Due to their physical importance in curve and surface theory, Galilean and Pseudo Galilean geometries have been widely studied in recent years. The Cayley Klein geometry with projective signature (0, 0, +, -) is an example of a Pseudo Galilean geometry, for detailed information see [5]. The absolute structure of a Pseudo Galilean geometry is represented by an ordered triple {w,f,I} consising of its ideal plane w, a line f in w and the fixed hyperbolic involution I of points of f. Pseudo Galilean 3-space, denoted as G31, is equipped with the scalar product g defined by

for any vectors X=x1,x2,x3, Y=y1,y2,y3∈G31. The Pseudo Galilean norm of a vector X defined by

A vector X=x1,x2,x3 in Pseudo Galilean 3-space is called a non-isotropic vector if x1≠0, and is otherwise X is called an isotropic vector.

The cross product is defined by

All unit non-isotropic vectors are of the form 1,x2,x3. The vector X is called an isotropic space-like vector if x22−x32>0 satisfies and X is called an isotropic time-like vector if x22−x32<0 satisfies. If x22−x32=0 then X is called an isotropic lightlike vector, in this case x2=±x3. If x22−x32=±1 then X is called a non-lightlike isotropic vector [6, 1, 3]. A curve γ:I⊂ℝ→G31 defined by γ(s)=x(s),y(s),z(s) is an admissible curve if none of the points are inflection points, all the tangents and the normal vectors are non-isotropic at each points of the curve. If the curve γ(s) is an admissible curve with the arc length parameter s then the position vector of γ(s) is

The curvature κ (s) and the torsion τ (s) are defined by

An admissible curve has no inflection points, no isotropic tangents or normals whose projections on the absolute plane would be light-like vectors. The Frenet trihedron is given by

where ϵ=∓1, under the condition detT,N,B=1. This requires that

Thus the principal normal vector, or simply normal, is space-like if ϵ =1 and time-like if ϵ =-1. The curve γ given by (1.1) is time-like (resp. space-like) if N(s) is a space-like (resp. time-like) vector. The following Serret-Frenet formulas hold

for derivatives of the tangent vector T(s), the normal vector N(s) and the binormal vector B(s), respectively [6, 1, 7, 3]. Karacan and Tunçer studied Weingarten and linear Weingarten type tubular surfaces in Galilean and Pseudo Galilean spaces [4]. They also studied also surfaces in the same spaces[8]. D.W. Yoon, studied the Gauss Map of Tubular Surfaces in Galilean space and classified them in [9]. For an open subset D⊆R2 and for a Cr-immersion X:D→G31, the set Φ=X(D) is called a regular Cr-surface(for r≥2) in Pseudo Galilean 3-space. If X is a Cr-embedding then the set Φ is called a simple Cr-surface(for r≥2). If the Cr-surface Φ does not have pseudo-Euclidean tangent planes then Φ is called admissible Cr-surface. Let us denote

and

then Φ is an admissible surface if and only if x,i≠0 for some i=1,2. Assume that Φ⊂G31 is a regular admissible surface. The unit normal vector field of Φ is

where W(u,v)=x1y2−x2y12−x1z2−x2z12. The function W(u,v) is equal to the Pseudo Galilean norm of the isotropic vector x,1X,2−x,2X,1. The vector

is called a side tangential vector. Throughout the study we will consider the surfaces with W ≠ 0 [8, 10]. Since we have gη, η=ϵ=±1, we consider two types of admissible surfaces: space-like surfaces having time-like surface normals (ϵ=−1), and time-like surfaces having space-like normals (ϵ=1). The first fundamental form (F.F.F.) of a surface in G31 is defined by

where

and

[8, 10]. For a vector x, x˜ denotes the projection the vector onto the pseudo-Euclidean plane yoz.

In this study, we denote the components of ds2 by g˜ij. Furthermore, according to the local coordinate system {u1,u2} of the surface X(u,v) the Laplacien operator Δ of the F.F.F. is defined by

where  g˜ij= g˜ij−1 [9, 10].

In this section, we will classify the admissible tubular surfaces in G31 satisfying the equations ΔX=0, ΔX=AX, ΔX=λX, ΔX=ΔG, ΔG=0, ΔG=AG and ΔG=λX where X is the position vector of tubular surface, G is the Gauss map of tubular surface, λ is nonzero constant, A∈Mat(3,ℝ) and Δ is the Laplacien operator of the surface. Y.Tunçer and M.K.Karacan defined the canal surfaces in Pseudo Galilean 3-Spaces in [8]. Generalising this, we definition tubular surfaces in pseudo galilean 3-space. Let γ:(a,b)→G31 be an admissible curve satisfying (1.1), and let M be a tubular surface with the centered curve γs. There are two types non-isotropic tubular surfaces in G31.

Type-1: If M is space-like (time-like) tubular surface and γs is space-like (time-like) curve then M is parametrized by

Type-2: If M is space-like (time-like) tubular surface and γs is time-like (space-like) curve then M is parametrized by

Let M be a type-1 tubular surface in G31 is parametrized by (2.1), then we have the natural frame {Xsμ,Xtμ} of M given by

and from (1.4), (1.5) and (1.6), we have

which are the components of F.F.F., so we obtain the g˜ij as

By a direct computation using the equation (1.7), the Laplacian operator Δ on M is

Suppose that M satisfies ΔXμs,t=AXμs,t, with the matrix A∈Mat(3,ℝ), then from (2.1) and (2.3) we obtain the equality

so it is easy to see that the equality ΔXμs,t=0 is not satisfied for type-1 tubular surface. Hence we give the following theorem.

Theorem 2.1. There is not any harmonic type-1 tubular surface given by (2.1) in G31.

For the other cases, we give the following theorem.

Theorem 2.2.Let M be a type-1 tubular surface given by (2.1) in G31. M satisfies ΔXμs,t=AXμs,t, A∈Mat(3,ℝ) if

Proof. Differentiating (2.1) with respect to t we get

Taking the derivative (2.6) with respect to t, we have

Combining (2.6) and (2.7) we can obtain the following two equation

On the other hand, from (2.4), (2.7) and (2.8)

and differentiating (2.9) with respect to s and by using (2.7) and (2.8), we have

By taking the derivative (2.11) with respect to s , we get

By considering ATs′=AT′s=κsANs in (2.12) and from (2.8), we obtain

Thus, this completes the proof.

From the first equation of (2.5), we can obtain

and by using the second equation of (2.5), we have

Equation (2.15) has complex solutions but in the case of κs and τs are constant, then (2.15) has the real solution. Thus we can give following corollary as a remark of theorem 2.2.

Corollary 2.3. Let M be a type-1 tubular surface given by (2.1) in G31. If M satisfies

then M is one of the following.

i. M is a type-1 surface determined by

where d1≠0,c1≠0,c2,c3,d2,d3 are constants (for time-like centered curve c1>d1 and for space-like centered curve c1<d1, see Figures 1 and 2).

Figure 1. (a)

Figure 2. (b)

ii. M is a type-1 surface determined by

or

where d1≠0,c1≠0,c2,c3,d2,d3 are constants (see Figures 3 and 4).

Figure 3. (c)

Figure 4. (d)

iii. M is a type-1 space-like surface determined by

where a>1 is constant (see Figure 3). The Gauss map G of type-1 tubular surface M is

and from (2.3), we find

Thus, it is easy to see that, following theorem holds.

Theorem 2.4. Let M be a type-1 tubular surface given by (2.1) in G31 then followings are true.

i. There are no type-1 tubular surface given by (2.1) in G31 with the Gauss map G being harmonic.

ii. All type-1 tubular surfaces satisfy ΔGs,t=λGs,t,λ≠0.

iii. All type-1 tubular surface satisfy ΔGs,t=AGs,t where A=1μr2I3.

As a result of Theorem 2.4., we can say M has a type-1 Gauss map Gs,t in the sense of Chen [2].

Assume that M satisfy ΔXμs,t=ΔGs,t. From (2.3) and (2.16)

and so κ =0. Thus we get following theorem.

Theorem 2.5. Let M be a type-1 space-like tubular surface given by (2.1) in G31, then M satisfies ΔXμs,t=ΔGs,t if its position vector is

where a>1∈ℝ and r>0.

Let M be a type-2 tubular surface G31 parametrized by (2.2), then we have the natural frame {Xsσs,t,Xtσs,t} of M given by

and we have

which are the components of F.F.F.

and the Laplacian operator Δ on M is obtained as

The Gauss map Gs,t of type-1 tubular surface M is

and Laplacians of Xσs,t and Gs,t are

and

respectively. We can also obtain similar results for type-2 surfaces in G31 by using (2.18), (2.21) and (2.22).

Example 2.2. Timelike tube with time-like centered curve satisfying ΔXμs,t=AXμs,t where A=1−4I3

Spacelike tube with space-like centered curve satisfying ΔXμs,t=AXμs,t where A=14I3

Timelike tube with time-like centered curve satisfying ΔXμs,t=AXμs,t where A=1−4I3

Spacelike tube with space-like centered curve satisfying ΔXμs,t=AXμs,t where A=14I3

The authors are indebted to the referees for helpful suggestions and insights concerning the presentation of this paper.