‘ƒπ‘˜(𝑝) can be simultaneously diagonalized: if {𝑒1,…,𝑒𝑛} are the eigenvectors of 𝑆(𝑝) corresponding to the eigenvalues πœ†1(𝑝),…,πœ†π‘›(𝑝), respectively, then they are also the eigenvectors of π‘ƒπ‘˜(𝑝) with corresponding eigenvalues given by πœ‡π‘–,π‘˜(𝑝)=𝑖1<β‹―<π‘–π‘˜,π‘–π‘—β‰ π‘–πœ†π‘–1(𝑝)β‹―πœ†π‘–π‘˜(𝑝),(2.13) for every 1≀𝑖≀𝑛. We have the following formulae for the Newton transformations, [9]: 𝑃trπ‘˜ξ€Έ=π‘π‘˜π»π‘˜,(2.14)ξ€·trπ‘†βˆ˜π‘ƒπ‘˜ξ€Έ=π‘π‘˜π»π‘˜+1,(2.15)𝑆tr2βˆ˜π‘ƒπ‘˜ξ€Έ=βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ ξ€·π‘˜+1𝑛𝐻1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έ,(2.16) where π‘π‘˜βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ .=(π‘›βˆ’π‘˜)=(π‘˜+1)π‘˜+1(2.17)

Associated to each Newton transformation π‘ƒπ‘˜, we consider the second-order linear differential operator πΏπ‘˜βˆΆπΆβˆž(𝑀)β†’πΆβˆž(𝑀) given by πΏπ‘˜ξ€·π‘ƒ(𝑓)=trπ‘˜βˆ˜βˆ‡2𝑓.(2.18) Here βˆ‡2π‘“βˆΆπœ’(𝑀)β†’πœ’(𝑀) denotes the self-adjoint linear operator metrically equivalent to the Hessian of 𝑓 and is given by ξ«βˆ‡2𝑓(𝑋),π‘Œ=βŸ¨βˆ‡π‘‹(βˆ‡π‘“),π‘ŒβŸ©,(2.19) where 𝑋,π‘Œβˆˆπœ’(𝑀), βˆ‡π‘“ is the gradient of 𝑓, and βˆ‡ is the Levi-Civita connections on 𝑀.

Now we recall the definition of an πΏπ‘˜-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 hypersurface π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 is said to be of πΏπ‘˜-finite type if π‘₯ has a finite decomposition βˆ‘π‘₯=π‘šπ‘–=0π‘₯𝑖, for some positive integer π‘š satisfying the condition that πΏπ‘˜π‘₯𝑖=πœ…π‘–π‘₯𝑖, πœ…π‘–βˆˆβ„, 1β‰€π‘–β‰€π‘š, where π‘₯π‘–βˆΆπ‘€π‘›β†’β„π‘›+1 are smooth maps, 1β‰€π‘–β‰€π‘š, and π‘₯0 is constant. If all πœ…π‘–'s are mutually different, 𝑀𝑛 is said to be of πΏπ‘˜-π‘š-type. An πΏπ‘˜-π‘š-type hypersurface is said to be null if some πœ…π‘–;1β‰€π‘–β‰€π‘š, is zero. The polynomial βˆπ‘(𝑑)=π‘šπ‘–=1(π‘‘βˆ’πœ…π‘–) is called the minimal polynomial of 𝑀 for πΏπ‘˜.

We should mention that similar to Proposition 1 of [13], if 𝑀 is of πΏπ‘˜-finite type, then 𝑝(πΏπ‘˜)(π‘₯βˆ’π‘₯0)=0.

Proposition 2.2 (see [5]). If the isometrically immersed hypersurface π‘€π‘›βŠ‚β„π‘›+1 is a generalized cylinder π‘†π‘š(π‘Ÿ)Γ—β„π‘›βˆ’π‘š, then 𝑀 is of πΏπ‘˜-null-1-type, if π‘˜+1>π‘š, and it is of πΏπ‘˜-null-2-type, if π‘˜+1β‰€π‘š.

3. πΏπ‘˜-2-Type Hypersurfaces

In this section we would like to follow [6] and consider πΏπ‘˜-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 πΏπ‘˜-1 and 2-type Euclidean hypersurfaces. In Theorem 3.5, we prove that there is no πΏπ‘›βˆ’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, FerrΓ‘ndez 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 πΏπ‘˜-null-2-type (π‘˜β‰ π‘›βˆ’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 π‘˜-minimal hypersurfaces and open parts of hyperspheres are the only πΏπ‘˜-1-type Euclidean hypersurfaces.

Theorem 3.1. π‘˜-minimal Euclidean hypersurfaces and open parts of hyperspheres are the only πΏπ‘˜-1-type hypersurfaces in ℝ𝑛+1.

Proof. In Theorem 1 of [9] AlΓ­as and GΓΌrbΓΌz have classified hypersurfaces in ℝ𝑛+1 satisfying the general condition πΏπ‘˜π‘₯=𝐴π‘₯+𝑏, where π΄βˆˆβ„(𝑛+1)Γ—(𝑛+1) is a matrix and π‘βˆˆβ„π‘›+1. Section 3 of [9] and Proposition 2.2 show that π‘˜-minimal hypersurfaces and open parts of hyperspheres are the only hypersurfaces satisfying the condition πΏπ‘˜π‘₯=𝑐𝐼π‘₯+𝑏, for some real constant 𝑐.

Now we can state the next theorem.

Theorem 3.2. Let π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 be an isometrically immersed hypersurface in the Euclidean space. If it is of πΏπ‘˜-1-type or πΏπ‘˜-2-type for some 0β‰€π‘˜β‰€π‘›βˆ’1 and the (π‘˜+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 πΏπ‘˜-null-2-type.

Proof. The case π‘˜=0 was proved by Chen and Lue [15, Theorem 1] and FerrΓ‘ndez and Lucas [6, Corollary 3.2], so that we may consider the case π‘˜β‰₯1. Here, we will follow the techniques introduced by Chen and Lue [15, Theorem 1] for our context. According to Theorem 3.1, π‘˜-minimal hypersurfaces and open parts of hyperspheres are the only πΏπ‘˜-1-type hypersurfaces. If 𝑀 is of πΏπ‘˜-2-type, the position vector π‘₯ of 𝑀 in ℝ𝑛+1 has the following spectral decomposition: π‘₯βˆ’π‘=π‘₯1+π‘₯2,πΏπ‘˜π‘₯1=πœ…1π‘₯1,πΏπ‘˜π‘₯2=πœ…2π‘₯2,forsomeπ‘βˆˆβ„π‘›+1,(3.1) so 𝐿2π‘˜ξ€·πœ…π‘₯=1+πœ…2ξ€ΈπΏπ‘˜π‘₯βˆ’πœ…1πœ…2(π‘₯βˆ’π‘).(3.2) We also have πΏπ‘˜π‘₯=π‘π‘˜π»π‘˜+1𝑁,(3.3)𝐿2π‘˜π‘₯=βˆ’π‘π‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1𝑛𝐻1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2𝑁.(3.4)
The formula (3.4) holds since π»π‘˜+1 is a nonzero constant, see [9]. Therefore, by using (3.2), (3.3), and (3.4), we obtain βˆ’π‘π‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1𝑛𝐻1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2𝑁=π‘π‘˜ξ€·πœ…1+πœ…2ξ€Έπ»π‘˜+1π‘βˆ’πœ…1πœ…2(π‘₯βˆ’π‘).(3.5)
From (3.5) we have either πœ…1πœ…2=0 or π‘₯βˆ’π‘ is normal to 𝑀 at every point of 𝑀. If πœ…1πœ…2=0, then 𝑀 is of πΏπ‘˜-null-2-type. If π‘₯βˆ’π‘ is normal to 𝑀, then ⟨π‘₯βˆ’π‘,π‘₯βˆ’π‘βŸ© is a positive constant. In this case, 𝑀 is an open piece of 𝑆𝑛 centered at 𝑐; therefore, 𝑀 is of πΏπ‘˜-1-type, which is not.

As a corollary we get the following result.

Corollary 3.3. If 𝑀𝑛 is an πΏπ‘˜-2-type Euclidean hypersurface with constant (π‘˜+1)th mean curvature, then 𝑀𝑛 is of null πΏπ‘˜-2-type, in particular, if 𝑛=π‘˜+1 then the mean curvature is a nonzero constant.

Proof. By Theorem 3.2, 𝑀 is null, and we have the following spectral decomposition for the position vector π‘₯ of 𝑀 in ℝ𝑛+1: π‘₯=𝑐+π‘₯1+π‘₯2,πΏπ‘˜π‘₯1=0,πΏπ‘˜π‘₯2=πœ…2π‘₯2.(3.6) Since 𝑛=π‘˜+1 and 𝑀 is null, from formula (3.5) we have βˆ’(π‘˜+1)𝐻1𝐻2π‘˜+1𝑁=πœ…2π»π‘˜+1𝑁.(3.7) From (3.7) we obtain πœ…2=βˆ’(π‘˜+1)𝐻1π»π‘˜+1. Since πœ…2 and π»π‘˜+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. Let π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 be an isometrically immersed Euclidean hypersurface which is of πΏπ‘›βˆ’1-null-2-type, then the Gauss-Kronecker curvature of 𝑀 is nonzero and constant.

Proof. Let us consider the open set 𝒰={π‘βˆˆπ‘€βˆΆβˆ‡π»2𝑛(𝑝)β‰ 0}, our objective is to show that 𝒰 is empty. From [9], we have 𝐿2π‘›βˆ’1π‘₯=βˆ’π‘π‘›βˆ’1π»π‘›βˆ‡π»π‘›βˆ’2π‘π‘›βˆ’1ξ€·π‘†βˆ˜π‘ƒπ‘›βˆ’1ξ€Έβˆ‡π»π‘›βˆ’π‘π‘›βˆ’1𝑛𝐻1𝐻2π‘›βˆ’πΏπ‘›βˆ’1𝐻𝑛𝑁.(3.8) From this relation and (3.2) we obtain that βˆ’π‘π‘›βˆ’1π»π‘›βˆ‡π»π‘›βˆ’2π‘π‘›βˆ’1ξ€·π‘†βˆ˜π‘ƒπ‘›βˆ’1ξ€Έβˆ‡π»π‘›βˆ’π‘π‘›βˆ’1𝑛𝐻1𝐻2π‘›βˆ’πΏπ‘›βˆ’1𝐻𝑛𝑁=π‘π‘›βˆ’1πœ…2𝐻𝑛𝑁.(3.9) Therefore on 𝒰 we get ξ€·π‘†βˆ˜π‘ƒπ‘›βˆ’1ξ€Έβˆ‡π»π‘›1=βˆ’2π»π‘›βˆ‡π»π‘›.(3.10) But by the Cayley-Hamilton theorem we have 𝑃𝑛=0, so π‘†βˆ˜π‘ƒπ‘›βˆ’1=𝐻𝑛𝐼,π‘†βˆ˜π‘ƒπ‘›βˆ’1ξ€Έβˆ‡π»π‘›=π»π‘›βˆ‡π»π‘›,(3.11) which jointly with (3.10) yields βˆ‡π»2𝑛=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 no πΏπ‘›βˆ’1-null-2-type hypersurface in ℝ𝑛+1, with at most two distinct principal curvatures.

Proof. If 𝑀 is an πΏπ‘›βˆ’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 πΏπ‘›βˆ’1-null-2-type, we conclude that 𝑀 has exactly two constant principal curvatures. From [16], 𝑀 is an open piece of β„π‘Γ—π‘†π‘›βˆ’π‘ for some 𝑝β‰₯1, so the Gauss-Kronecker curvature of 𝑀 is zero; therefore, 𝑀 is of πΏπ‘›βˆ’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 no 𝐿1-null-2-type surface in ℝ3.

By Theorem 3.5 and the following theorems from [17, 18], we can prove that there is no πΏπ‘›βˆ’1-null-2-type compact hypersurface in ℝ𝑛+1 with everywhere nonzero sectional curvature.

Theorem 3.7 (see [17]). Let π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 be an isometrically immersed compact hypersurface in the Euclidean space, then the following conditions are equivalent:(i)π»π‘Ÿ=constβ‰ 0 for some π‘Ÿβˆˆ{2,…,𝑛} and 𝐾β‰₯0,(ii)𝐻=const and 𝐾β‰₯0,(iii)𝑀 has parallel second fundamental form (i.e., 𝑀 is isoparametric with at most two principle curvatures),
where 𝐾 is the sectional curvature of 𝑀.

The next theorem shows that there is no compact isometrically immersed hypersurface 𝑀𝑛 in ℝ𝑛+1 with everywhere nonpositive sectional curvature.

Theorem 3.8 (see [18]). Let 𝑀 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 no πΏπ‘›βˆ’1-null-2-type compact hypersurface in the Euclidean space ℝ𝑛+1 with everywhere nonzero sectional curvature.

Proof. By Theorem 3.8, 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 that πΏπ‘˜-null-2-type Euclidean hypersurfaces (π‘˜β‰ π‘›βˆ’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). Let 𝑀 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. Let π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 be an isometrically immersed hypersurface with at most two distinct principal curvatures and multiplicities are greater than one. Then 𝑀𝑛 is of πΏπ‘˜-null-2-type (π‘˜β‰ π‘›βˆ’1) if only if 𝑀 is isoparametric so locally isometric to π‘†π‘šΓ—β„π‘›βˆ’π‘š, π‘šβ‰₯π‘˜+1.

Proof. Let 𝑀𝑛 is of πΏπ‘˜-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 πΏπ‘˜-1-type, so 𝑀 is not totally umbilical. Therefore 𝑀 has two distinct principal curvatures of multiplicities π‘ž and π‘›βˆ’π‘ž, (π‘ž,π‘›βˆ’π‘ž>1).
Let us consider the open set 𝒰=π‘βˆˆπ‘€βˆΆβˆ‡π»2π‘˜+1ξ€Ύ.(𝑝)β‰ 0(3.12) 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=πœ†π‘–π‘’π‘– for every 𝑖=1,…,𝑛. We assume that πœ†1=πœ†2=β‹―=πœ†π‘ž=πœ†,πœ†π‘ž+1=β‹―=πœ†π‘›=πœ‡.(3.13) Therefore from (2.13) we have π‘ƒπ‘˜+1𝑒𝑖=πœ‡π‘–,π‘˜+1𝑒𝑖,(3.14) with πœ‡π‘–,π‘˜+1=𝑖1<β‹―<π‘–π‘˜+1,π‘–π‘—β‰ π‘–πœ†π‘–1β‹―πœ†π‘–π‘˜+1.(3.15)
So we get πœ‡1,π‘˜+1=β‹―=πœ‡π‘ž,π‘˜+1=ξ“π‘ βŽ›βŽœβŽœβŽπ‘ βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π‘žβˆ’1π‘›βˆ’π‘žπ‘˜+1βˆ’π‘ π‘ πœ‡π‘˜+1βˆ’π‘ ,πœ‡π‘ž+1,π‘˜+1=β‹―=πœ‡π‘›,π‘˜+1=ξ“π‘ βŽ›βŽœβŽœβŽπ‘žπ‘ βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’π‘žβˆ’1π‘˜+1βˆ’π‘ π‘ πœ‡π‘˜+1βˆ’π‘ .(3.16) We obtain from (2.10) that βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1=ξ“π‘ βŽ›βŽœβŽœβŽπ‘žπ‘ βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’π‘žπ‘˜+1βˆ’π‘ π‘ πœ‡π‘˜+1βˆ’π‘ .(3.17) Since 𝑀 is of πΏπ‘˜-null-2-type, the position vector field π‘₯ satisfies the following equation for some constant π‘šβ‰ 0, 𝐿2π‘˜π‘₯=π‘šπΏπ‘˜π‘₯.(3.18) So by using the formulae of πΏπ‘˜π‘₯ and 𝐿2π‘˜π‘₯ from [9], we can write βˆ’π‘π‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1βˆ‡π»π‘˜+1βˆ’2π‘π‘˜ξ€·π‘†βˆ˜π‘ƒπ‘˜ξ€Έξ€·βˆ‡π»π‘˜+1ξ€Έβˆ’π‘π‘˜βŽ›βŽœβŽœβŽβŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1𝑛𝐻1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έβˆ’πΏπ‘˜π»π‘˜+1βŽžβŽŸβŽŸβŽ π‘=π‘šπ‘π‘˜π»π‘˜+1𝑁.(3.19) From (3.19) we get ξ€·π‘†βˆ˜π‘ƒπ‘˜ξ€Έξ€·βˆ‡π»2π‘˜+1ξ€Έ1=βˆ’2βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1βˆ‡π»2π‘˜+1.(3.20) Since π‘˜β‰ π‘›βˆ’1, it follows from the inductive definition of π‘ƒπ‘˜+1 that (3.20) is equivalent to π‘ƒπ‘˜+1ξ€·βˆ‡π»2π‘˜+1ξ€Έ=32βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1βˆ‡π»2π‘˜+1on𝒰.(3.21) Therefore, writing βˆ‡π»2π‘˜+1=𝑛𝑖=1ξ«βˆ‡π»2π‘˜+1,𝑒𝑖𝑒𝑖,(3.22) we see that (3.21) is equivalent to ξ«βˆ‡π»2π‘˜+1,π‘’π‘–ξ¬βŽ›βŽœβŽœβŽπœ‡π‘–,π‘˜+1βˆ’32βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1⎞⎟⎟⎠=0on𝒰(3.23) for every 𝑖=1,…,𝑛. So there is no loss of generality, assuming that πœ‡1,π‘˜+1=β‹―=πœ‡π‘ž,π‘˜+1=32βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1.(3.24)
Let us denote the integral submanifolds through π‘₯βˆˆπ’° corresponding to πœ† and πœ‡ by π’°π‘ž1(π‘₯) and 𝒰1π‘›βˆ’π‘ž(π‘₯), respectively. From Lemma 3.10, we know that πœ† is constant on π’°π‘ž1(π‘₯). Equations (3.16), (3.17), and (3.24) imply that πœ‡ is constant on π’°π‘ž1(π‘₯). Again by Lemma 3.10, we get that πœ‡ is constant on 𝒰1π‘›βˆ’π‘ž(π‘₯). It now follows from [20, page 182, I] that 𝒰 is locally isometric to the Riemannian product of the maximal integral manifolds π’°π‘ž1(π‘₯) and 𝒰1π‘›βˆ’π‘ž(π‘₯). Therefore, πœ‡ is constant on 𝒰. By the same assertion, we know that πœ† is constant on 𝒰, so π»π‘˜+1 is constant on 𝒰, which is a contradiction. Hence π»π‘˜+1 is constant and nonzero on 𝑀. From (3.19), we obtain that tr(𝑆2βˆ˜π‘ƒπ‘˜)=π‘š; therefore, tr(𝑆2βˆ˜π‘ƒπ‘˜) is constant. By the fact that 𝑀 has two principal curvatures and π»π‘˜+1, tr(𝑆2βˆ˜π‘ƒπ‘˜) 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(π‘Ÿ) and circular cylinder. On the other hand, since 𝑀𝑛 is of πΏπ‘˜-null-2-type, by using Proposition 2.2, we conclude that 𝑀𝑛 is locally isometric to π‘†π‘šΓ—β„π‘›βˆ’π‘š, π‘šβ‰₯π‘˜+1.

Theorem 3.12. Let π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 be an isometrically immersed Euclidean hypersurface with at most two distinct principal curvatures, one of them is simple. Then 𝑀𝑛 is of πΏπ‘˜-null-2-type, (π‘˜β‰ π‘›βˆ’1), if only if 𝑀 is isoparametric, so locally isometric to β„Γ—π‘†π‘›βˆ’1(π‘Ÿ)0β‰€π‘˜<π‘›βˆ’1 or β„π‘›βˆ’1×𝑆1(π‘Ÿ) for π‘˜=0.

Proof. Let 𝑀𝑛 be of πΏπ‘˜-null-2-type (π‘˜β‰ π‘›βˆ’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 πΏπ‘˜-1-type, so 𝑀 cannot be totally umbilic. Therefore suppose 𝑀 has two distinct principal curvatures of multiplicities 1 and π‘›βˆ’1.
Let us consider the open set. 𝒰=π‘βˆˆπ‘€βˆΆβˆ‡π»2π‘˜+1ξ€Ύ.(𝑝)β‰ 0(3.25)
We want to prove that 𝒰 is empty. If π’°β‰ βˆ…, then we express π»π‘˜+1 as a polynomial in πœ† (the nonsimple principal curvature of 𝑀) with constant coefficients, after that we express πœ† 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 πΏπ‘˜-null-2-type, we get that πœ† satisfies a polynomial with constant coefficients. So πœ† is constant; hence, π»π‘˜+1 is constant, a contradiction with π’°β‰ βˆ…. So 𝒰 is empty.
Here is the detailed treatment of the proof.
With the assumption that π’°β‰ βˆ…, consider {𝑒1,…,𝑒𝑛} to be a local orthonormal frame of principal directions of 𝑆 on 𝒰 such that 𝑆𝑒𝑖=πœ†π‘–π‘’π‘– for every 𝑖=1,…,𝑛. We assume πœ†1=πœ†2=β‹―=πœ†π‘›βˆ’1=πœ†,πœ†π‘›=πœ‡.(3.26) Therefore from (2.10) we have π‘ƒπ‘˜+1𝑒𝑖=πœ‡π‘–,π‘˜+1𝑒𝑖,(3.27) with πœ‡π‘–,π‘˜+1=𝑖1<β‹―<π‘–π‘˜+1,π‘–π‘—β‰ π‘–πœ†π‘–1β‹―πœ†π‘–π‘˜+1.(3.28) So we get πœ‡1,π‘˜+1=β‹―=πœ‡π‘›βˆ’1,π‘˜+1=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’2π‘˜+1π‘˜+1+βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’2π‘˜πœ‡πœ‡,𝑛,π‘˜+1=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’1π‘˜+1π‘˜+1.(3.29) We obtain from (2.13) that βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’1π‘˜+1π‘˜+1+βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’1π‘˜πœ‡.(3.30) Since 𝑀 is of πΏπ‘˜-null-2-type, the position vector field π‘₯ satisfies the following equation for some constant π‘šβ‰ 0, 𝐿2π‘˜π‘₯=π‘šπΏπ‘˜π‘₯.(3.31) So by using the formulae of πΏπ‘˜π‘₯ and 𝐿2π‘˜π‘₯ from [9], we can write βˆ’π‘π‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1βˆ‡π»π‘˜+1βˆ’2π‘π‘˜ξ€·π‘†βˆ˜π‘ƒπ‘˜ξ€Έξ€·βˆ‡π»π‘˜+1ξ€Έβˆ’π‘π‘˜βŽ›βŽœβŽœβŽβŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1𝑛𝐻1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έβˆ’πΏπ‘˜π»π‘˜+1βŽžβŽŸβŽŸβŽ π‘=π‘šπ‘π‘˜π»π‘˜+1𝑁.(3.32) From (3.32) we get ξ€·π‘†βˆ˜π‘ƒπ‘˜ξ€Έξ€·βˆ‡π»2π‘˜+1ξ€Έ1=βˆ’2βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1βˆ‡π»2π‘˜+1.(3.33) Since π‘˜β‰ π‘›βˆ’1, it follows from the inductive definition of π‘ƒπ‘˜+1 that (3.33) is equivalent to π‘ƒπ‘˜+1ξ€·βˆ‡π»2π‘˜+1ξ€Έ=32βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1βˆ‡π»2π‘˜+1on𝒰.(3.34) Therefore, by the formula βˆ‡π»2π‘˜+1=𝑛𝑖=1ξ«βˆ‡π»2π‘˜+1,𝑒𝑖𝑒𝑖,(3.35) we see that (3.34) is equivalent to ξ«βˆ‡π»2π‘˜+1,π‘’π‘–ξ¬βŽ›βŽœβŽœβŽπœ‡π‘–,π‘˜+1βˆ’32βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1⎞⎟⎟⎠=0on𝒰(3.36) for every 𝑖=1,…,𝑛. Hence, for every 𝑖 such that βŸ¨βˆ‡π»2π‘˜+1,π‘’π‘–βŸ©β‰ 0 on 𝒰 we get πœ‡π‘–,π‘˜+1=32βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1.(3.37) So for the expression βˆ‡π»2π‘˜+1 in (3.35) we consider two cases.
Case  1. βŸ¨βˆ‡π»2π‘˜+1,π‘’π‘›βŸ©β‰ 0, by (4.31), we obtain that π»π‘˜+1=23(π‘›βˆ’π‘˜βˆ’1)π‘›πœ†π‘˜+1.(3.38)Case  2. βŸ¨βˆ‡π»2π‘˜+1,π‘’π‘›βŸ©=0, so on 𝒰 we have βŸ¨βˆ‡π»2π‘˜+1,π‘’π‘—βŸ©β‰ 0 for some 𝑗=1,…,π‘›βˆ’1. By (4.31) and using the fact that πœ‡π‘–,π‘˜ are the eigenvalues of π‘ƒπ‘˜ and the formula of tr(π‘ƒπ‘˜+1), we obtain that π»π‘˜+1=(π‘›βˆ’π‘˜βˆ’1)πœ†π‘›(βˆ’(1/2)π‘›βˆ’π‘˜+1/2)π‘˜+1.(3.39) Both cases require the same calculation, so we consider just Case I.
By Lemma 3.10, let us denote the maximal integral submanifold through π‘₯βˆˆπ’°, corresponding to πœ† by 𝒰1π‘›βˆ’1(π‘₯). We write ξ“π‘‘πœ†=π‘–πœ†,π‘–πœ”π‘–ξ“π‘‘πœ‡=π‘—πœ‡,π‘—πœ”π‘—.(3.40) Then Lemma 3.10 implies that πœ†,1=β‹―=πœ†,π‘›βˆ’1=0. We can assume that πœ†>0 on 𝒰, then (3.30) and (3.38) yield πœ‡=π‘˜+1βˆ’π‘›3π‘˜+3πœ†.(3.41) By means of (2.7) and (2.9), we obtain that ξ“π‘˜β„Žπ‘–π‘—π‘˜πœ”π‘˜=π›Ώπ‘–π‘—π‘‘πœ†π‘—+ξ€·πœ†π‘–βˆ’πœ†π‘—ξ€Έπœ”π‘–π‘—.(3.42) We adopt the notational convention that 1β‰€π‘Ž,𝑏,𝑐,β€¦β‰€π‘›βˆ’1.
From (3.41) and (3.42), we have β„Žπ‘–π‘—π‘˜=0,ifβ„Žπ‘–β‰ π‘—,π‘Žπ‘Žπ‘=0,β„Žπ‘Žπ‘Žπ‘›=πœ†,𝑛,β„Žπ‘›π‘›π‘Ž=0,β„Žπ‘›π‘›π‘›=πœ‡,𝑛.(3.43) Combining this with (2.8) and the formula ξ“π‘–β„Žπ‘Žπ‘›π‘–πœ”π‘–=π‘‘β„Žπ‘Žπ‘›+ξ“π‘–β„Žπ‘–π‘›πœ”π‘–π‘Ž+ξ“π‘–β„Žπ‘Žπ‘–πœ”π‘–π‘›=(πœ†βˆ’πœ‡)πœ”π‘Žπ‘›,(3.44) we obtain from (3.41) πœ”π‘Žπ‘›=πœ†,π‘›πœ”πœ†βˆ’πœ‡π‘Ž=(3π‘˜+3)πœ†,π‘›πœ”(2π‘˜+2+𝑛)πœ†π‘Ž.(3.45) Therefore we have π‘‘πœ”π‘›=ξ“π‘Žπœ”π‘›π‘Žβˆ§πœ”π‘Ž=0.(3.46)
Notice that we may consider πœ† 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 πœ†. We may put πœ”π‘›=𝑑𝑠.
Thus, for πœ†=πœ†(𝑠), we have π‘‘πœ†=πœ†,𝑛𝑑𝑠,πœ†,𝑛=πœ†β€²(𝑠),(3.47) so from(3.45), we get πœ”π‘Žπ‘›=πœ†,π‘›πœ”πœ†βˆ’πœ‡π‘Ž=(3π‘˜+3)πœ†β€²(𝑠)πœ”(2π‘˜+2+𝑛)πœ†π‘Ž.(3.48) According to the structure equations of ℝ𝑛+1 and (3.48), we may compute π‘‘πœ”π‘Žπ‘›=π‘›βˆ’1𝑏=1πœ”π‘Žπ‘βˆ§πœ”π‘π‘›+πœ”π‘Žπ‘›+1βˆ§πœ”π‘›+1𝑛=ξ‚΅(3π‘˜+3)πœ†β€²ξ‚Ά(2π‘˜+2+𝑛)πœ†π‘›βˆ’1𝑏=1πœ”π‘Žπ‘βˆ§πœ”π‘βˆ’πœ†πœ‡πœ”π‘Žβˆ§π‘‘π‘ ,(3.49)π‘‘πœ”π‘Žπ‘›ξ‚»=𝑑(3π‘˜+3)πœ†β€²πœ”(2π‘˜+2+𝑛)πœ†π‘Žξ‚Ό=ξ‚΅(3π‘˜+3)πœ†β€²ξ‚Ά(2π‘˜+2+𝑛)πœ†π‘‘π‘ βˆ§πœ”π‘Ž+ξ‚΅(3π‘˜+3)πœ†β€²ξ‚Ά(2π‘˜+2+𝑛)πœ†π‘‘πœ”π‘Ž=ξƒ―ξ‚΅(3π‘˜+3)πœ†ξ…žξ‚Ά(2π‘˜+2+𝑛)πœ†ξ…ž+ξ‚΅(3π‘˜+3)πœ†β€²ξ‚Ά(2π‘˜+2+𝑛)πœ†2ξƒ°πœ”π‘Žξ‚΅βˆ§π‘‘π‘ +(3π‘˜+3)πœ†β€²ξ‚Ά(2π‘˜+2+𝑛)πœ†π‘›βˆ’1𝑏=1πœ”π‘Žπ‘βˆ§πœ”π‘.(3.50) Then we obtain from the two equalities above that ξ‚΅(3π‘˜+3)πœ†ξ…ž(ξ‚Ά2π‘˜+2+𝑛)πœ†ξ…žβˆ’ξ‚΅(3π‘˜+3)πœ†β€²ξ‚Ά(2π‘˜+2+𝑛)πœ†2βˆ’πœ†πœ‡=0.(3.51) Combining (3.51) with (3.41), we have ξ‚΅(3π‘˜+3)πœ†ξ…ž(ξ‚Ά2π‘˜+2+𝑛)πœ†ξ…žβˆ’ξ‚΅(3π‘˜+3)πœ†β€²ξ‚Ά(2π‘˜+2+𝑛)πœ†2βˆ’ξ‚€π‘˜+1βˆ’π‘›ξ‚πœ†3π‘˜+32=0.(3.52) Let us define a function 𝛽(𝑠), π‘ βˆˆ(βˆ’βˆž,+∞) by 𝛽=(1/πœ†)(3π‘˜+3)/(2π‘˜+2+𝑛), then (3.52) reduces to π›½ξ…žξ…ž=ξ‚€π‘›βˆ’π‘˜βˆ’1𝛽3π‘˜+3(βˆ’7π‘˜βˆ’7βˆ’2𝑛)/(3π‘˜+3).(3.53) Integrating (3.53), we obtain ξ€·π›½ξ…žξ€Έ2=ξ‚€π‘˜+1βˆ’π‘›ξ‚π›½2π‘˜+2+𝑛(βˆ’4π‘˜βˆ’4βˆ’2𝑛)/(3π‘˜+3)+𝐢,(3.54) where 𝐢 is the constant of integration.
Equation (3.54) is equivalent to (πœ†ξ…ž)2=ξ‚΅(𝑛+2π‘˜+2)(π‘˜+1βˆ’π‘›)(3π‘˜+3)2ξ‚Άπœ†(14π‘˜+4𝑛+14)/(2π‘˜+2+𝑛)ξ‚€+𝐢2+2π‘˜+𝑛3π‘˜+32πœ†(10π‘˜+10+2𝑛)/(2π‘˜+2+𝑛).(3.55) Now we use the definition of πΏπ‘˜π»π‘˜+1=tr(π‘ƒπ‘˜βˆ˜βˆ‡2π»π‘˜+1) to compute πΏπ‘˜π»π‘˜+1. So we need to compute βˆ‡π‘’π‘Žβˆ‡π»π‘˜+1, βˆ‡π‘’π‘›βˆ‡π»π‘˜+1, π‘ƒπ‘˜(π‘’π‘Ž), and π‘ƒπ‘˜(𝑒𝑛).
From (3.51) we have βˆ‡π»π‘˜+1=2(π‘˜+1)(π‘›βˆ’π‘˜βˆ’1)πœ†3π‘›π‘˜πœ†ξ…žπ‘’π‘›.(3.56) By using (3.48) and (3.56) we obtain βˆ‡π‘’π‘Žβˆ‡π»π‘˜+1=2(π‘˜+1)(π‘›βˆ’π‘˜βˆ’1)πœ†3π‘›π‘˜πœ†β€²βˆ‡π‘’π‘Žπ‘’π‘›=2(π‘˜+1)(π‘›βˆ’π‘˜βˆ’1)πœ†3π‘›π‘˜ξ“πœ†β€²π‘πœ”π‘›π‘ξ€·π‘’π‘Žξ€Έπ‘’π‘=βˆ’2(π‘›βˆ’π‘˜βˆ’1)(π‘˜+1)2πœ†π‘›(2π‘˜+2+𝑛)π‘˜βˆ’1πœ†β€²2π‘’π‘Žβˆ‡π‘’π‘›βˆ‡π»π‘˜+1=2(π‘˜+1)(π‘›βˆ’π‘˜βˆ’1)βˆ‡3π‘›π‘’π‘›πœ†π‘˜πœ†ξ…žπ‘’π‘›=2π‘˜(π‘˜+1)(π‘›βˆ’π‘˜βˆ’1)πœ†3π‘›π‘˜βˆ’1πœ†ξ…ž2𝑒𝑛+2(π‘˜+1)(π‘›βˆ’π‘˜βˆ’1)πœ†3π‘›π‘˜πœ†ξ…žξ…žπ‘’π‘›.(3.57) Now we compute π‘ƒπ‘˜(π‘’π‘Ž) and π‘ƒπ‘˜(𝑒𝑛): π‘ƒπ‘˜ξ€·π‘’π‘Žξ€Έ=πœ‡π‘Ž,π‘˜π‘’π‘Ž=βŽ›βŽœβŽœβŽξ“π‘–1<β‹―<π‘–π‘˜,π‘–π‘—β‰ π‘Žπœ†π‘–1β‹―πœ†π‘–π‘˜βŽžβŽŸβŽŸβŽ π‘’π‘Ž=βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘›βˆ’22π‘˜+3πœ†3π‘˜+3π‘˜π‘’π‘Ž,π‘ƒπ‘˜ξ€·π‘’π‘›ξ€Έ=βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’1π‘˜π‘’π‘›.(3.58) From (3.57) and (3.58), we get πΏπ‘˜π»π‘˜+1=π‘π‘˜π»π‘˜+1ξ‚΅(βˆ’2π‘˜βˆ’3)(π‘˜+1)(π‘›βˆ’π‘˜βˆ’1)πœ†π‘›(2π‘˜+2+𝑛)π‘˜βˆ’2πœ†β€²2+π‘˜(π‘˜+1)π‘›πœ†π‘˜βˆ’2πœ†β€²2+π‘˜+1π‘›πœ†π‘˜βˆ’1πœ†ξ…žξ…žξ‚Ά.(3.59) Since 𝑀𝑛 is of πΏπ‘˜-null-2-type, hence from (3.32), we get πΏπ‘˜π»π‘˜+1=π‘π‘˜π»π‘˜+1ξ€·ξ€·π‘†π‘šβˆ’tr2βˆ˜π‘ƒπ‘˜ξ€Έξ€Έ=π‘π‘˜π»π‘˜+1βŽ›βŽœβŽœβŽβŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘šβˆ’π‘›βˆ’12π‘›π‘˜+3π‘›βˆ’2π‘˜βˆ’2π‘˜2πœ†3π‘˜+3π‘˜+2⎞⎟⎟⎠.(3.60) Combining (3.58) and (3.60), we have πœ†πœ†ξ…žξ…ž+ξ‚΅(π‘˜+βˆ’2π‘˜βˆ’3)(π‘›βˆ’π‘˜βˆ’1)ξ‚Άπœ†2π‘˜+2+π‘›ξ…ž2+βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ π‘›ξ€·π‘›βˆ’12π‘›π‘˜+3π‘›βˆ’2π‘˜βˆ’2π‘˜2ξ€Έπœ†(π‘˜+1)(3π‘˜+3)4βˆ’π‘šπ‘›πœ†π‘˜+12βˆ’π‘˜=0.(3.61) Equation (3.52) is equivalent to πœ†πœ†ξ…žξ…ž=5π‘˜+5+𝑛2π‘˜+2+π‘›πœ†β€²2+(2π‘˜+2+𝑛)(π‘˜+1βˆ’π‘›)(3π‘˜+3)2πœ†4.(3.62) Thus, putting together (3.61) and (3.62) one has 4π‘˜2+12π‘˜βˆ’π‘˜π‘›βˆ’2𝑛+82π‘˜+2+π‘›πœ†β€²2+ξ€·(2π‘˜+2+𝑛)(π‘˜+1βˆ’π‘›)+3π‘˜π‘›βˆ’1𝑛2π‘›π‘˜+3π‘›βˆ’2π‘˜βˆ’2π‘˜2ξ€Έ(3π‘˜+3)2πœ†4βˆ’π‘šπ‘›πœ†π‘˜+12βˆ’π‘˜=0.(3.63) We deduce, using (3.55), (3.63), and (3.38), that π»π‘˜+1 is locally constant on 𝒰, which is a contradiction with the definition of 𝒰. Hence π»π‘˜+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 hypersurfaces 𝑀𝑛 of πΏπ‘˜-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. For 𝑛=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 π‘›βˆ’1 (see [21] for more details). This significant fact is crucial in our classification of πΏπ‘˜-null-2-type conformally flat Euclidean hypersurfaces 𝑀𝑛 for 𝑛>3.

As a simple consequence of Theorem 3.12, we obtain the following nice corollary which is an extension of the result of [22].

Corollary 3.14. Let 𝑀𝑛 be a conformally flat hypersurface of ℝ𝑛+1, 𝑛>3. Then 𝑀𝑛 is of πΏπ‘˜-null-2-type, if only if 𝑀 is locally isometric to the cylinder β„Γ—π‘†π‘›βˆ’1(π‘Ÿ)(0<π‘˜<π‘›βˆ’1).

4. πΏπ‘›βˆ’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 FerrΓ‘ndez 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 πΏπ‘˜-null-3-type with constant π»π‘˜+1 and at most two distinct principal curvatures.

Theorem 4.1. There is no 𝐿1-3-type surface in ℝ3 with constant Gaussian curvature.

Proof. Let π‘₯βˆΆπ‘€2→ℝ3 be the position vector of an 𝐿1-3-type surface 𝑀, and set |π‘₯|=π‘Ÿ for the corresponding distance function. Let 𝑁 be the unit normal vector field of 𝑀. We decompose the position vector π‘₯ of 𝑀 as follows π‘₯=π‘₯𝑑+<π‘₯,𝑁>𝑁,(4.1) where π‘₯𝑑 is the tangential component, then the gradient of ⟨π‘₯,π‘₯⟩=π‘Ÿ2 on 𝑀 is given by βˆ‡π‘Ÿ2=2π‘₯𝑑. By taking covariant derivative of (4.1) and using the Gauss and Weingarten formula, we have βˆ‡π‘‹βˆ‡π‘Ÿ2=βˆ‡π‘‹2π‘₯𝑑=2𝑋+2⟨π‘₯,π‘βŸ©π‘†π‘‹,(4.2) for every tangent vector field π‘‹βˆˆπœ’(𝑀). Therefore by using (2.14) and (2.15) we have 𝐿1π‘Ÿ2𝑃=tr1βˆ˜βˆ‡2π‘Ÿ2ξ€Έ=4𝐻1+4𝐻2⟨π‘₯,π‘βŸ©.(4.3) 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 𝐿𝑖1π‘₯, 𝑖=1,2,3, one finds that 𝐿1π‘₯=2𝐻2𝑁,𝐿21π‘₯=βˆ’4𝐻1𝐻22𝑁,(4.4)𝐿31π‘₯=8𝐻22ξ€·π‘†βˆ˜π‘ƒ1ξ€Έβˆ‡π»1βˆ’4𝐻22𝐿1𝐻1βˆ’2𝐻21𝐻2𝑁.(4.5) Since 𝑀 is of 𝐿1-3-type, we have 𝐿31ξ€·πœ…π‘₯=1+πœ…2+πœ…3𝐿21ξ€·πœ…π‘₯βˆ’1πœ…2+πœ…2πœ…3+πœ…1πœ…3𝐿1π‘₯+πœ…1πœ…2πœ…3π‘₯,(4.6) where π‘₯=π‘₯1+π‘₯2+π‘₯3, 𝐿1π‘₯𝑖=πœ…π‘–π‘₯𝑖, 1≀𝑖≀3. Taking into account (4.4), and comparing the tangential and normal components of 𝐿31π‘₯ in (4.5), (4.6) we obtain the following useful equations: ξ€·π‘†βˆ˜π‘ƒ1ξ€Έβˆ‡π»1=πœ…1πœ…2πœ…38𝐻22π‘₯𝑑,(4.7)𝐿1𝐻1=2𝐻2𝐻21+𝐻1ξ€·πœ…1+πœ…2+πœ…3ξ€Έ+12𝐻2ξ€·πœ…1πœ…2+πœ…2πœ…3+πœ…1πœ…3ξ€Έβˆ’πœ…1πœ…2πœ…34𝐻22⟨π‘₯,π‘βŸ©.(4.8) For brevity we set πœ…=πœ…1πœ…2πœ…3/8𝐻22. We distinguish the following two cases.
Case  I (πœ…=0). That is 𝑀 is of 𝐿1-null-3-type, then, by (4.7), (π‘†βˆ˜π‘ƒ1)βˆ‡π»1=0. By the Cayley-Hamilton theorem, we have 𝑃2=0, and from the inductive definition of 𝑃2 we get that π‘†βˆ˜π‘ƒ1=𝐻2𝐼.(4.9) So ξ€·π‘†βˆ˜π‘ƒ1ξ€Έβˆ‡π»1=𝐻2βˆ‡π»1=0.(4.10) Since 𝐻2β‰ 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 (πœ…β‰ 0). Then from (4.7) and (4.9) we obtain that 𝐻2βˆ‡π»1=πœ…π‘₯𝑑,(4.11) so we have βˆ‡(𝐻2𝐻1βˆ’(πœ…/2)π‘Ÿ2)=0; hence, 𝐻2𝐻1βˆ’(πœ…/2)π‘Ÿ2 is constant, therefore 𝐿1𝐻1=πœ…2𝐻2𝐿1π‘Ÿ2,(4.12) from which, by using (4.3), one finds that 𝐿1𝐻1=2πœ…π»2𝐻1+2πœ…βŸ¨π‘₯,π‘βŸ©,(4.13) from which, by using (4.8) we obtain π›Όβˆ‡π»1=4πœ…βˆ‡βŸ¨π‘₯,π‘βŸ©,(4.14) where we set ξ‚΅βˆ’π›Ό=2πœ…π»2𝐻1+4𝐻2𝐻1+ξ€·πœ…1+πœ…2+πœ…3ξ€Έξ‚Ά.(4.15) Since βˆ‡βŸ¨π‘₯,π‘βŸ©=βˆ’π‘†π‘₯𝑑, by (4.11), (4.14) we have 𝑆π‘₯𝑑=βˆ’(𝛼/4𝐻2)π‘₯𝑑. If π‘₯𝑑 is identically zero on an open subset π‘‰βŠ†π‘€, then 𝑉 should be a sphere, thus of 𝐿1-1-type, a contradiction. Hence π‘₯𝑑 is a principal direction with corresponding principal curvature βˆ’π›Ό/4𝐻2. Therefore, the other principal curvature is 2𝐻1+𝛼/4𝐻2, thus the Gaussian curvature has to be 𝐻2=βˆ’(𝛼/4𝐻2)(2𝐻1+𝛼/4𝐻2) or 16𝐻32+8𝐻1𝐻2𝛼+𝛼2=0.(4.16) Substituting 𝛼 from (4.15) in (4.16), we get the following polynomial equation for 𝐻1: ξ€·βˆ’48𝐻22𝐻21+ξ€·24πœ…βˆ’12𝐻2ξ€·πœ…1+πœ…2+πœ…3𝐻1+ξƒ©βˆ’4πœ…2𝐻22+2πœ…π»2ξ€·πœ…1+πœ…2+πœ…3ξ€Έβˆ’16𝐻32ξƒͺ=0.(4.17) 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]). Let 𝑀 be an 𝑛-dimensional hypersurface in a Euclidean space ℝ𝑛+1, with constant π‘šth mean curvature π»π‘š and two distinct principal curvatures. If the multiplicities are greater than one, then M is locally isometric to β„π‘˜Γ—π‘†π‘›βˆ’π‘˜(𝑐), 2β‰€π‘˜β‰€π‘›βˆ’2.

The next theorem will generalize the FerrΓ‘ndez and Lucas result in [6].

Theorem 4.3. There is no hypersurface of πΏπ‘˜-null-3-type in ℝ𝑛+1 with constant π»π‘˜+1 and at most two distinct principal curvatures.

Proof. Let π‘₯βˆΆπ‘€π‘›β†’β„π‘›+1 be the position vector of an πΏπ‘˜-null-3-type hypersurface 𝑀, and let 𝑁 be the unit normal vector field of 𝑀. If 𝑀 has constant π»π‘˜+1, it is nonzero. In fact, if π»π‘˜+1=0, then according to Theorem 3.1, 𝑀 must be of πΏπ‘˜-1-type. So, we assume that 𝑀 has nonzero constant π»π‘˜+1 and is of πΏπ‘˜-null-3-type.
By computing πΏπ‘–π‘˜π‘₯, 𝑖=1,2,3 one finds that πΏπ‘˜π‘₯=π‘π‘˜π»π‘˜+1𝑁,𝐿2π‘˜π‘₯=βˆ’π‘π‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1𝑛𝐻1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2𝑁,(4.18)𝐿3π‘˜π‘₯=2π‘π‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1ξ€·π‘†βˆ˜π‘ƒπ‘˜ξ€Έβˆ‡ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έβˆ’π‘π‘˜βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1ξ‚€πΏπ‘˜ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έβˆ’ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έ2𝑁.(4.19) Since 𝑀 is of πΏπ‘˜-null-3-type, we have 𝐿3π‘˜ξ€·πœ…π‘₯=1+πœ…2+πœ…3𝐿2π‘˜ξ€·πœ…π‘₯βˆ’1πœ…2+πœ…2πœ…3+πœ…1πœ…3ξ€ΈπΏπ‘˜π‘₯,(4.20) where π‘₯=π‘₯1+π‘₯2+π‘₯3, πΏπ‘˜π‘₯𝑖=πœ…π‘–π‘₯𝑖, 1≀𝑖≀3.
We prove the theorem in three steps.
Step I (π‘˜=π‘›βˆ’1). From (4.18) and comparing the tangential and normal components of 𝐿3π‘›βˆ’1π‘₯ in (4.19), (4.20) and by the Cayley-Hamilton theorem we obtain the following useful equation: ξ€·π‘†βˆ˜π‘ƒπ‘›βˆ’1ξ€Έβˆ‡π»1=π»π‘›βˆ‡π»1=0.(4.21) 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 β„π‘Γ—π‘†π‘›βˆ’π‘ for some 𝑝β‰₯1, so Gauss-Kronecker curvature of 𝑀 is zero, therefore 𝑀 is of πΏπ‘›βˆ’1-null-1-type, which is a contradiction.
Step II (π‘˜β‰ π‘›βˆ’1). 𝑀 has at most two distinct principal curvatures with multiplicities greater than one.
The conclusion is directly obtained from Theorem 4.2 and Proposition 2.2.
Step III (π‘˜β‰ π‘›βˆ’1). 𝑀 has two distinct principal curvatures, one of them is simple.
By (4.18) and comparing the tangential and normal components of 𝐿3π‘˜π‘₯ in (4.19), (4.20) and using the definition of π‘ƒπ‘˜+1, we obtain that π‘†βˆ˜π‘ƒπ‘˜βˆ‡ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έ=βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1βˆ‡ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έβˆ’π‘ƒπ‘˜+1βˆ‡ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έ=0.(4.22) It gives π‘ƒπ‘˜+1βˆ‡ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έ=βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1βˆ‡ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έ.(4.23) Consider {𝑒1,…,𝑒𝑛} to be a local orthonormal frame of principal directions of 𝑆 on 𝑀 such that 𝑆𝑒𝑖=πœ†π‘–π‘’π‘– for every 𝑖=1,…,𝑛. We assume that πœ†1=πœ†2=β‹―=πœ†π‘›βˆ’1=πœ†,πœ†π‘›=πœ‡.(4.24) Therefore from (2.13) we have π‘ƒπ‘˜+1𝑒𝑖=πœ‡π‘–,π‘˜+1𝑒𝑖,(4.25) with πœ‡π‘–,π‘˜+1=𝑖1<β‹―<π‘–π‘˜+1,π‘–π‘—β‰ π‘–πœ†π‘–1β‹―πœ†π‘–π‘˜+1.(4.26)
So we get πœ‡1,π‘˜+1=β‹―=πœ‡π‘›βˆ’1,π‘˜+1=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’2π‘˜+1π‘˜+1+βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’2π‘˜πœ‡πœ‡,𝑛,π‘˜+1=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’1π‘˜+1π‘˜+1.(4.27) We obtain from (2.10) that βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1=βŽ›βŽœβŽœβŽβŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’1π‘˜+1π‘˜+1+βŽ›βŽœβŽœβŽπ‘˜βŽžβŽŸβŽŸβŽ πœ†π‘›βˆ’1π‘˜πœ‡.(4.28) We can write βˆ‡ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έ=𝑛𝑖=1ξ«βˆ‡ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έ,𝑒𝑖𝑒𝑖,(4.29) we see that (4.29) is equivalent to ξ«βˆ‡ξ€·π‘›π»1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2ξ€Έ,π‘’π‘–ξ¬βŽ›βŽœβŽœβŽπœ‡π‘–,π‘˜+1βˆ’βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1⎞⎟⎟⎠=0,(4.30) for every 𝑖=1,…,𝑛.
If (𝑛𝐻1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2) is constant, by the fact that 𝑀 has two principal curvature and π»π‘˜+1, (𝑛𝐻1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+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 βŸ¨βˆ‡(𝑛𝐻1π»π‘˜+1βˆ’(π‘›βˆ’π‘˜βˆ’1)π»π‘˜+2),π‘’π‘–βŸ©β‰ 0 we get πœ‡π‘–,π‘˜+1=βŽ›βŽœβŽœβŽπ‘›βŽžβŽŸβŽŸβŽ π»π‘˜+1π‘˜+1.(4.31) So by using (4.27) and (4.28), we conclude that 𝑀 is isoparametric and this is a contradiction.

From Theorem 4.3, we can easily obtain the following corollary.

Corollary 4.4. There is no πΏπ‘˜-null-3-type conformally flat Euclidean hypersurface with constant π»π‘˜+1.

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