Ruled surfaces with lightlike ruling in 3-Minkowski space

Journal of Geometry and Physics 59 (2009) 74–78

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Ruled surfaces with lightlike ruling in 3-Minkowski space Huili Liu ∗ Department of Mathematics, Northeastern University, Shenyang 110004, PR China

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Article history: Received 24 January 2008 Received in revised form 6 August 2008 Accepted 12 October 2008 Available online 19 October 2008

In this paper we discuss ruled surfaces with lightlike ruling in 3-Minkowski space and give some characterizations and examples of so called B-scroll surfaces. © 2008 Elsevier B.V. All rights reserved.

Dedicated to Professor Udo Simon on the occasion of his 70th birthday MSC: 53B30 53A05 Keywords: Ruled surface B-scroll Associated curve Associated surface

L.K. Graves introduced the surfaces in 3-Minkowski space E31 called B-scrolls ([6]; see also [1]). A ruled surface x(u, v) = a(u) + v b(u) in 3-Minkowski space E31 is called a B-scroll if a(u) and b(u) satisfy ha0 (u), a0 (u)i = hb(u), b(u)i = 0, ha0 (u), b(u)i = 1 and

 α 0 (u) = λ(u)β(u), β 0 (u) = −µα(u) − λ(u)γ (u), γ 0 (u) = µβ(u),

(1)

where a0 (u) = α(u), γ (u) = b(u), β(u) = γ (u) × α(u), hβ(u), β(u)i = 1, µ is constant, h, i is the inner product and × the vector product in E31 . B-scrolls are included in many classification results of some special ruled surfaces etc. ([1,2,4,5,7,8], etc.). However, there are very few conclusions on the properties of these B-scroll surfaces. From [9] we know that any cone curve with cone Frenet frames satisfies condition (1). Therefore, the B-scrolls form a much larger class of ruled surfaces in 3Minkowski space E31 and it is meaningful to study the properties of B-scrolls. In this paper we give some characterizations and also some examples of these B-scroll surfaces. For example in Theorem 3 we present a very simple necessary and sufficient condition for a surface to be a B-scroll. Let E31 be 3-Minkowski space with the inner product

hx, yi = x1 y1 + x2 y2 − x3 y3

(2)

and the vector product

 x x × y = 2 y2 ∗



x3 x3 , y3 y3





x x1 , − 1 y1 y1



x2 , y2

Tel.: +86 24 83680913; fax: +86 24 83680913. E-mail address: [email protected].

0393-0440/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.geomphys.2008.10.003

(3)

H. Liu / Journal of Geometry and Physics 59 (2009) 74–78

75

where x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ) ∈ E31 . For any a, b, c and d ∈ E31 , we have the Lagrange formula

ha × b, c × di = − (ha, c ihb, di − ha, dihb, c i).

(4)

Let x(u, v) = a(u) + v b(u) be a ruled surface with lightlike ruling in 3-Minkowski space b(u) ∈ E31 , hb(u), b(u)i = hb(u), b0 (u)i = 0, u ∈ I, I is an open interval, −∞ < v < ∞. We define xu =

∂ x(u,v) ∂u

∂ 2 x(u,v) ∂ u∂v

and xuv =

E31 .

This means that a(u),

etc. From

 xu = a0 (u) + v b0 (u)     x v = b ( u) 0 0 xuu = a0 (u) + v b0 (u)  0   xuv = b (u) xvv = 0

(5)

we have the first fundamental form of x(u, v) I = hdx, dxi = Edu2 + 2F dudv + Gdv 2 , where

  2  E = a0 (u) + v b0 (u) = ha0 (u) + v b0 (u), a0 (u) + v b0 (u)i,   0 F = ha (u), b(u)i, G=0   p, p  D = |EG − F 2 | = ha0 (u), b(u)i2 = |ha0 (u), b(u)i|.

(6)

When ha0 (u), b(u)i = 0 the surface x(u, v) is degenerate. When ha0 (u), b(u)i 6= 0 the surface x(u, v) is nondegenerate and timelike. In the following we consider only nondegenerate surfaces; then ha0 (u), b(u)i 6= 0. Putting n = n(u, v) =

xu × xv |xu × xv |

we know that the vector field n(u, v) is spacelike and the normal vector field of x(u, v). We have also the second fundamental form of x(u, v)



II = hd2 x, ni = d2 x,



xu × xv = Ldu2 + 2Mdudv + Ndv 2 , |xu × xv |

where

 L = D−1 (a0 (u) + v b0 (u), b(u), a0 0 (u) + v b0 0 (u)), M = D−1 (a0 (u), b(u), b0 (u)),  N = 0.

(7)

Then the Gauss curvature K of x(u, v) is K =

LN − M 2

|EG −

F2

|

=

−(a0 (u), b(u), b0 (u))2

(8)

D4

and the mean curvature H of x(u, v) is H =

EN − 2FM + GL 2(|EG − F 2 |)

=

−ha0 (u), b(u)i(a0 (u), b(u), b0 (u)) D3

.

(9)

For a timelike surface x(u, v) = a(u) + v b(u) with lightlike ruling b(u) in E31 , the striction line of x(u, v) is given by x(u) = a(u) −



 ha0 (u), b0 (u)i b(u). hb0 (u), b0 (u)i

(10)

For the timelike surface x(u, v) = a(u) + v b(u) with lightlike ruling b(u) in E31 , putting

(a0 (u) + v b0 (u))2 Dα(u, v) = a (u) + v b (u) − 2ha0 (u), b(u)i  0  0 (a (u) + v b (u))2 = xu − xv , 2ha0 (u), b(u)i 0

0





b(u) (11)

76

H. Liu / Journal of Geometry and Physics 59 (2009) 74–78

we have hα(u, v), α(u, v)i = 0, hα(u, v), b(u)i = ±1. If we put

Z 

A(u) = ε

a0 (u) −

a0 (u)2







b(u) du,

(12)

 −(a0 (u) + v b0 (u))2 , A(u) = x(u, v)|dv : du = 2ha0 (u), b(u)i

(13)

2ha0 (u), b(u)i

where ε = signha0 (u), b(u)i = ±1, or



we have

( 0 hA (u), A0 (u)i = 0, hb(u), b(u)i = 0, hA0 (u), b(u)i = 1.

(14)

Therefore the surface x(u, v) = A(u) + v b(u)

(15)

is a null-scroll. Theorem 1. Any ruled surface x(u, v) = a(u) + v b(u) with lightlike ruling b(u) in 3-Minkowski space E31 can be written as a null-scroll such that ha0 (u), a0 (u)i = hb(u), b(u)i = 0, ha0 (u), b(u)i = 1. Let x(u, v) = a(u) + v b(u) be a null-scroll in E31 ; by a parameter transformation (u, v) → (u, ha0 (u), b(u)iv) we can assume that ha0 (u), b(u)i = 1. Putting α(u) = a0 (u), γ (u) = b(u) and β(u) = γ (u) × α(u) we have hα, αi = hγ , γ i = hα, βi = hγ , βi = 0 and hβ, βi = hα, γ i = 1. Assuming that α 0 (u) = λ(u)β(u) and γ 0 (u) = µ(u)β(u) we have

 0 a (u) = α(u),   0 α (u) = λ(u)β(u), 0  β 0(u) = −µ(u)α(u) − λ(u)γ (u), γ (u) = µ(u)β(u).

(16)

By the definition, if λ(u) 6= 0 and µ(u) = constant, x(u, v) is called a B-scroll ([2,5,6]). For the B-scroll, from (8) and (9) we know that the Gauss curvature K and the mean curvature H are constant. du We define a new parameter u1 = u1 (u) satisfying c du1 = µ(u), where c 6= 0 is constant. Then (16) can be written as

 da(u1 ) du c c da(u1 ) 0   a˙ (u1 ) = du1 = du du1 = a (u) µ(u) = µ(u1 ) α(u1 ),    dα(u1 ) λ(u1 )   α( ˙ u1 ) = =c β(u1 ), du1 µ(u1 ) dβ(u1 ) λ(u1 )   ˙ u1 ) = = −c α(u1 ) − c γ (u1 ), β(   du µ( u1 ) 1    γ˙ (u ) = dγ (u1 ) = c β(u ). 1 1

(17)

du1

That means that, if we assume that a˜ (u1 ) =

1

u1

Z

c

µ(u)

u0

da(u) du

du,

the surface x(u1 , v) = a˜ (u1 ) + v b(u1 ) is a B-scroll. Theorem 2. Let x(u, v) = a(u) + v b(u) be a null-scroll in E31 and µ(u) = ±|b0 (u)|. For any nonzero constant c, assume that

Z  1    u1 =

u

µ(u)du, Z u1 1 da(u)   µ(u) du. a˜ (u1 ) = c

u0

c

u0

(18)

du

Then the surface x(u1 , v) = a˜ (u1 ) + v b(u1 ) is a B-scroll.

(19)

H. Liu / Journal of Geometry and Physics 59 (2009) 74–78

77

Theorem 3. Let x(u, v) = a(u) + v b(u) be a null-scroll in E31 . The surface x(u, v) is a B-scroll if and only if the parameter u of x(u, v) is an affine arc length parameter of the ruling b(u) as a cone curve. Proof. From Theorem 2, when µ(u) = |b0 (u)| = c we have a˜ (u) = a(u) − a(u0 ). The surface x(u, v) is a B-scroll. By |b0 (u)| = c we know that u is an affine arc length parameter of the cone curve b(u). Inversely, for the null-scroll x(u, v) = a(u) + v b(u) in E31 , if u is an affine arc length parameter of b(u) we may assume that |b0 (u)| = 1. Then putting α(u) = a0 (u), β(u) = b0 (u), γ (u) = b(u) we have (cf. [9], Section 2, (2.1))

 0 a (u) = α(u),   0 α (u) = λ(u)β(u), 0  β 0(u) = −α(u) − λ(u)γ (u), γ (u) = β(u). Therefore x(u, v) is a B-scroll.

(20)



Proposition 1. For any null curve a(u) in E31 with Frenet frame {α(u), β(u), γ (u)} such that α(u) = a0 (u), hγ (u), γ (u)i = 0, hα(u), γ (u)i = 1, β(u) = γ (u) × α(u), if u is an affine arc length parameter of b(u) = γ (u) as a cone curve, the surface x(u, v) = a(u) + v b(u) is a B-scroll. Proof. It is easy to get this proposition from Theorem 3.



Proposition 2. For any continuous function λ(u), by a translation if necessary we can find a B-scroll x(u, v) = a(u) + v b(u) in E31 such that α(u) = a0 (u), γ (u) = b(u), β(u) = γ (u) × α(u), and

 α 0 (u) = λ(u)β(u), β 0 (u) = −α(u) − λ(u)γ (u), γ 0 (u) = β(u).

(21)

Proof. For any continuous function λ(u), from the fundamental theorem of the cone curve (cf. [9,3]) we know that there are cone curves α(u), γ (u) (called a cone curve pair in [9]) such that

 α 0 (u) = λ(u)β(u), β 0 (u) = −α(u) − λ(u)γ (u), γ 0 (u) = β(u),

(22)

where β(u) = γ (u) × α(u) and u is the arc length parameter of γ (u). Then putting b(u) = γ (u), a0 (u) = α(u), the surface x(u, v) = a(u) + v b(u) is a B-scroll.  Proposition 3. For the function λ(u), and the vector fields α(u), β(u), γ (u) as in Proposition 2, putting du˜ = λ(u)du, then the surface x˜ (˜u, v) = a˜ (˜u) + v b˜ (˜u), a˜ 0 (˜u) = γ (˜u), b˜ (˜u) = α(˜u), is a B-scroll. Proof. From [9], Section 2, (2.4), we know that the parameter u˜ defined by du˜ = λ(u)du is the arc length parameter of cone curve α(u). Therefore by Proposition 1 we have Proposition 3.  Definition 1. We call the surface x˜ (˜u, v) the associated surface or associated B-scroll of the surface (B-scroll) x(u, v). The surfaces x(u, v) and x˜ (˜u, v) are called the B-scroll pair. From [9], by a long calculation we have the following examples. Example 1. The curve b(s) =

s c2



1,

1 2

1



(s − s ), (s + s ) , c

−c

c

−c

(23)

2

for any constant c 6= 0, ±1, is a cone curve with the arc length parameter s. b (s) = 0

1 c2



1,

1 2


1
c −c (1 + c )s − (1 − c )s , (1 + c )s + (1 − c )s . c

−c

2

(24)

The cone curve a0 (s) a0 (s) = (X (s), Y (s), Z (s)),
1 X ( s) = c2 − 1 , 2s

(25)

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H. Liu / Journal of Geometry and Physics 59 (2009) 74–78

(1 + c )2

Y (s) = − Z ( s) = −

4

(1 + c )2 4

s c −1 + s c −1 −

(1 − c )2 4

(1 − c )2 4

s−c −1 , s−c −1 ,

is the associate curve of b(s). Then the surface x(s, v) = a(s) + v b(s) is a B-scroll. By a translation we can take a(s) =



1 2

(c 2 − 1) log s, −

(1 + c )2 4c

sc −

(1 − c )2 4c

s−c , −

(1 + c )2 4c

sc +

(1 − c )2 4c



s −c .

(26)

Example 2. The curve b(s) =



s c2

1 2

 (c − log s), c log s, (c + log s) , 2

1

2

2

2

(27)

2

for any constant c 6= 0 is a cone curve with the arc length parameter s. b0 ( s) =



1 c2

1 2

 1 (c 2 − log2 s) − log s, c log s + c , (c 2 + log2 s) + log s .

(28)

2

The cone curve a0 (s) a0 (s) = (X (s), Y (s), Z (s)), X (s) =

Y (s) = − Z ( s) =

(29)

4 + 4 log s − c 2 + log2 s 4s c (2 + log s)

=

(2 + log s)2 − c 2 4s

,

,

2s −4 − 4 log s − c 2 − log2 s 4s

=

−(2 + log s)2 − c 2 4s

,

is the associate curve of b(s). Then the surface x(s, v) = a(s) + v b(s) is a B-scroll. By a translation we can take a(s) =



1 12

(2 + log s) − 3

c2 4

c

1

c2



log s, − (2 + log s) , − (2 + log s) − log s . 4 12 4 2

3

(30)

Remark. From Proposition 3, by a calculation we can also get the associated B-scroll for the surfaces given in Examples 1 and 2. Acknowledgements The author thanks the referee for valuable comments and suggestions. The author was partially supported by NSFC; Joint Research of NSFC and KOSEF; Chern Institute of Mathematics; NEU. References [1] [2] [3] [4] [5] [6] [7] [8] [9]

L.J. Alías, A. Ferrández, P. Lucas, 2-type surfaces in S31 and H31 , Tokyo J. Math. 17 (1994) 447–454. L.J. Alías, A. Ferrández, P. Lucas, M.A. Meroño, On the Gauss map of B-scrolls, Tsukuba J. Math. 22 (1998) 371–377. W.B. Bonnor, Null curves in a Minkowski spacetime, Tensor, N. S. 20 (1969) 229–242. A. Ferrández, P. Lucas, On surfaces in the 3-dimensional Lorentz–Minkowski space, Pacific J. Math. 152 (1992) 93–100. A. Ferrández, P. Lucas, On the Gauss map of B-scrolls in 3-dimensional Lorentzian space forms, Czechoslovak Math. J. 50 (2000) 699–704. L.K. Graves, Codimension one isometric immersions between Lorentz spaces, Trans. Amer. Math. Soc. 252 (1979) 367–392. Dong-Soo Kim, Young Ho Kim, B-scrolls with non-diagonalizable shape operators, Rocky Mountain J. Math. 33 (1) (2003) 175–190. Dong-Soo Kim, Young Ho Kim, Dae Won Yoon, Extended B-scrolls and their Gauss maps, Indian J. Pure Appl. Math. 33 (7) (2002) 1031–1040. Huili Liu, Curves in the lightlike cone, Contributions Algebra Geom. 45 (2004) 291–303.