ON THE GENERALIZED B-SCROLLS WITH P th DEGREE IN n-DIMENSIONAL MINKOWSKI SPACES AND STRICTION (CENTRAL) SPACES

15

ON THE GENERALIZED B-SCROLLS WITH P th DEGREE IN n-DIMENSIONAL MINKOWSKI SPACES AND

STRICTION (CENTRAL) SPACES

Şeyda KILIÇOĞLU

Başkent Üniversitesi Eğitim Fakültesi, ANKARA E-mail:[email protected]

ABSTRACT

In this paper, generalized b-scrolls with p^th degree are introduced in the n- dimensional Minkowski space

R

₁ⁿ. Asymptotic bundle and tangential bundle are defined. In the case of space-like or time-like Frenet vectors , the equation of central space is computed.

Keywords: B-scroll, time-like, ruled surfaces, central spaces.

n-BOYUTLU MİNKOWSKİ UZAYINDA P. DERECEDEN GENELLEŞTİRLMİŞ B-SCROLLAR VE

STRİKSİYON(MERKEZ) UZAYLAR

ÖZET

Bu çalışmada, n-boyutlu Minkowski uzayında, p.mertebeden b-scrollar tanımlandı.

Asimptotik ve teğetsel demetler yardımı ile Frenet vektörlerinin space-like veya time- like olması durumlarında oluşan merkez uzayın denklemi ifade edildi.

Anahtar Kelimeler: B-scroll, time-like, regle yüzeyler, merkez uzaylar 1. INTRODUCTION

First of all b-scrolls were introduced in the 3-dimensional Minkowski space

3

,

R

1 [1] and [2]. For an integer q with 0 < q< n , changing the first plus signs above to minus gives a metric tensor

16

1 1

, =

q n

i i j j

p p

i j q

v w v w v w

of index q. The resulting semi-Euclidean space

R

_qⁿ reduces to

R

ⁿ if q=0.

For n> 2,

R

₁ⁿ is called Minkowski n-space ,[3] . In the n-dimensional Minkowski space

R

₁ⁿ, lorentz metric is

1 1 2

, =

n j j

p p

j

v w v w v w

In the n-dimensional semi-euclidean space Rⁿ_q

,

if the Frenet vectors of curve

)

(I

with arc length t are

V

₁

, V

₂

, ..., V

_r, the Frenet formulas can be given by the following equations

.

=

=

=

1 1 1 2

1 1

1 1 2 2 1 1

r r r r r

j j j j j j j

V k V

V k V k V

V k V











Here _{i 1}

= V

_i

, V

_i

and i r for k

_i

0

, [4] and [5].

In the n- dimensional Minkowski space, since the index q is

1

, only one of the _{i 1}

= V

_i

, V

_i

, 1 < i < r ,

will take the value

1

. Here, since

(I )

is time-like curve, then

V

₁ is a time-like vector. Hence, only ₀

= 1

. As

V

r

V

V

₂

,

₃

,...,

are space-like, then ₁

=

₂

=

₃

= ... =

_{r 1}

= 1

.

If

V

₁ is a time-like vector , then the Frenet formulas can be given by the following matrix form ,

17

r 1 r

2 r 4 3 2 1

1 r

1 r 2

r

2 r 3

3 2

2 1

2

r 1 r

2 r 4 3 2 1

V V V V V V V

0 k 0

0

k 0

k

0 k

0 0

k 0

0

k 0 k 0

0 k 0 k

0 0

0 k 0

=

V V V V V V V









































Similary, if

V

₂ is a time-like vector, then

r 1 r

2 r 4 3 2 1

1 r

1 r 2

r

2 r 3

3 2

2 1

2

r 1 r

2 r 4 3 2 1

V V V V V V V

0 k 0

0

k 0

k

0 k

0 0

k 0 0

k 0 k 0

0 k 0 k

0 0

0 k 0

=

V V V V V V V









































is the matrix form of the Frenet formulas. Similary, for each time-like vector

V

i, matrix form of the Frenet formulas can be obtained.

Definition 1. In the

n

dimensional Minkowski space R₁ⁿ ,

(I )

is a time- like curve with arc length t. If the Frenet vectors are

V

₁

, V

₂

,..., V

_r , then

n r p V V V

Sp

₁

,

₂

,...,

_p

; <

is the time-like oskulator space with p th degree. In this case,

) t ( V u )

t (

= ) u ,..., u , u , t

(

_{j j}

r

1 p

= j r

2 p 1 p

is the parametrization of generalized

b

scroll with

p

^th degree.The directrix

18

of this generalized

b

scroll with

p

^th degree, is the time-like curve

(I )

. That is

 ( t ) = V

₁ is a time-like vector. The space-like generating space of generalized

b

scroll with

p

^th degree has span with subvectors

r 2 p 1

p

, V ,..., V V

Since this generating space is

r p

dimensional , it can be shown by

E

_{r p} . The dimension of this special surface b-scroll is

r p 1

.

Figure 1: The generalized b-scrolls with p^th degree.

Let M be this surface whose ordered basis tangent vectors at the point

(t )

are given as follows:

19

.

=

=

=

) (

= ) ( )

(

=

2 2 1 1

1

= 1 1

=

r r u

p p u

p p u

j j r

p j j

j r

p j t

V V V

t V u V

t V u t





 

Definition 2. In the

n

dimensional Minkowski space R₁ⁿ, the asymptotic bundle, [6], of generalized

b

scroll with

p

^th degree, is denoted by

r 2 p 1 p r 2 p 1

p

, V ,..., V , V , V ,..., V V

Sp

= ) t (

A   

. Since







3 p 2 p 1 p 1 p 2

p

2 p 1 p p p 1

p

V k V k

= V

V k V k

= V

Then only the vector

V

_{p 1} is linearly independent from vectors

r p

p

V V

V

₁

,

₂

,...,

. On the other hand, the vectors

V 

_{p 2}

,..., V 

_r are dependent on the vectors

V

_{p 1}

, V

_{p 2}

,..., V

_r . All these vectors are space-like vectors.

r 2 p 1 p

p

, V , V ,..., V V

is an orthonormal basis of

A (t )

and dim

A ( t ) = r p 1

. The asymptotic bundle

A (t )

is space-like because, unique time-like vector

V

₁ of Frenet vectors is not an element of

A (t )

.

Definition 3. In the

n

dimensional Minkowski space R₁ⁿ, denote the tangential bundle ,[6], of the generalized

b

scroll with

p

^thdegree, by

 



 , V ,..., V , V

, V ,..., V , V Sp

= ) t (

T

_{p 1 p 2 r p 1 p 2 r}

Since

20







3 p 2 p 1 p 1 p 2

p

2 p 1 p p p 1

p

V k V k

= V

V k V k

= V

only the two vectors

 = V

₁ and

V

_{p 1} are independent from vectors

r p

p

V V

V

₁

,

₂

,...,

. The vectors

V 

_{p 2}

,..., V 

_r are dependent on the vectors

r p

p

V V

V

₁

,

₂

,...,

. The vectors

V

_p1

, V

_{p 2}

,..., V

_r are space-like, but

 = V

₁ is time-like.

r 2 p 1 p p

1

, V , V , V ,..., V V

is the orthonormal basis vectors of

T (t )

and dim

T ( t ) = r p 2

.

T (t )

is time-like because, the time-like vector

V

₁ is an element of

T (t )

.

Definition 4. In the

n

dimensional Minkowski space R₁ⁿ, since dim

) (t

A

dim

T (t )

, the generalized

b

scroll with

p

^th degree and with time-like directrix, has not an edge space but , there is a striction (central) space, [6]. The dimension of this striction (central) space is

r p

. Vectors

for

V

_{p i}

, 1 < i ,

are space-like thus we can calculate the striction space as in the Euclidean space. That is, the position vectors of the striction space are the solutions of the differantial equation system , which has the following matrix form:

r 1 r

2 r

3 p

2 p

1 p

1 r

1 r 2

r

2 r 2

p

2 p 1

p

1 p

r 1 r

2 r

3 p

2 p

1 p

u u u u u u

0 k

0 0

k 0 k

0 k

0 0

k 0

k 0 k

0 0

k 0

=

u u u u u u



































(1).

Corollary: In the

n

dimensional Minkowski space R₁ⁿ , if one of the vectors

21

V

p

V V

V

₁

,

₂

,

₃

,...,

is time-like, the position vectors of the striction space of the generalized

b

scroll with

p

^th degree will be the same with the solutions of the equation system which has the matrix form given above (1) .

Definition 5. In the

n

dimensional Minkowski space R₁ⁿ,

(I )

is a space- like curve with arc length t. If

V

₁

, V

₂

,..., V

_r are the Frenet vectors, then

n r p V V V

Sp

₁

,

₂

,...,

_p

; < ,

is the space-like osculator space with p^th degree. In this case,

) t ( V u )

t (

= ) u ,..., u , u , t

(

_{j j}

r

1 p

= j r

2 p 1 p

is the parametrization of generalized

b

scroll with

p

^thdegree.The directrix of this generalized

b

scroll with

p

^thdegree, is the space-like curve

(I )

, that is

 ( t ) = V

₁ a space-like vector .

r 2 p 1 p p

r

= Sp V , V ,..., V E

is the time-like generating space of the generalized

b

scroll with

p

^th

degree. Only one of the vectors

V

_p1

, V

_{p 2}

,..., V

_r is a time-like vector, since the index q is 1.

First of all, let

V

_{p 1} be a time-like vector. It means that

1

= ,

=

_{p 1 p 1}

p

V V

and _{p 1}

= V

_{p 2}

, V

_{p 2}

,...,

_{r 1}

= V

_r

, V

_r

= 1

. According to the definitions of the asymptotic bundle and the tangential bundle of generalized

b

scroll with

p

^th degree,

22











4 p 3 p 2 p 2 p

4 p 3 p 2 p 2 p 2 p 1 p 3

p

3 p 2 p 1 p 1 p

3 p 2 p 1 p 1 p 1 p p 2

p

2 p 1 p p p

2 p 1 p p p p 1 p 1

p

1 p p 1 p 1 p 1 p 2 p p

V k V k

=

V k V k

= V

V k V k

=

V k V k

= V

V k V k

=

V k V k

= V

V k V k

= V

are obtained by using Frenet formulas. If

V

_{p 1}is time-like , then only first terms of vectors

V

_{p 1} and

V

_{p 2} will change their signs. However, other signs will not change.

) (t

p

is any curve family with equation

) t ( V ) t ( u ) t (

= ) t (

p

_{j j}

r

1 p

= j

and it has the derivative

. ...

=

...

...

...

=

=

=

= ) (

1 1 1

1 2 2 1

2 2 1 3 3 2 3

3 4 2 2 3

2 2 3 1 1 2 1 1 2 1 1

1

1 1 1

1 1 2 2 2 3 3

3 2 2 2 1 1 2 2 1 3 3 2

3 3 4 2 2 3 1 1 2 1

1 1 2 2 3

3 2 2 1 1 1

1 1 1 2 1 1

1

= 1 1 1

1

= 1 2 1

= 1

1 1 1 2 1 1 1 1 2 1

1

= 1

= 1

1

= 1

=

r r r r r r r r r r

r r r r r r p

p p p p p

p p p p p p p p p p p p p

r r r r r r r r r r r r

p p p p p p r r r r r r

p p p p p p p p p p p p

r r r r r r p

p p p p p

r r r r r j j j r

p j j j j r

p j j j j j r

p j

r r r r r j j j j j j j r

p j j j r

p j

j j r

p j j j r

p j

V k u u V k u k u u

V k u k u u V

k u k u u

V k u k u u V k u u V k u V

V k u V k u V k u V k u

V k u V k u V k u V k u

V k u V k u V k u V k u

V u V u V u V

u V u V u V

V k u V

k u V

k u V

u V

V k u V

k V k u

V u V

V u V u t

p





























 

 

If there exist a common perpendicular to two constructive rullings in the

23

skew surface , then the foot of common perpendicular on the main rulling is called the central point. The locus of central points is called the striction space,[7].

Under the condition of orthonormalizm , the solution vectors u of the equation

0

= ) t ( V ) t ( dt u

, d ) t (

p

_{i i}

r

1 p

= i



are the position vectors of the striction space. This equation implies that

. 0

= ...

2 1 1 2

1 2

2 1

2 2 1 3 3 2 2

3 4 2 2 3

2 2 3 1 1 2 2

1 2 1 2

1

r r r r

r r r r

r r r r r p

p p p p

p p p p p p

p p p

p

k u u k

u k u u

k u k u u k

u k u u

k u k u u k

u u k

u













If

u

_{p 1}

k

_p

= 0

, then,

u

_{p 1}

0

then

k

_p

= 0

or if

k

_p

0

and

0

1

=

u

p . In the other terms, we can continue on the similiar way. Let assume that all of the curvatures ki be different from zero. In this condition , if

u

_{p 1}

= 0,

we can take

u

_{p 1}

= 0

,

u

_{p 2}

= 0 u 

_{p 2}

= 0 u

_{p 3}

= 0 ...

So, the space-like directrix

(I )

of this generalized

b

scroll with p^th degree, is the striction space. Under the special condition

0

= k u u 

_{p 1 p 2 p 1}

we can solve the differential equation system. Using the equations

1 r 1 r r

2 r 2 r r 1 r 1

r

3 r 3 r 1 r 2 r 2

r

2 p 2 p 4 p 3 p 3

p

1 p 1 p 3 p 2 p 2

p

2 p 1 p 1

p

u k

= u

u k u k

= u

u k u k

= u

u k u k

= u

u k u k

= u

u k

= u















24 we can obtain Lyapunov matrix

r r p p p

r r p

p p

p

r r p p p

u u u u u

k k k

k k

k

u u u u u

1 3 2 1

1 1 2

2 1

1

1 3 2 1

0 0

0 0 0

0

0

0 0

0

= 































That is, the position vectors of the striction space are the solutions of the homogeneous differantial equation

).

( ) (

= )

( t A t U t U

In further studies, it is possible to seek for other solutions , except these special solutions.

Now let

V

_{p 2} be a time-like vector , in the time-like generating space

p

E

r of generalized

b

scroll with

p

^thdegree. It means that

1

= V , V

=

_{p 2 p 2}

1 p

and

1

= V , V

= 1,...,

= V , V

= 1,

= V , V

=

_{p 1 p 1 p 2 p 3 p 3 r 1 r r}

p

are obtained. According to the definitions of the asymptotic bundle and the tangential bundle of generalized

b

scroll with

p

^th degree ,

.

25











5 p 4 p 3 p 3 p

5 p 4 p 3 p 3 p 3 p 2 p 4

p

4 p 3 p 2 p 2 p

4 p 3 p 2 p 2 p 2 p 1 p 3

p

3 p 2 p 1 p 1 p

3 p 2 p 1 p 1 p 1 p p 2

p

2 p 1 p p p

2 p 1 p p p p 1 p 1

p

V k V k

=

V k V k

= V

V k V k

=

V k V k

= V

V k V k

=

V k V k

= V

V k V k

=

V k V k

= V

are obtained by using Frenet formulas. It is obvious that, if

V

_{p 2} is time-like , then only the first terms of vectors

V

_{p 2} and

V

_{p 3} will change their

signatures. The others will not change.

) (t

p

is any curve family with equation

) t ( V ) t ( u ) t (

= ) t (

p

_{j j}

r

1 p

= j

and it has the differantial form

1 1 1 2 1 1 2 1 1 3 2 2

3 2 2 4 3 3 2 3 3 1 2 2

1 2 2 1 1 1 1

( ) =

...

.

p p p p p p p p p p p p p

p p p p p p r r r r r r

r r r r r r r r r r

p t V u k V u u k V u u k u k V

u u k u k V u u k u k V

u u k u k V u u k V

  

 

 

Under the condition of orthonormalism, the solution vectors u of the equation

0

= ) t ( V ) t ( dt u

, d ) t (

p

_{i i}

r

1 p

= i



are the position vectors of the striction curve (space). This equation implies that

26

0

= k u u k

u k u u

k u k u u ...

k u k u u

k u k u u k

u u k u

2 1 r 1 r r 2 1 r r 2 r 2 r 1 r

2 2 r 1 r 3 r 3 r 2 r 2

3 p 4 p 2 p 2 p 3 p

2 2 p 3 p 1 p 1 p 2 p 2 1 p 2 p 1 p 2 p 1 p













If

u

_{p 1}

k

_p

= 0

, then

u

_{p 1}

0

then,

k

_p

= 0

or if

k

_p

0

and

0

1

=

u

p . In the other terms we can continue on the similiar way. Let assume that all of the curvatures kⁱ be different from zero. In this condition , if

u

_{p 1}

= 0,

we can take

u

_{p 1}

= 0

,

u

_{p 2}

= 0 u 

_{p 2}

= 0 u

_{p 3}

= 0 ...

So, the space-like directrix

(I )

of this generalized

b

scroll with p th degree, is the striction space.

Under the special condition

0

= k u k u

u 

_{p 2 p 1 p1 p 3 p 2}

we can solve the differential equation system. Using the equations

1 r 1 r r

2 r 2 r r 1 r 1

r

3 r 3 r 1 r 2 r 2

r

2 p 2 p 4 p 3 p 3

p

3 p 2 p 1 p 1 p 2

p

2 p 1 p 1

p

u k

= u

u k u k

= u

u k u k

= u

u k u k

= u

u k u k

= u

u k

= u















, we can obtain Lyapunov matrix

r r r p p p

r r r

r p

p p

p p

p

r r r p p p

u u u u u u

k k k

k k

k k

k k

k

u u u u u u

1 2 3 2 1

1 1 2

2 3

3 2

2 1

1

1 2 3 2 1

0 0

0

0 0 0

0

0 0

0 0

0 0

= 



































That is, the position vectors of the striction space are the solutions of the

.

27 homogeneous differantial equation

).

( ) (

= )

( t A t U t U

In further studies, it is possible to seek for the other solutions, except these special solutions.

Finally, let

V

_r be the time-like vector of the time-like generating space

E

_{r p} of generalized

b

scroll with p th degree. It means that

1

= V , V

=

_{r r}

1 r

and

1

= V , V

= 1,...,

= V , V

= 1,

= V , V

=

_{p 1 p 1 p 1 p 2 p 2 r 2 r1 r 1}

p

are obtained. According to the definitions of the asymptotic bundle and the tangential bundle of generalized

b

scroll with p th degree,

1 r 1 r

1 r 1 r 1 r 2 r r

r 1 r 2 r 2 r

r 1 r 2 r 2 r 2 r 3 r 1

r

V k

=

V k

= V

V k V k

=

V k V k

= V





are obtained by using Frenet formulas. It is obvious that, if

V

_r is time-like , then only

V

_r will change its signature.The others will not change.

) (t

p

is any curve family with equation

) t ( V ) t ( u ) t (

= ) t (

p

_{j j}

r

1 p

= j

and it has the differential form

. ...

= ) (

1 1 1

1 2 2 1

2 2 1 3 3 2 3

3 4 2 2 3

2 2 3 1 1 2 1 1 2 1 1

1

r r r r r r r r r r

r r r r r r p

p p p p p

p p p p p p p p p p p p p

V k u u V k u k u u

V k u k u u V

k u k u u

V k u k u u V k u u V k u V t p















Under the condition of orthonormalism, the solution vectors u of the equation

28

0

= ) t ( V ) t ( dt u , d ) t (

p _{i i}

r

1 p

= i



are the position vectors of the striction curve (space).This equation implies that

0

= k u u k

u k u u

k u k u u ...

k u k u u

k u k u u k

u u k

u

2 1 r 1 r r 2 1 r r 2 r 2 r 1 r

2 2 r 1 r 3 r 3 r 2 r 2

3 p 4 p 2 p 2 p 3 p

2 2 p 3 p 1 p 1 p 2 p 2 1 p 2 p 1 p 2 p 1 p













. If

u

_{p 1}

k

_p

= 0

, then

u

_{p 1}

0

then

k

_p

= 0

or if

k

_p

0

and

0

1

=

u

p . In the other terms we can continue on the similiar way. Let assume that all of the curvatures ki be different from zero. In this condition , if

u

_{p 1}

= 0,

we can take

u

_{p 1}

= 0

,

u

_{p 2}

= 0 u 

_{p 2}

= 0 u

_{p 3}

= 0 ...

So, the space-like directrix

(I )

of this generalized

b

scroll with p th degree, is the striction space. Under the special condition.

0

= k u u 

_{r r1 r 1}

we can solve the differential equation system.Using the equations

1 r 1 r r

2 r 2 r r 1 r 1

r

1 r 2 r 3 r 3 r 2

r

4 p 3 p 2 p 2 p 3

p

3 p 2 p 1 p 1 p 2

p

2 p 1 p 1

p

u k

= u

u k u k

= u

u k u k

= u

u k u k

= u

u k u k

= u

u k

= u















we can obtain Lyapunov matrix

29

r 1 r

2 r

3 p

2 p

1 p

1 r

1 r 2

r 2 r 3

p

3 p 2

p

2 p 1

p 1 p

r 1 r

2 r

3 p

2 p

1 p

u u u u u u

0 k 0

0

k 0

k

0 k k

k 0

k 0

0 k

0 k

0 0

0 k

0

=

u u u u u u





































. That is, the position vectors of the striction space are the solutions of the homogeneous differantial equation

).

( ) (

= )

( t A t U t U

In further studies, it is possible to seek for the other solutions, except these special solutions.

REFERENCES

[1]. Alias, L.J., Ferrandez, A., Lucas, P. 2-type surfaces in

S

₁³ and

H

₁³ Tokyo J. Math. 17 (1994) 447-454

[2]. Graves, L.K. 1979. Codimension one isometric immersions between Lorentz spaces.Trans. Amer. Math. Soc., 252; 367-392

[3]. O'Neill, B. 1983. Semi-Riemannian geometry with applications to relativity.Academic Press, 468 p., New York.

[4]. Ekmekçi, N. and İlarslan, K. 1998. Higher curvatures of a regular curve in Lorentzian space. Jour. of Inst. of Math & Camp. Sci. (Math. Ser) Vol. 11, No.2; 97-102.

[5]. İlarslan, K. 2002. Öklid olmayan manifoldlar üzerindeki bazı özel eğriler.

Doktora tezi, Ankara Üniversitesi Fen Bilimleri Enstitüsü, 118 s., Ankara [6]. Sabuncuoğlu, A. 1982. Genelleştirilmiş regle yüzeyler. Doçentlik tezi,

Ankara Üniversitesi Fen Bilimleri Enstitüsü, 60 s., Ankara

[7]. Turgut, A. and Hacısalihoğlu, H.H., 1997, Time like ruled surfaces in the Minkowski space. Far East. J. Math. Sci., 5(1); 83-90.