Rotational Hypersurfaces with L r-Pointwise 1-Type Gauss Map

abstract: In this paper, we study hypersurfaces in E which Gauss map G satisfies the equation LrG = f(G + C) for a smooth function f and a constant vector C, where Lr is the linearized operator of the (r + 1)th mean curvature of the hypersurface, i.e., Lr(f) = tr(Pr ◦ ∇f) for f ∈ C∞(M), where Pr is the rth Newton transformation, ∇f is the Hessian of f , LrG = (LrG1, . . . , LrGn+1), G = (G1, . . . , Gn+1). We show that a rational hypersurface of revolution in a Euclidean space E has Lr-pointwise 1-type Gauss map of the second kind if and only if it is a right n-cone.


Introduction
An isometrically immersed submanifold x : M n → E n+k is said to be of finite type if x has a finite decomposition as x − x 0 = p i=1 x i , for some positive integer p < +∞ such that ∆x i = λ i x i , λ i ∈ R, 1 ≤ i ≤ p, x 0 is constant, x i : M n → E n+k , 1 ≤ i ≤ p are non-constant smooth maps and ∆ is the Laplace operator of M , (see the excellent survey of B. Y. Chen [7]).In [8], this definition was similarly extended to differentiable maps, in particular, to Gauss map of submanifolds.The notion of finite type Gauss map is especially a useful tool in the study of submanifolds (cf.[2,3,4,5,9,10,12,13]).If an oriented submanifold M of a Euclidean space has 1-type Gauss map G, then G satisfies ∆G = λ(G + C) for a constant λ ∈ R and a constant vector C.In [8], Chen and Piccinni made a general study on compact submanifolds of Euclidean spaces with finite type Gauss map, and for hypersurfaces they proved that a compact hypersurface M of E n+1 has 1-type Gauss map G if and only if M is a hypersphere in E n+1 .
As is well known, the Laplace operator of a hypersurface M immersed into E n+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 ∆ can be seen as the first one of a sequence of n operators L 0 = ∆, L 1 , . . ., L n−1 , where L r stands for the linearized operator of the first variation of the (r + 1)th mean curvature arising from normal variations of the hypersurface (see [19]).These operators are given by L r (f ) = tr(P r • ∇ 2 f ) for any f ∈ C ∞ (M ), where P r denotes the rth Newton transformation associated to the second fundamental form of the hypersurface and ∇ 2 f is the hessian of f (see the next section for details).
From this point of view, as an extension of finite type theory, S.M.B. Kashani ( [11]) introduced the notion of L r -finite type hypersurface in the Euclidean space, which has been followed in the author's doctoral thesis.One can see our results in the last section of the last chapter of B. Y. Chen's book ( [6]), second edition.
Recently, in [15] the notion of pointwise 1-type Gauss map for the surfaces of Euclidean 3-space E 3 was extended in a natural way in terms of the Chen-Yau operator .Based on this definition rotational, helicoidal and canal surfaces in E 3 with L 1 -pointwise 1-type Gauss map were discussed in [16,18].Motivated by such an idea, the following definition was given by the author in [17].
Definition 1.1.An oriented hypersurface M of Euclidean space E n+1 is said to have L r -pointwise 1-type Gauss map if its Gauss map satisfies for a smooth function f ∈ C ∞ (M ) and a constant vector C ∈ E n+1 .An L rpointwise 1-type Gauss map is called proper if the function f is non-constant.More precisely, an L r -pointwise 1-type Gauss map is said to be of the first kind if (1.1) is satisfied for C = 0; otherwise, it is said to be of the second kind.Moreover, if (1.1) is satisfied for a constant function f , then we say that M has-(global) 1-type Gauss map.
In the same paper, we focused on the hypersurfaces with constant (r + 1)th mean curvature, constant mean curvature.We obtain some classification and characterization theorems for such hypersurfaces with L r -pointwise 1-type Gauss map.Therefore, it seems natural and interesting to propose the following problem.Open Problem.Classify hypersurfaces in E n+1 with L r -1-type Gauss map.
On the other hand, rotational surfaces of Euclidean spaces and pseudo-Euclidean spaces with pointwise 1-type Gauss map have been studied in several papers [9,12,14,20].For example, in [9] the rotational surfaces of E 3 with pointwise 1-type Gauss map have been studied by B.Y. Chen, M. Choi and Y.H. Kim.They proved that rotational surfaces of Euclidean spaces with pointwise 1-type Gauss map of the first kind coincide with rotational surfaces with constant mean curvature; and the right cones are the only rational surfaces of revolution with pointwise 1-type Gauss map of the second kind.Fortheremore, Dursun in [10] extentend the results given by B.Y. Chen, M. Choi and Y.H. Kim for surfaces of revolution with pointwise 1-type Gauss map in E 3 ( [9]) to the hypersurfaces of revolution with pointwise 1-type Gauss map in E n+1 .He proved that a rational hypersurface of revolution of Euclidean space E n+1 has pointwise 1-type Gauss map if and only if it is an open portion of a hyperplane, a generalized cylinder, or a right n-cone.
In this paper, our aim is to study the hypersurfaces of revolution of a Euclidean space E n+1 in terms of L r -pointwise 1-type Gauss map of the first and second kind.We first give some examples of hypersurfaces of revolution with proper L r -pointwise 1-type Gauss map of the first kind and the second kind, respectively.Then, we classify rational rotational hypersurfaces of E n+1 with L r -pointwise 1-type Gauss map which extend the results given in [10] on rational hypersurfaces of revolution with pointwise 1-type Gauss map to the rational hypersurfaces of revolution with L r -pointwise 1-type Gauss map.

Preliminaries
In this section, we recall preliminary concepts from [1,17].Let x : M n → E n+1 be an isometrically immersed hypersurface in the Euclidean space, with the Gauss map G.We denote by ∇ 0 and ∇ the Levi-Civita connections on E n+1 and M n , respectively.Then, the basic Gauss and Weingarten formulae of the hypersurface are written as As is well-known, for every point p ∈ M n , S defines a linear self-adjoint endomorphism on the tangent space T p M n , and its eigenvalues λ 1 (p), λ 2 (p), . . ., λ n−1 (p), λ n (p) are the principal curvatures of the hypersurface.The characteristic polynomial Q S (t) of S is defined by where a k is given by The rth mean curvature H r of M n in E n+1 is defined by n r H r = a r , H 0 = 1.If H r+1 = 0, then, we say that M n is a r-minimal hypersurface.The r-th Newton transformation of M n is the operator P r : (−1) j a r−j S j . Equivalently, Associated to each Newton transformation P r , we consider the second-order linear differential operator L r : Here, ∇ 2 f : χ(M n ) → χ(M n ) denotes the self-adjoint linear operator metrically equivalent to the Hessian of f and is given by < Now we state the following lemma from [1], which is useful in the present paper.. Lemma 2.1.Let x : M n → E n+1 be a connected orientable hypersurface immersed into the Euclidean space, with Gauss map G.Then, the Gauss map G of M satisfies Now, from the definition (1.1) and the lemma 2.1, we state the following theorem which characterize the hypersurfaces of Euclidean spaces with L r -pointwise 1-type Gauss map of the first kind.
Theorem 2.1.An oriented hypersurface M in E n+1 has proper L r -pointwise 1type Gauss map of the first kind if and only if H r+1 is constant and We can have the following corollary on hypersurfaces with L r -1-type Gauss map.
Corollary 2.1.All oriented isoparametric hypersurfaces of a Euclidean space E n+1 has L r 1-type Gauss map.

So, hyperplanes, hyperspheres and the generalized cylinder
We can also state that: Theorem 2.2.If an oriented hypersurface M in E n+1 has proper L r -pointwise 1-type Gauss map of the second kind, then the (r + 1)th mean curvature of M is non-constant.

Rotational hypersurfaces
Let x 1 = ϕ(v), x n+1 = ψ(v) be a curve in the x 1 x n+1 -half plane lying in halfspace x 1 = φ(v) > 0. Rotating this curve around the x n+1 -axis we obtain a rotational hypersurface M in E n+1 .Let {η 1 , . . ., η n+1 } be the standard orthonormal basis of E n+1 and S n−1 (1) be the unit sphere in E n spanned by {η 1 , . . ., η n }.We can have an orthogonal parametrization of S n−1 (1) ⊂ E n as which is the position vector of the sphere where η n+1 = (0, 0, . . ., 0, 1) is the axis of the rotation.Taking derivative we have the orthogonal coordinate vector fields on M as Hence the Gauss map of the hypersurface of revolution is given by By straightforward calculation we can have the Weingarten map as where I n−1 is the (n − 1) × (n − 1) identity map.Thus the (r + 1)th mean curvature is Since the (r + 1)th mean curvature H r+1 is the function of v, using (3.4) we can have the gradient of H r+1 as

Examples of rotational hypersurfaces with L r -pointwise 1-type Gauss map
We can have the following examples of rotational hypersurfaces with proper L r -pointwise 1-type Gauss map of the first kind and the second kind, respectively.Example 3.1.Let M be the rotational hypersurface in E n+1 parameterized by where a is nonzero constant, q = 2(n−r−1)

Rotational hypersurfaces of rational kind with L r -pointwise 1-type Gauss map
Let M be a rotational hypersurfaces in E n+1 parameterized by taking ϕ(t) = t, t > 0 and ψ(t) = g(t) in (3.3) where Y is given by (3.2).The Gauss map G of M parameterized by (3.10) is given by where tr (S 2 • P r ) = (−1) Suppose that M has L r -pointwise 1-type Gauss map of the second kind.Then (1.1) holds for some function f and some vector C. When the Gauss map is not L r -harmonic (i.e.L r G = 0), (1.1), (3.1), (3.11) and (3.12) imply that the first n components of C must be zero and tr (S 2 • P r ) + n r + 1 (3.15)

Theorem 3 . 1 .Example 3 . 2 .
An oriented hypersurface M in E n+1 with at most 2 distinct principal curvatures has L n−1 -(global) 1-type Gauss map of the first kind if and only if it is either an n-minimal hypersurface or an open part of a hypersphere, a hyperplane or a generalized cylinder.Consider the right n-cone C a based on the sphere S n−1 (1) which is parameterized by x(u 1 , . . ., u n−1 , v) = vY (u 1 , . . ., u n−1 ) + avη n+1 , a ≥ 0,