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B-tubular surfaces in Lorentzian Heisenberg Group [H.sup.3]/Superficies B-tubulares no Grupo [H.sup.3] de Lorentz-Heisenberg.

Introduction

Tubular surfaces are very useful for representing long thin objects, for instance, poles, 3D fonts, brass instrument or internal organs of the body in solid modeling. It includes natural quadrics (cylinder, cone and sphere), revolute quadrics, tori, pipes and Dupin cyclide. Also, canal surfaces are among the surfaces which are easier to describe both analytically and operationally, (CARMO, 1976), (O'NEIL, 1983), (FAROUKI; NEFF, 1990), (LU, 1994), (GRAY, 1998), (PETERNELL; POTTMANN, 1997), (ZHU et al., 2005), (XU et al., 2006), (KORPINAR; TURHAN, 2011), (TURHAN; KORPINAR, 2011a).

We remind that, if [gamma] is a space curve, a tubular surface associated to this curve is a surface swept by a family of spheres of constant radius (which will be the radius of the tube), having the center on the given curve. Alternatively, as we shall see in the next section, for them we can construct quite easily a parameterization using the Frenet frame associate to the curve. The tubular surfaces are used quite often in computer graphics, but we think they deserve more attention for several reasons. For instance, there is the problem of representing the curves themselves. Usually, the space curves are represented by using solids rather then tubes. There are, today, several very good computer algebra system (such as Maple, or Mathematica) which allow the vizualisation of curves and surfaces, in different kind of representations, (KORPINAR; TURHAN, 2012), (POTTMANN; PETERNELL, 1998).

The aim of this paper is to study tubular surfaces surrounding biharmonic spacelike B slant helices according to Bishop frame in the Lorentzian Heisenberg group [H.sup.3].

Let (M, g) and (N, h) be Lorentzian manifolds and [phi] : M [right arrow] N a smooth map. Denote by [[nabla].sup.[phi]] the connection of the vector bundle [[phi].sup.*] TN induced from the Levi-Civita connection [[nabla].sup.h] of (N,h) . The 'second fundamental form' [nabla]d[phi] is defined by

([nabla]d[phi]X,Y) = [[nabla].sup.[phi]][x.sub.d][phi]{Y) - d[phi]([nabla] [sub.x]Y), X,Y [member of] [GAMMA](TM).

Here [nabla] is the Levi-Civita connection of (M, g). The tension field [tau]([phi]) is a section of [[phi].sup.*]TN defined by [tau]([phi]) = tr[nabla]d[phi].

A smooth map [phi] is said to be harmonic if its tension field vanishes. It is well known that [phi] is harmonic if and only if [phi] is a critical point of the 'energy':

E ([phi]) = 1/2 [integral] h (d[phi], d[phi])[dv.sub.g]

over every compact region of M. Now let [phi]: M [right arrow] N be a harmonic map. Then the Hessian H of E is given by

[H.sub.[phi]] (V, W) = [integral]h([J.sub.[phi]](V) W)[dv.sub.g], V, W [member of] [GAMMA]([[phi].sup.*]TN)

Here the Jacobi operator [J.sub.[phi]] is defined by

[J.sub.[phi]](V):= [[nabla].sup.[phi]]V - [R.sub.[phi]](V) [nabla] [member of] [GAMMA]([[phi].sup.*]TN)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [R.sup.N] and {[e.sub.i]} are the Riemannian curvature of N , and a local orthonormal frame field of M , respectively, (EELLS; SAMPSON, 1964), (JIANG, 1986a, 1986b), (TURHAN; KORPINAR, 2010).

Let [phi]:(M, g) [right arrow] (N, h) be a smooth map between two Lorentzian manifolds. The bienergy [E.sub.2] ([phi]) of [phi] over compact domain [OMEGA] [subset] M is defined by

[E.sub.2] ([phi]) = [[integral].sub.a] h ([tau]([phi]), [tau]([phi]))[dv.sub.g].

A smooth map [phi]: (M, g) [right arrow] (N, h) is said to be 'biharmonic' if it is a critical point of the [E.sub.2]([phi]) .

The section [[tau].sub.2]([phi]) is called the bitension field of [phi] and the Euler-Lagrange equation of [E.sub.2] is

[[tau].sub.2]([phi]):= -[J.sub.[phi]]([tau]([phi])) = 0

This study is organised as follows: Firstly, we give necessary and sufficient conditions for new type B--slant helices to be biharmonic. We characterize B--tubular surfaces in the Lorentzian Heisenberg group [H.sup.3]. Secondly, we study B--tubular surfaces in terms of biharmonic spacelike new type B--slant helices according to Bishop frame in the Lorentzian Heisenberg group [H.sup.3] . Finally, we illustrate our results in Figures 1, 2, 3 and 4.

The Lorentzian Heisenberg Group [H.sup.3]

The Heisenberg group [Heis.sup.3] is a Lie group which is diffeomorphic to [R.sup.3] and the group operation is defined as

(x, y, z) x ([bar.x], [bar.y], [bar.z]) = (x + [bar.x], y + [bar.y], z + [bar.z] - [bar.x]y + x[bar.y]).

The identity of the group is (0,0,0) and the inverse of (x, y, z) is given by (-x,-y,-z) (RAHMANI; RAHMANI, 2006). The left-invariant Lorentz metric on [Heis.sup.3] is

g = [dx.sup.2] + [dy.sup.2] + [(xdy + dz).sup.2].

The following set of left-invariant vector fields forms an orthonormal basis for the corresponding Lie algebra:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2.1)

The characterising properties of this algebra are the following commutation relations:

g([e.sub.1],[e.sub.1]) = g([e.sub.2],[e.sub.2]) = 1, g ([e.sub.3],[e.sub.3]) = -1

Proposition 2.1.

For the covariant derivatives of the Levi-Civita connection of the left-invariant metric g , defined above the following is true:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.2)

where the (i, j)--element in the table above equals [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. for our basis

{[e.sub.k], k = 1,2,3}.

Moreover we put

[R.sub.ijk] = R([e.sub.i], [e.sub.j])[e.sub.k], [R.sub.ijkl] = R([e.sub.i], [e.sub.j], [e.sub.k], [e.sub.l]),

where the indices i, j, k and l take the values 1, 2 and 3.

[R.sub.121] = [e.sub.2], [R.sub.131] = -[e.sub.3], [R.sub.232] = 3[e.sub.3]

and

[R.sub.1212] = 1, [R.sub.131]3 = 1, [R.sub.2323] = -3.

Spacelike biharmonic new type B-slant helices with Bishop frame in the Lorentzian Heisenberg group [H.sup.3]

Let [gamma] : I [right arrow] [H.sup.3] be a non geodesic spacelike curve on the Lorentzian Heisenberg group [H.sup.3] parametrized by arc length. Let {t,n,b} be the Frenet frame fields tangent to the Lorentzian Heisenberg group [H.sup.3] along [gamma] defined as follows:

t is the unit vector field [gamma]' tangent to [gamma], n is the unit vector field in the direction of [[nabla].sub.t] t (normal to [gamma]), and b is chosen so that {t, n, b} is a positively oriented orthonormal basis. Then, we have the following Frenet formulas, (TURHAN; KORPINAR, 2011b):

[[nabla].sub.t] t = [kappa]n,

[[nabla].sub.t]n = [kappa]t + [tau]b, (3.1)

[[nabla].sub.t]B = [tau]n, (3.1)

where [kappa] is the curvature of [gamma] and [tau] is its torsion and

g (t,t)=1, g (n,n )= -1, g (b,b) = 1, g (t,n )= g (t,b)= g (n,b)= 0

the rest of the paper, we suppose everywhere [kappa] [not equal to] 0 and [tau] [not equal to] 0.

The Bishop frame or parallel transport frame is an alternative approach to defining a moving frame that is well defined even when the curve has vanishing second derivative, (BISHOP, 1975). The Bishop frame is expressed as

[[nabla].sub.t]t = [k.sub.1][m.sub.1] - [k.sub.2][m.sub.2], [[nabla].sub.t][m.sub.1] = [k.sub.1]t, [[nabla].sub.t][m.sub.2] = [k.sub.2]t, (3.2)

where

g(t,t )=1, g ([m.sub.1], [m.sub.1])= -1, g ([m.sub.2], [m.sub.2])=1, g (T, [M.sub.1]) = g (t, [m.sub.2])= g ([m.sub.1], [m.sub.2]) = 0

Here, we shall call the set {t, [m.sub.1], [m.sub.2]} as Bishop trihedra, [k.sub.1] and [k.sub.2] as Bishop curvatures.

Also, [tau](s) = [gamma]' (s) and [kappa](s) = [square root of [absolute value of [k.sup.2.sub.2] - [k.sup.2.sub.1]].

With respect to the orthonormal basis {[e.sub.1], [e.sub.2], [e.sub.3]} we can write

t = [t.sup.1][e.sub.1] + [t.sup.2][e.sub.2] + [t.sup.3][e.sub.3], [m.sub.1] = [m.sup.1.sub.1][e.sub.1] + [m.sup.2.sub.2] [e.sub.2] + [m.sup.3.sub.2][e.sub.3], (3.3)

Theorem 3.1.

[gamma] : I [right arrow] [H.sup.3] is a spacelike biharmonic curve with Bishop frame if and only if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

Proof.

Using Equation(2.2) and Jacobi operator, we obtain above system. This completes the proof.

Definition 3.2.

A regular spacelike curve [gamma] : I [right arrow] [H.sup.3] is called a new type slant helix provided the spacelike unit vector m2 of the curve [gamma] has constant angle Q with spacelike vector u, that is

g ([m.sub.2] (s), u) = cos Q for all s [member of] I. (3.5)

The condition is not altered by reparametrization, so without loss of generality we may assume that slant helices have unit speed. The slant helices can be identified by a simple condition on natural curvatures.

To separate a spacelike new type slant helix according to Bishop frame from that of Frenet-Serret frame, in the rest of the paper, we shall use notation for the curve defined above as spacelike new type B-slant helix.

B-tubular surfaces in terms of spacelike biharmonic new type B-Slant helices according to Bishop frame in the Heisenberg group [H.sup.3]

The envelope of a 1-parameter family of the spheres in the Lorentzian Heisenberg group [H.sup.3] is called a tubular surface in the Lorentzian Heisenberg group [H.sup.3]. The curve formed by the centers of the spheres is called center curve of the tubular surface. The radius of the tubular surface is the function r such that r is the constant radius of the sphere.

To separate a tubular surface according to Bishop frame from that of Frenet-Serret frame, in the rest of the paper, we shall use notation for the surface defined above as B-tubular surface.

Theorem 4.1.

Let [gamma]: I [right arrow] [H.sup.3] be a unit speed biharmonic spacelike new type B--slant helix with non-zero curvatures. Then the equation of B - tubular surface [M.sup.B] (s, t) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (4.1)

where [C.sub.0], [C.sub.1], [C.sub.2], [C.sub.3] are constants of integration and

[C.sub.0] = [square root of [k.sup.2.sub.2] - [k.sup.2.sub.1]/cos Q - 2 sin Q.

Proof.

Assume that the center curve of B - tubular surface [M.sup.B] (s, t) is a unit speed spacelike biharmonic B--slant helix [gamma] and [M.sup.B] denote a patch that parametrizes the envelope of the spheres defining the tubular surface.

Thus, it is seen that

[M.sup.B] (s, t) = [gamma] (s) + a (s, t)t (s) + b (s, t) [m.sub.1] (s) + c (s, t) [m.sub.2] (s), (4.2)

where a , b and c are differentiable on the interval on which [gamma] is defined.

So, without loss of generality, we take the axis of [parallel] is parallel to the spacelike vector [e.sub.1] . Then,

g([m.sub.2], [e.sub.1]) = [m.sup.1.sub.2] = cos Q, (4.3)

where Q is constant angle.

On the other hand, the vector [m.sub.2] is a unit spacelike vector, we reach

[m.sub.2] = cos Q [e.sub.1] + sin Q cosh A (s )[e.sub.2] + sin Q sinh A (s) [e.sub.3]. (4.4)

On the other hand, using Bishop formulas Equation (3.2) and Equation (2.1), we have

[m.sub.1] = sinh A(s)[e.sub.2] + cosh A(s)[e.sub.3]. (4.5)

It is apparent that

t = sin Q[e.sub.1] + cos Q cosh A(s)[e.sub.2] + cos Q sinh A(s) [e.sub.3]. (4.6)

A straightforward computation shows that

[[nabla].sub.t]t = ([t.sub.1])[e.sub.1] + ([t.sub.2] + 2[t.sub.1][t.sub.3] )[e.sub.2] + ([t.sub.3] + 2[t.sub.1][t.sub.2])[e.sub.3] .

Therefore, we use Bishop formulas Equation (3.2) and above equation we get

A(s) = [[square root of [k.sup.2.sub.2] - [k.sup.2.sub.1]/cos Q] -2sin Q]s + [C.sub.1], (4.7)

where [C.sub.1] is a constant of integration.

From Equation(4.6), we get

t = (cos Q sinh [[C.sub.0] s + [C.sub.1] ], cos Q cosh [[C.sub.0]s + [C.sub.1]], (4.8) sin Q - x cos Q cosh [[C.sub.0] s + [C.sub.1]]). (4.8)

Since,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

On the other hand, using definition of tubular surface, we have

g ([M.sup.B] (s, t) - [gamma](s), [M.sup.B] (s, t) = [gamma] (s)) [r.sup.2] (s) (4.9)

Since [M.sup.B](s,t) - [gamma](s) is a normal vector to the tubular surface, we get

g ([M.sup.B] (s, t (-[gamma](s ), [M.sup.B.sub.s] (s, t))= 0. (4.10)

Using now Equation (4.9) and Equation (4.10), we get

[a.sup.2] - [b.sup.2]s) + [C.sup.2](s) = [r.sup.2](s), a(s) [a.sub.s](s)-b(s)[b.sub.s](s) + c(s)[c.sub.s](s)= 0. (4.11)

When we differentiate Equation (4.2) with respect to s and use the Bishop formulas, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4.12)

Combining Equation (4.9) and Equation (4.10) we have

a(s) = - r (s )r'(s) (4.13)

A further computation gives

- [b.sup.2] (s) + [c.sup.2] (s)= [r.sup.2] (s)(1 - [(r(s)).sup.2]) (4.14)

The solution of Equation (4.14) can be written in the following form

b(s) = r sinh t, c (s) = r cosh t. (4.15)

Thus Equation (4.2) becomes

[M.sup.B] (s, t) = [gamma](s )+ r sinh [tm.sub.1] (s)+ r cosh [tm.sub.2] (s) (4.16)

Therefore, by Equations.(4.4)-(4.6) and taking into account Equation(4.16), we obtain the system Equation(4.1). This completes the proof.

Theorem 4.2.

Let [gamma]. I [right arrow] [H.sup.3] be a unit speed spacelike biharmonic new type B--slant helix with non-zero Bishop curvatures. Then, the parametric equations of B--tubular surface [M.sup.B] (s, t) are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sub.0], [C.sub.1], [C.sub.2], [C.sub.3] are constants of integration and

C0 = [square root of [k.sup.2.sub.2] - [k.sup.2.sub.1/cosQ] - 2 sin Q

Proof.

Using Equation (2.1) and Equation (4.1), we have Equation (4.17). Thus proof is complete.

The obtained parametric equations for Equation (4.17) is illustrated in Figures 1, 2 and 3 with helping the programme of Mathematica as follow:

Corollary 4.3.

Let [gamma] : I [right arrow] [H.sup.3] be a unit speed biharmonic spacelike new type B--slant helix with non-zero Bishop curvatures. Then the equations of [gamma] are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [C.sub.0], [C.sub.1], [C.sub.2], [C.sub.3] are constants of integration and

[C.sub.0] = [square root of [k.sup.2.sub.2] - [k.sup.2.sub.1]/cos Q] - 2 sin Q.

If we use Mathematica in above system, we get (Figure 4):

Conclusion

A tubular surface associated to a curve is a surface swept by a family of spheres of constant radius (which will be the radius of the tube), having the center on the given curve. Alternatively, as we shall see in the next section, for them we can construct quite easily a parameterization using the Frenet frame associate to the curve.

In this work, B -tubular surfaces in the Lorentzian Heisenberg group [H.sup.3] are characterized.

Doi: 10.4025/actascitechnol.v37i1.16670

Acknowledgements

The author wish to thank the referees for providing constructive comments and valuable suggestions.

References

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TURHAN, E.; KORPINAR, T. On Characterization of timelike horizontal biharmonic curves in the Lorentzian Heisenberg group Heis3. Zeitschrift fur Naturforschung A- A Journal of Physical Sciences, v. 65, n. a, p. 641-648, 2010.

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Received on April 4, 2012.

Accepted on May II, 2012.

Talat Korpinar

Department of Mathematics, University of Mus Alparslan, 49250, Mu, Turkey. E-mail: talatkorpinar@gmail.com
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