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Kyungpook Mathematical Journal 2022; 62(3): 497-507

Published online September 30, 2022

On the Gauss Map of Tubular Surfaces in Pseudo Galilean 3-Space

Yılmaz Tunçer∗, Murat Kemal Karacan, Dae Won Yoon

Department of Mathematics, Usak University, Usak 64200, Turkey
e-mail : yilmaz.tuncer@usak.edu.tr

Department of Mathematics, Usak University, Usak 64200, Turkey
e-mail : murat.karacan@usak.edu.tr

Department of Mathematics Education and RINS, Gyeongsang National University, Jinju 52828, Republic of Korea
e-mail : dwyoon@gsnu.ac.kr

Received: March 21, 2021; Revised: December 7, 2021; Accepted: December 10, 2021

In this study, we define tubular surfaces in Pseudo Galilean 3-space as type-1 or type-2. Using the X(s,t) position vectors of the surfaces and G(s,t) Gaussian transformations, we obtain equations for the two types of tubular surfaces that satisfy the conditions ΔX(s,t) =0, ΔX(s,t)=AX(s,t), ΔX(s,t)=λX(s,t), ΔX(s,t)=ΔG(s,t), ΔG(s,t)=0, ΔG(s,t)=AG(s,t) and ΔG(s,t)=λG(s,t).

Keywords: Tubular surfaces, Gauss map, pointwise 1-type Gauss map, Pseudo Galilean 3-Space

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

gX,Y=x1y1 ifx10y10x2y2x3y3 ifx1=0y1=0.

for any vectors X=x1,x2,x3, Y=y1,y2,y3G31. The Pseudo Galilean norm of a vector X defined by

X=x1 ifx10x22x32ifx1=0.

A vector X=x1,x2,x3 in Pseudo Galilean 3-space is called a non-isotropic vector if x10, and is otherwise X is called an isotropic vector.

The cross product is defined by

XG31Y=0e2 e3 x1 x2 x3 y1 y2 y3 ifx10y10

All unit non-isotropic vectors are of the form 1,x2,x3. The vector X is called an isotropic space-like vector if x22x32>0 satisfies and X is called an isotropic time-like vector if x22x32<0 satisfies. If x22x32=0 then X is called an isotropic lightlike vector, in this case x2=±x3. If x22x32=±1 then X is called a non-lightlike isotropic vector [6, 1, 3]. A curve γ:IG31 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

γ(s)=s,p(s),q(s).

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

κ(s)=p(s)2q(s)2,τ(x)=p(s)q(s)p(s)q(s)κ2(s).

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

T(s)=γ(s)=1,p(s),q(s)N(s)=1κ(s)0,p(s),q(s)B(s)=1κ(s)0,ϵq(s),ϵp(s)

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

p(s)2q(s)2=ϵp(s)2q(s)2.

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

T(s)=κ(s)N(s),N(s)=τ(s)B(s),B(s)=τ(s)N(s)

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 DR2 and for a Cr-immersion X:DG31, the set Φ=X(D) is called a regular Cr-surface(for r2) in Pseudo Galilean 3-space. If X is a Cr-embedding then the set Φ is called a simple Cr-surface(for r2). If the Cr-surface Φ does not have pseudo-Euclidean tangent planes then Φ is called admissible Cr-surface. Let us denote

X=X(x(u1,u2),y(u1,u2),z(u1,u2)),

and

x,i=xui, y,i=yui, z,i=zui

then Φ is an admissible surface if and only if x,i0 for some i=1,2. Assume that ΦG31 is a regular admissible surface. The unit normal vector field of Φ is

η(u,v)=0,x1z2x2z1,x1y2x2y1W(u,v),

where W(u,v)=x1y2x2y12x1z2x2z12. The function W(u,v) is equal to the Pseudo Galilean norm of the isotropic vector x,1X,2x,2X,1. The vector

ρ(u,v)=x,1X,2x,2X,1W

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

ds2=g1du1+g2du22+δ(h11du2+2h12dudv+h22dv2),

where

gi=x,i,
hij=g X˜,i,X˜,j

and

δ=0;ifdirectiondu1:du2 is non-isotropic1; if direction du1:du2 is isotropic.

[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

Δ=1det g˜ij i,j=12uidet g˜ijg˜ijuj,

where  g˜ij= g˜ij1 [9, 10].

2. Tubular Surfaces in Pseudo Galilean 3-Space

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, AMat(3,) 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

Xμs,t=γs+rcosh(t)Ns+rsinh(t)Bs,
μ=+11if M is a space-like canal surface with space-like centered curveif M is a time-like canal surface with time-like centered curve.

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

Xσs,t=γs+rsinh(t)Ns+rcosh(t)Bs,
σ=+11if M is a space-like canal surface with time-like centered curveif M is a time-like canal surface with space-like centered curve.

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

Xsμs,t=Ts+rτssinh(t)Ns+rτscosh(t)BsXtμs,t=rsinh(t)Ns+rcosh(t)Bs

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

g1=1,g2=0,h11=μr2τs2,h21=h12=μr2τs,h22=μr2

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

g˜11=1+μr2τs2,g˜12=g˜21=μr2τs,g˜22=μr2.

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

Δ=μr2τ s2+1μr22t2τst2τs2ts+2s2.

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

1μrμrκs+cosh(t)Ns+sinh(t)Bs=Aγs+rcosh(t)ANs+rsinh(t)ABs,

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, AMat(3,) if

2κsτs+κsτs=0,κsκs+τs2=1μr2.

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

ΔXμs,tt=1μrsinh(t)Ns+cosh(t)Bs=rsinh(t)ANs+rcosh(t)ABs.

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

ΔXμs,ttt=1μrcosh(t)Ns+sinh(t)Bs=rcosh(t)ANs+rsinh(t)ABs.

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

ANs=1μr2Ns,
ABs=1μr2Bs.

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

Aγs=κsNs

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

ATs=κsNs+κsτsBs.

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

ATs=κs+κsτs2Ns+κsτs+κsτsBs.

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

2κsτs+κsτs=0,κsκs+τs2=1μr2.

Thus, this completes the proof.

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

κ2s=aτs

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

κsκs+aκ4s=1μr2.

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

ΔXμs,t=00001μr2 0001μr2 Xμs,t

then M is one of the following.

i. M is a type-1 surface determined by

Xμs,t=s,c1s2+c2s+c3+2rc1cosh(t)+2rd1sinh(t)    ,d1s2+d2s+d3+2rd1cosh(t)+2rc1sinh(t)

where d10,c10,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

Xμs,t=s,c1s2+c2s+c3+2rc1cosh(t),d1s+d2+2rc1sinh(t)

or

Xμs,t=s,c1s+c2+2d1rsinh(t),d1s2+d2s+d3+2d1rcosh(t)

where d10,c10,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

Xμs,t=s,ra21cosh(t)+rasinh(t),racosh(t)+ra21sinh(t)

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

Gs,t=cosh(t)Ns+sinh(t)Bs

and from (2.3), we find

ΔGs,t=1μr2cosh(t)Ns+sinh(t)Bs.

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,λ0.

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)

1μr2μrκs+cosh(t)Ns+sinh(t)Bs=1μr2cosh(t)Ns+sinh(t)Bs,

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

Xμs,t=s,ra21cosh(t)+rasinh(t),racosh(t)+ra21sinh(t)

where a>1 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

Xsσs,t=Ts+rτscosh(t)Ns+rτssinh(t)BsXtσs,t=rcosh(t)Ns+rsinh(t)Bs

and we have

g1=1,g2=0,h11=σr2sτs2 , h12=σr2sτs,h12=σr2s

which are the components of F.F.F.

g˜11=1+σr2sτs2,g˜12=g˜21=σr2sτs,g˜11=σr2s

and the Laplacian operator Δ on M is obtained as

Δ=σr2τs2+1σr22t2τst2τs2ts+2s2.

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

Gs,t=sinh(t)Ns+cosh(t)Bs

and Laplacians of Xσs,t and Gs,t are

ΔXσs,t=1σrσrκs+sinh(t)Ns+cosh(t)Bs

and

ΔGs,t=1σr2sinh(t)Ns+cosh(t)Bs

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=14I3

Xμs,t=s,2s2+2s+2cosh(t)+sinh(t)3,s2+s+1+cosh(t)+2sinh(t)3

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

Xμs,t=s,s2+2s+cosh(t)2sinh(t)3,2s2+s+1+2cosh(t)sinh(t)3

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

Xμs,t=s,2s2+2s+1+12cosh(t),s+1+12sinh(t)

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

Xμs,t=s,2s+112sinh(t),2s2+s+1+12cosh(t)

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

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