Finite type immersions are first given by Chen [6]. Let M be a submanifold in m-dimensional Euclidean space Em. An isometric immersion x:M→ Em is of finite type if it can be written as a finite sum of eigenvectors of the Laplacian Δ of M for a constant map x₀, and non-constant maps x1,x2,...,xk, i.e.,

Here, Δx=λixi,λi∈ℝ, 1≤i≤k. The submanifold is said to be of k-type if the numbers λis are different [6].

Chen and Piccinni generalised these immersions to the Gauss map G of M

for a constant vector C and a real number a in [7]. A submanifold that satisfies the last equality are said to have a 1-type Gauss map.

In the last equality, one can take a non-constant differentiable function f instead of a. Namely, one can generalise the last equality to

A submanifold that satisfies the equation (1.1) is said to have a pointwise 1-type Gauss map. Also, if the vector C is zero, the pointwise 1-type Gauss map is said to be of the first kind. Otherwise, it is of the second kind. If Δ G=0, the Gauss map is harmonic. Surfaces satisfying the equation (1.1) are the subject of many studies such as [3, 4, 13].

In [2, 10], the notion of finite type submanifolds is generalised by replacing the Laplacian operator with operators Lk(k=1,2,...,n−1) that represent the linear operators of the first variation of the (k+1)-th mean curvature of a submanifold. Here, L0=−Δ and L₁ is the Cheng-Yau operator. Recently, some papers have been published about surfaces having L₁-pointwise 1-type Gauss map in some spaces, such as [11, 12, 18].

Tubular surfaces are special cases of canal surfaces which are the envelopes of a family of spheres. In canal surfaces, the center of the spheres are on a given space curve (spine curve), and the radius of the spheres are different. In tubular surfaces, the radius functions are constant. These surfaces have been widely studied in recent times [5, 13, 14, 15, 16]. In Galilean 3-space, tubular surfaces are studied in [9].

Here, some preliminaries about Galilean geometry are given. For more detailed information, the studies [19, 20] can be examined.

The scalar product and the cross product of the two vectors a=a1,a2,a3 and b=b1,b2,b3 in G3 are defined as

and

respectively. Here, e2=(0,1,0) and e3=(0,0,1) are the orthonormal unit vectors. The length (norm) of the vector a=(a1,a2,a3) is given as follows:

[17].

An admissible unit speed curve α:I⊂ ℝ→G3 is given with the parametrization

The associated Frenet frame on the curve is given as

where κ(u)=y′′(u)2+z′′(u)2 and τ(u)=detα′(u),α′′(u),α′′′(u)κ2(u) are the curvature and the torsion of the curve, respectively. Thus, the famous Frenet formulas can be written as

Definition 2.1. ([1]) A regular curve in Galilean space G3 with constant curvature and non-constant torsion is called a Salkowski curve.

For an isometric immersion X:M→M˜ from a hypersurface M from an (n+1)-dimensional Riemannian manifold M˜, and for the Levi-Civita connections ∇˜ of M˜ and ∇ of M, the Gauss formula is given by

where X,Y∈χ(M) and S is the shape operator of M. It is known that the eigenvalues κ1,κ2,...,κn of S are the principal curvatures of M. For a smooth function f on M, linear operators L_k are defined

where ∇ is the gradient, div is the divergence operator and

is the Newton k-th transformation, sk=nkHk is the k-th mean curvature [8]. Thus, for k=0, P0=In (I_n is the identity matrix), and for k=1, P1=tr(S)In−S.

Now, let M be a surface, e1,e2 be the principal directions correspond to the curvatures k₁,k₂ of M. From (2.1), for a smooth function f the Cheng-Yau operator L₁f can be given as

Hence, the Cheng-Yau operator L₁ can be given

[11].

Let the surface M parametrized with

in G3. To represent the partial derivatives, we use

If x,i≠0 for some i=1,2, then the surface is admissible (i.e. having not any Euclidean tangent planes). The first fundamental form I of the surface M is defined as

where gi=x,i, hij=y,iy,j+z,iz,j; i,j=1,2 and

Let a function W is given by

Then, the unit normal vector field is given as

Similarly, the second fundamental form II of the surface M is defined as

where

or

The Gaussian and the mean curvatures of M are defined as

A surface is flat (resp. minimal) if its Gaussian (resp. mean) curvatures vanish [19].

Lemma 2.2. ([11]) Let M be an oriented surface in E3 and K and H be the Gaussian and the mean curvatures of M, respectively. Then the Gauss map G

Definition 2.3. ([11]) Let M be an oriented surface in E3. Then, M is said to have an L₁-harmonic Gauss map if its Gauss map satisfies L₁G=0.

Definition 2.4. ([11]) Let M be an oriented surface in E3. Then, M is said to have an L₁-pointwise 1-type Gauss map if its Gauss map satisfies

for a smooth function f and a constant vector C. If the vector C is zero, the pointwise L₁-type Gauss map is of the first kind, otherwise, it is of the second kind.

A tubular surface M in G3 at a distance r from the points of spine curve α(u)=(u,y(u),z(u)) is given with

Writing the Frenet vectors of α(u) in (3.1), the parametrization can be given as

From (3.2),

An orthonormal frame e1,e2,G of M is given by

and

Here W=r. The coefficients of the second fundamental form are obtained as

From, (3.3) and (3.6), the curvature functions of M are obtained as

[9].

Corollary 3.5. ([9]) Tubular surfaces are constant mean curvature surfaces in Galilean space.

By (3.7), we write the gradient of the Gaussian curvature

Thus, from (3.4), (3.7) and (3.7), we obtain the Cheng-Yau operator of the Gauss map as

Now, we consider the surface M has L₁-harmonic Gauss map, i.e. L₁G=0. Then, from (3.9), we have

and

Writing κ′rcosv=0 in (3.10), we get

Multiplying the first equation with cosv and the second with sinv, we obtain κcosv=0, which implies κ =0 or cosv=0. If cosv=0, again from (3.10), κ =0.

Then, we give the following theorem:

Theorem 3.6. Let M be a tubular surface given with the parametrization (3.1) in G3. M has L₁-harmonic Gauss map if and only if the spine curve α is a straight line and M is an open part of a cylinder. Thus, the surface is flat.

Example 3.7. Let us consider the tubular surface M, which has L₁-harmonic Gauss map with the parametrization (3.1) in G3. Taking the straight line α(u)=(u,u+1,u+2) and writing the Frenet vectors of it n(u)=(0,1,0), b(u)=(0,0,1) and r=4 in (3.1), we write the parametrization of the surface M as

By using the software Maple, we plot the graph of the surface in (3.11).

Figure 1. Tubular surfaces M which has L₁-harmonic Gauss map with the spine curve α(u)=(u,u+1,u+2) and the radius r=4.

Now, we assume that the tubular surface M has L₁-pointwise 1-type Gauss map of the first kind, i.e., L₁G=fG for a smooth function f. Then, from (3.5) and (3.9),

From (3.12), we have

and

Similar to above, writing κ′cosv=0 in (3.13), we get

Multiplying the first equation with cosv, the second with sinv, and combining them, we obtain f=κcosvr2. Moreover, since κ′cosv=0, we have two cases: κ =0 or κ is a constant. If κ =0, the tubular surface M has L₁-harmonic Gauss map. Thus, κ is a nonzero constant.

Theorem 3.8. Let M be a tubular surface given with the parametrization (3.1) in G3. M has L₁-pointwise 1-type Gauss map of the first kind if and only if the curvature κ of the curve is constant and f=−Kr.

Corollary 3.9. The spine curve of the surface which has L₁-pointwise 1-type Gauss map of the first kind is a Salkowski curve in G3.

Example 3.10. Let us consider the tubular surface M, which has L₁-pointwise 1-type Gauss map of the first kind with the parametrization (3.1) in G3. For the curves α1(u)=(u,cosu,sinu), α2(u)=(u,u22,0), and the radius r=2, we write the parametrizations of the surfaces M₁ and M₂ as

We again use Maple to plot the graphs of the surfaces in (3.15).

Figure 2. Tubular surfaces M₁ and M₂ which have L₁-harmonic Gauss map with the spine curves α1(u)=(u,cosu,sinu), and the radius r=2.

Lastly, we consider that the tubular surface M has L₁-pointwise 1-type Gauss map of the second kind, i.e., L1G=fG+C for a smooth function f and a nonzero constant vector C. From the equations (2.6) and (3.9), we can write the vector C as

where

Since C is a nonzero constant vector, ∇˜e1C=0 and ∇˜e2C=0. Thus, we have

and

Since

we get

From the last differential equation system,

where a is a real constant. By the equation (3.17),

which means

Here h₁(u) and h₂(u) are any functions of u. Moreover, from (3.16),

Writing (3.19) in the last equation, we obtain

which has a solution as

From (3.16), we have

Again writing (3.19) in the last equation, we obtain

which has a solution as

Then, we give the following theorem:

Theorem 3.11. Let M be a tubular surface given with the parametrization (3.1) in G3. M has L₁-pointwise 1-type Gauss map of the second kind if and only if f=aκ′cosv for a real constant a and the equations (3.19)-(3.21) are hold.