Let M be a hypersurface of the Euclidean space . A smooth mapping φ: M → is said to be of k-type if it can be expressed as a sum of eigenvectors of Laplace operator Δ corresponding to k distinct eigenvalues of Δ ([7]). If φ is an immersion from M into is of k-type, then M itself is said to be of k-type ([3]). The study of finite type mappings was summed up in a report by B.-Y. Chen ([4]).

On the other hand, if the Gauss map G of M is of 1-type, then it satisfies

for a constant λ ∈ ℝ and a constant vector C. In this case, M is said to have 1-type Gauss map, [8]. However, Gauss map of some important submanifolds such as a helicoid and a catenoid in and several rotational surfaces in satisfy a very similar equation to (1.1), namely

for some function f ∈ C^∞(M) and a constant vector C, ([11, 12]). These submanifolds whose Gauss map G satisfying (1.2) is said to have pointwise 1-type Gauss map. Submanifolds with pointwise 1-type Gauss map have been worked in several papers (cf. [5, 11, 16, 17, 18, 20, 21]).

In the recent years, the definition of being k-type of an hypersurface is extended in a natural way by replacing Laplace operator Δ with a sequence operators L₀, L₁, L₂, …, L_k such that L₀ = −Δ, where L_k is the linearized operator of the first variation of the (k + 1)-th mean curvature arising from normal variations of a hypersurface M of the Euclidean space . For convenience, the notation □ is used to denote the operator L₁ which is called as theCheng-Yau operator introduced in [9]. The authors Alías et al. studied an isometric immersion x: Mⁿ → Rⁿ⁺¹ satisfying L_k(x) = Ax + b for a constant matrix A and a constant vector b, where k is a positive integer.

In [15], the authors give the following definition.

Definition 1.([15])

An oriented surface M of Euclidean space is said to have □-pointwise 1-type Gauss map if its Gauss map satisfies

for a smooth function f ∈ C^∞(M) and a constant vector C ∈ . More precisely, a □-pointwise 1-type Gauss map is said to be of the first kind if (1.3) is satisfied for C = 0; otherwise, it is said to be of the second kind. Moreover, if (1.3) is satisfied for a constant function f, then we say M has □-(global) 1-type Gauss map.

In the same paper, authors states

Open Problem

Classify surfaces in with □-1-type Gauss map.

On the other hand, there are many studies done on rotational surfaces, ruled surfaces and translation surfaces in terms of being finite type or having pointwise 1-type Gauss map. For example, in [5] and [14], the rotational surfaces of the Euclidean 3-space and the Minkowski 3-space E13 with (Δ-)pointwise 1-type Gauss map have been studied. Also, a classification of ruled surfaces in terms of their Gauss map was studied in [6] and [16].

In this paper, we study rotational surfaces, ruled surfaces and translation surfaces in and E13 with □-pointwise 1-type Gauss map.

Let q ∈ {0, 1} and Eq3 denote the 3-dimensional semi-Euclidean space with the canonical semi-Euclidean metric tensor of index q given by

A non-zero vector u in the Minkowski space E13 is called space-like (resp., time-like or light-like) if 〈u, u〉 > 0 (resp., 〈u, u〉 < 0 or 〈u, u〉 = 0). Furthermore, a curve β is called space-like (resp., time-like or light-like) if its tangent vector β′ is space-like (resp., time-like or light-like) at every point.

On the other hand, a two dimensional subspace U of E13 is called non-degenerate if U ∩ U^⊥ = {0} and a non-degenerate subspace U of index r is called space-like (resp., time-like) if r = 0 (resp., r = 1). Eventually, a surface M in E13 is called non-degenerate, (resp., degenerate, space-like or time-like) if its tangent space T_pM is non-degenerate, (resp., degenerate, space-like or time-like) at every point p ∈ M.

The following lemmas are well-known and useful (see, for instance [13]):

Lemma 2.1

Let u, v be two orthogonal vectors in E13. If u is time-like, then v is space-like.

Lemma 2.2

Two light-like vectors are orthogonal if and only if they are linearly dependent.

Lemma 2.3

A two dimensional subspace U of E13is a time-like space if and only if it contains two linearly independent lightlike vectors.

2.1. Surfaces in 3-dimensional Euclidean and Minkowski spaces

Let M be an oriented surface in Eq3. We denote the Levi-Civita connections of E13 and M by ∇̃ and ∇, respectively and D stands for the normal connection of M. We put

Then, we have 〈N, N〉 = (−1)^qɛ, where N is the unit normal vector field associated with the orientation of M. The mapping G:M→Eq3 which assigns every point p to N(p) is called the Gauss map of M.

The well-known Gauss and Weingarten formulas are given by

for tangent vector fields X, Y of M, where h is the second fundamental form and S is the shape operator of M. The covariant derivative of h is defined by

Then, the Codazzi equation is given by

for tangent vector fields X, Y, Z of M. Note that S and h satisfy 〈S(X), Y 〉 = 〈h(X, Y ), N〉.

The functions , H and K defined by (λ) = det(S − λI) = λ² − 2Hλ + K are called the characteristic polynomial of S, the mean curvature of M and the Gaussian curvature of M, respectively. M is said to be minimal (resp., flat) if H (resp., K) vanishes identically. Sometimes the (complex valued) functions λ₁ and λ₂ satisfying (λ_i) = 0, i = 1, 2 are called the principal curvatures of M.

The Gauss equation is given by

where R is the curvature tensor associated with connection ∇ and e₁, e₂ are orthonormal vector fields on M.

We will use χ(M) to denote the space of all smooth functions from M into Eq3 and C^∞(M) the space of all smooth functions defined on M. Let = {e₁, e₂, e₃} be an orthonormal frame field defined on M, i.e., 〈e₁, e₁〉 = ɛ, 〈e₂, e₂〉 = 1, e₃ = N and 〈e_i, e_j〉 = 0 for i ≠ j (i, j = 1, 2, 3). If X ∈ χ(M) is tangent to M, its divergence divX is defined by divX = ɛ〈∇_e1X, e₁〉 + 〈∇_e2X, e₂〉. On the other hand, the gradient of a function f ∈ C^∞(M) is given by ∇f = ɛe₁(f)e₁ + e₂(f)e₂ and the Laplace operator acting on M is given as Δ = −ɛ∇_e1e₁ −∇_e2e₂ + e₁e₁ + e₂e₂.

2.2. Surfaces with □-pointwise 1-type Gauss map

Let M be a surface in Eq3 and P₀, P₁ the Newton transformations given by P₀ = I, P₁ = 2HI – S, where I is the identity operator acting on the tangent bundle of M. Then, the second order differential operators L_k: C^∞(M) → C^∞(M) associated with P_k are given by L_k(f) = tr(P_k ∘∇²f), k = 1, 2. Note that we have L₀ = −Δ and L₁ = □, where □ is the Cheng-Yau operator introduced in [9]. As a matter of fact, it turns out to be

for f ∈ C^∞(M) ([2]).

We will use following lemma and theorems in [15].

Lemma 2.4.([15])

Let M be an oriented surface in with Gaussian curvature K and mean curvature H. Then, the Gauss map G of M satisfies

Theorem 2.5.([15])

An oriented surface M in has □-harmonic Gauss map if and only if it is flat, i.e, its Gaussian curvature vanishes identically.

Theorem 2.6.([15])

An oriented surface M in has □-pointwise 1-type Gauss map of the first kind if and only if it has constant Gaussian curvature.

Theorem 2.7.([15])

An oriented minimal surface M in has □-pointwise 1-type Gauss map if and only if it is an open part of a plane.

Let M be a ruled surface in given by (2.5). Then, as a surface, we have x_t = β ≠ 0 and x(s, t) = α + t̃ β̃ where t̃ = t〈β, β〉^1/2 and β̃ = β/〈β, β〉^1/2. Thus, without loss of generality, we may assume 〈β, β〉 = 1. By re-defining s appropriately, we also suppose that 〈β′, β′〉 = 1. Moreover, for another base curve ᾱ of M given by ᾱ(s) = α(s)+g(s)β(s) with g′(s)+〈α′(s), β(s)〉 = 0 we have 〈ᾱ′, β〉 = 0. Hence, without loss of generality, we may also assume 〈α′, β〉 = 0.

Because of these assumptions, we have

and

for some smooth functions a = a(s), b = b(s) and c = c(s).

We choose an orthonormal frame field as

where

By a direct calculation, we obtain the connection form ω12 as

where {θ₁, θ₂} is the dual base of {e₁, e₂}. Moreover, the Gaussian curvature K and the mean curvature H are given by

where

On the other hand, from the Codazzi equation (2.3) and the Gauss equation (2.4) we obtain

By using (3.5) and (3.11), we obtain

3.1. Ruled surfaces with □-pointwise 1-type Gauss map of the first kind

We first give the following theorem: