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Erratum to: “Ruled surfaces with time like rulings” [Appl. Math. Comput. 147 (2004) 241–248]
Applied Mathematics and Computation 174 (2006) 1660–1667
www.elsevier.com/locate/amc
Erratum
Erratum to: ‘‘Ruled surfaces with time like rulings’’ [Appl. Math. Comput. 147 (2004) 241–248] _ Emin Kasap *, Ismail Aydemir, Nuri Kuruog˘lu Ondokuz Mayis University, Faculty of Arts and Sciences, Department of Mathematics, 55139 Kurupelit-Samsun, Turkey
Abstract In this article, a new type of ruled surfaces in a Lorentz 3-space R31 is obtained by a strictly connected time-like oriented line moving with FrenetÕs frame along a space-like curve. These surfaces are classified into time-like and space-like surfaces. The wellknown theorems due to Bonnet and Chasles in the 3-dimensional Euclidean space are proved for a time-like ruled surface. Ó 2005 Published by Elsevier Inc.
1. Introduction Definition 1. Let R31 be a 3-dimensional Lorentzian space with the pseudo-met~; ~ ~ and ~ ~ and ~ ric ds2 = dx2 dy2 + dz2. If hX Y i ¼ 0 for all X Y , the vectors X Y are called perpendicular in the sense of Lorentz, where h , i is the induced inner product in R31 [1]. *
DOI of original article: 10.1016/S0096-3003(02)00664-1. Corresponding author. E-mail address: [email protected] (E. Kasap).
0096-3003/$ - see front matter Ó 2005 Published by Elsevier Inc. doi:10.1016/j.amc.2005.07.004
E. Kasap et al. / Appl. Math. Comput. 174 (2006) 1660–1667
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~ 2 R3 is denoted by kX ~k and defined as Definition 2. The norm of X 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~k ¼ jhX ~; X ~ij. kX ~ 2 R3 is called a space-like, time-like and null (light-like) vector if The vector X 1 ~ ~ ~ ¼ 0, hX ~; X ~i < 0, and hX ~; X ~i ¼ 0 for X ~ 6¼ 0, respectively [5]. hX ; X i > 0 or X Definition 3. A regular curve aðsÞ : I ! R31 , I R in R31 is said to be a spacelike, time-like and null curve if the velocity vector ~ a0 ðsÞ ¼ da=ds is a space-like, time-like or null vector respectively [4]. Definition 4. A surface in a 3-dimensional Lorentz space is called a time-like surface if the induced metric on the surface is a Lorentz metric, i.e., the normal on the surface is a space-like vector [7]. As revealed from the foregoing definitions, one can prove the following: Lemma 1. In the Lorentz space R31 , the following properties are satisfied: (i) Two time-like vectors are never orthogonal. (ii) Two null vectors are orthogonal if and only if they are linearly dependent. (iii) A time-like vector is never orthogonal to a null (light-like) vector [3]. Let ~ a ¼~ aðsÞ be a unit speed space-like curve in R31 ; by j(s), s(s) we denote the natural curvature and torsion of ~ aðsÞ respectively. Consider the Frenet frame f~ e1 ;~ e2 ;~ e3 g attached to the space-like curve ~ a ¼~ aðsÞ such that ~ e2 ¼ ~ e2 ðsÞ is the principal normal vector field of type time-like and ~ e3 ¼ ~ e3 ðsÞ is the binormal vector field of type space-like and ~ e1 ¼ ~ e1 ðsÞ is the unit tangent vector field with the normalization [4]. h~ e1 ;~ e1 i ¼ h~ e2 ;~ e2 i ¼ h~ e3 ;~ e3 i ¼ 1; h~ ei ;~ ej i ¼ 0
ð1:1Þ
for i 6¼ j; detð~ e1 ;~ e2 ;~ e3 Þ ¼ 1.
The infinitesimal displacements of the frame are given as ~ e01 ðsÞ ¼ jðsÞ~ e2 ;
~ e02 ðsÞ ¼ jðsÞ~ e1 þ sðsÞ~ e3 ;
~ e03 ðsÞ ¼ sðsÞ~ e2 .
ð1:2Þ
One can easily see the following: Lemma 2. Assume that ~ aðsÞ is a unit speed space-like curve in R31 and f~ e1 ;~ e2 ;~ e3 g be its FrenetÕs frame field. Then ~ e2 ¼ ~ e3 ; e1 ^~
~ e1 ^~ e3 ¼ ~ e2 ;
~ e2 ^~ e3 ¼ ~ e1 .
ð1:3Þ
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2. Ruled surfaces with time-like generators in R31 A time-like straight line ~ L in R31 such that it is strictly connected to FrenetÕs frame of the space-like curve ~ a ¼~ aðsÞ is represented, uniquely with respect to this frame, in the form ~ LðsÞ ¼
3 X
‘i ðsÞ~ ei ðsÞ;
h~ LðsÞ; ~ LðsÞi > 0;
ð2:1Þ
i¼1
where the components ‘i = ‘i(s) (i = 1, 2, 3) are scalar functions of the arc length parameter of the base curve ~ a ¼~ aðsÞ. Without loss of generality the family of ruled surfaces under investigation are characterized by the regular parametrization. M : Uðs; mÞ ¼ ~ aðsÞ þ m~ LðsÞ; ‘21 ‘22 þ ‘23 ¼ 1;
) ðIÞ
0 ~ L ðsÞ 6¼ 0.
Definition 5. If there exists a common perpendicular to two constructive rulings in the skew surface, then the foot of the common perpendicular on the main ruling is called a central point. The locus of the central point is called the striction curve [7]. Using (1.2) and (2.1), it is easy to see that the parametrization of the striction curve on a ruled surface of type (I) is given by ~ bðsÞ ¼ ~ aðsÞ /ðsÞ~ LðsÞ;
where /ðsÞ ¼
‘01 þ ‘2 j . 0 0 h~ L ðsÞ; ~ L ðsÞi
ð2:2Þ
The unit normal vector ~ n on the ruled surface of type (I) is given by 0 ~ LðsÞ þ m~ L ðsÞ ^ ~ LðsÞ a0 ðsÞ ^ ~ ~ n¼ 0 . 0 ~ ~ ~ k~ a ðsÞ ^ LðsÞ þ mL ðsÞ ^ LðsÞk
ð2:3Þ
Thus, from (2.3) and (1.2) the unit normal vector to the surface M at the point (s, o) is ‘2~ e3 þ ‘3~ e2 ~ nðs; oÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ‘22 ‘23
ð2:4Þ
From (2.4), it follows that the base curve of the surface (I) to be a geodesic curve (~ nðs; oÞ ¼ ~ e2 ) if ‘2 = 0, ‘3 5 0. From (2.2) and (2.4) it follows that the BonnetÕs theorem for the ruled surface (I) can be formulated as
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Theorem 1. If a space-like curve on a ruled surface (I) in R31 has two of the following properties, it has the third also (i) It is a geodesic (‘2 = 0 or j = 0). (ii) It cut the rulings at a constant angle (‘1 = const.). (iii) It is a striction curve (/(s) = 0). The induced metric on the ruled surfaces (I) is given by [2] 0 0 g ¼ ð1 þ ‘21 Þ ð2m/ðsÞ þ m2 Þh~ L ;~ L i.
Therefore, one can give a classification to the ruled surfaces (I) into time-like or space-like ruled surfaces as follows: The class of ruled surfaces (I) is of type time-like (g < 0) if one of the following conditions: (i) m and /(s) have the same sign, (ii) /(s) = 0, (iii) m and /(s) have opposite sign and 0 0 ð1 þ ‘21 Þ þ ðm2 2m/ðsÞÞh~ L ðsÞ; ~ L ðsÞi > 0
is satisfied. The ruled surfaces (I) are space-like ruled surfaces if and only if m and /(s) have opposite sign and 0 0 ð1 þ ‘21 Þ þ ðm2 2m/ðsÞÞh~ L ðsÞ; ~ L ðsÞi < 0.
Theorem 2. The class of time-like ruled surfaces is divided into three subclasses according to the conditions (i), (ii) and (iii) respectively. 2.1. Time-like ruled surface in R31 Here, we study a subclass of time-like ruled surfaces which is characterized by the condition (ii). For this type, the striction curve is the base curve of a ruled surface of type (I), i.e., M t : Uðs; mÞ ¼ ~ aðsÞ þ m~ LðsÞ; ‘21 ‘22 þ ‘23 ¼ 1;
0 k~ L ðsÞk 6¼ 0;
‘01 þ ‘2 j ¼ 0.
) ðIIÞ
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The distribution parameter k(s) of the time-like ruled surface Mt is defined as kðsÞ ¼
0 ½~ a0 ðsÞ; ~ LðsÞ; ~ L ðsÞ 0 0 h~ L ðsÞ; ~ L ðsÞi
and from (1.2) and (2.1) we have kðsÞ ¼
‘2 ‘03 ‘3 ‘02 þ ð1 þ ‘21 Þs ‘1 ‘3 j . 0 0 h~ L ðsÞ; ~ L ðsÞi
ð2:5Þ
From (2.3) and (II) it is easy to see that, the unit normal vector to the time-like ruled surface Mt at (s, m) and (s, o) are ~ nðs; mÞ ¼
k~ lðsÞ þ m~ lðsÞ ^ ~ LðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 k þm
~ lðsÞ ¼ no ¼ ~
0 ~ L ðsÞ . 0 k~ L ðsÞk
ð2:6Þ
From Definition 4 and Lemma 1, one can see that ~ n and ~ no are unit space-like vectors. The angle h of rotation from the normal ~ no to the normal ~ n is given from m sin h ¼ k~ no ^ ~ nk ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 k þ m2 Hence Chasles theorem is valid for a time-like ruled surface Mt in R31 . As an immediate result we have the following: Corollary 1. The tangent plane turns evidently through 180° along a ruling in a non-developable (k 5 0) time-like ruled surface Mt. The Gaussian curvature K of the surface in R31 is given by K = (ehab)/gab, where e ¼ h~ n;~ ni and gab, hab are the first and second fundamental quantities, respectively [6]. After simple calculation, one can see that the Gaussian curvature of the time-like ruled surface Mt is K¼
k2 2
ðk2 þ m2 Þ
.
ð2:7Þ
Thus we have the following: Theorem 3. The Gaussian curvature K of a time-like ruled surface Mt in R31 is non negative and K is equal to zero only along the ruling which meet the striction curve at a singular point (k = 0, m 5 0). Eq. (2.7) allows us to give a geometric interpretation of the (regular) central points of a time-like ruled surface. Indeed, the points of a ruling except perhaps the central point, are regular points of the surface if k 5 0. The function K(s, m) is a continuous function on the ruling (s = const.) and the central point is characterized by the fact that K(s, m) has a maximum there.
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2.2. Time-like ruled surfaces with constant parameter of distribution From (2.5), one can see that a time-like ruled surface Mt with a constant distribution parameter satisfies the following differential equation: 0 0 L ðsÞ; ~ L ðsÞi; ‘2 ‘03 ‘3 ‘02 þ ð1 þ ‘21 Þs ‘1 ‘3 j ¼ Ch~
where C is constant. Using (II), one can see that R 9 ‘1 ¼ ‘2 j ds þ c1 ; > > > = R Ch~L0 ðsÞ;~L0 ðsÞi ‘3 ¼ ‘2 s ds þ c ; ‘ ¼ 6 ‘ ; 2 2 3 2 1þl1 > > > ; 2 2 2 ‘1 ‘2 þ ‘3 ¼ 1.
/ðsÞ ¼ 0;
ð2:8Þ
The second equation of the triple (2.8) is an integral equation for the unknown ‘3 = ‘3(S). Therefore, if we given ‘2 = ‘2(s) we get ‘1 = ‘1(s) and ‘3 = ‘3(s). Thus, we have the existence theorem: Theorem 4. The range of existence of a one parametric time-like ruled surfaces {Mt} with constant parameter of distribution comprises within one arbitrary function of one variable. Developable time-like ruled surface are special class of the ruled surfaces described by Theorem 4 (k = c = 0). If a point P be displaced orthogonally along ~ L to a neighbouring point Po, we have an orthogonal trajectory. The condition that the point P be displaced orthogonally to the ruling is hd~ c=ds, ~ Li ¼ 0. One can see that m¼
Z ‘1 ds.
ð2:9Þ
From (2.9) and Theorem 1, for the time-like ruled surface Mt, one can give an interpretation to the parameters m, ‘1, as Theorem 5. For a time-like ruled surface Mt in R31 . If the base curve is geodesic, then the distance m along a ruling from the base curve ~ a ¼~ aðsÞ to an orthogonal trajectory is proportional to the arc length of the base curve. 2.3. The geodesic and normal curvature Let ~ c ¼~ cðsÞ be a curve on the time-like ruled surface Mt, then it can be represented in the form ~ cðsÞ ¼ ~ aðsÞ þ mðsÞ~ LðsÞ.
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Using (1.2), it is easy to see that the unit tangent vector along the curve ~ c ¼~ cðsÞ is ~ c0 ðsÞ ¼
n1~ e1 þ n2~ e2 þ n3~ e3 pffiffiffi ; R
where n1 ¼ 1 þ m0 ‘1 ;
n2 ¼ mu1 þ m0 ‘2 ;
u1 ¼ ‘1 j þ ‘02 þ ‘3 s;
n3 ¼ mu2 þ m0 ‘3 ;
u2 ¼ ‘2 s þ ‘03 ;
R ¼ jn21 n22 þ n23 j. Therefore, one can see that ~ e1 þ ðn02 þ n1 j þ n3 sÞ~ e2 þ ðn03 þ n2 sÞ~ e3 R c00 ðsÞ ¼ R3=2 ð½ðn01 þ n2 jÞ~
R0 ðn ~ e1 þ n2~ e2 þ n3~ eÞÞ. 2 1
Using (2.6), the unit normal vector field on the time-like ruled surface Mt along the curve (m = m(s)) is mð‘3 u1 ‘2 u2 Þ~ e1 þ ð‘3 m‘1 u2 Þ~ e2 þ ð‘2 m‘1 u1 Þ~ e3 ffi. ~ nðs; mðsÞÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 m2 ð‘3 u1 ‘2 u2 Þ þ ð‘3 m‘1 u2 Þ þ ð‘2 m‘1 u1 Þ The geodesic curvature of the curve ~ c ¼~ cðsÞ is given by n1 n2 n3 1 kg ¼ n02 þ n1 j þ n3 s n03 þ n2 s . n01 þ n2 j Rk~ nðs; mðsÞÞk mð‘3 u ‘2 u Þ ‘3 m‘1 u2 ‘2 m‘1 u1 1 2 ð2:10Þ Then, the differential equation of the geodesic curves on the time-like ruled surface Mt is given by 2
f ðsÞm00 þ hðsÞðm0 Þ þ f ðsÞm0 þ rðsÞm þ ‘2 j ¼ 0; where f ðsÞ ¼ 1 þ ‘21 þ m2 ðu22 u21 Þ; hðsÞ ¼ m½u21 u22 þ ‘02 u1 ‘03 u2 ‘2 su2 þ ð‘1 j þ ‘3 sÞu1 ; gðsÞ ¼ m‘1 ½u22 u21 2u1 ð‘1 j þ ‘3 sÞ þ ‘03 u2 ‘3 u02 þ 2‘2 su2 ‘02 u1 þ m2 ðu1 u01 u2 u02 Þ þ ‘1 ‘2 j; rðsÞ ¼ ‘2 u01 ‘3 u02 ‘1 ju1 þ ‘2 su2 ‘3 su1 þ m½‘1 ðu2 u02 u1 u01 Þ þ ‘2 jðu22 u21 Þ þ m2 ½ðu01 u2 u1 u02 Þð‘2 u2 ‘3 u1 Þ þ ðu31 u1 u22 Þð‘1 j þ ‘3 sÞ.
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From (2.10), the geodesic curvature of the base curve (m = 0) is ‘2 j ðk g Þb ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ‘22 ‘23 Then, the normal curvature of a curve ~ c ¼~ cðsÞ is 0 1 R n1 ð‘3 u1 ‘2 u2 Þ m n01 þ n2 j k n ¼ 1=2 2R nðs; mðsÞÞk R k~ 0 R n02 þ n1 j þ n3 s n2 ð‘3 m‘1 u2 Þ 2R R0 0 n ð‘2 m‘1 u1 Þ . þ n3 þ n2 s 2R 3
ð2:10aÞ
ð2:11Þ
Then, the normal curvature of the base curve (m = 0) is ‘3 j ðk n Þb ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ‘22 ‘23
ð2:11aÞ
Therefore, from (2.10a), (2.11a) and (I) we have Corollary 2. The geodesic curvature kg and the normal curvature kn of a spacelike base curve on the time-like ruled surface Mt in R31 space satisfy (kg)2 (kn)2 = j2.
References [1] J.K. Beem, P.E. Ehrlich, Global Lorentzian Geometry, Marcel Dekker Inc., New York, 1981. [2] K.L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi Riemannian Manifolds and Applications, Kluwer Academic Publishers, 1996. [3] W. Greub, Linear Algebra, fourth ed., Springer-Verlag, New York, 1975. [4] T. Ikaw, On curves and submanifolds in indefinite Riemannian manifold, Tsukuba J. Math. 9 (2) (1985) 353–371. [5] B. OÕNeill, Semi-Riemannian Geometry, Academic Press, New York, 1983. [6] G. Tsagas, B. Papanyoniou, On the rectilinear congruences of Lorentz manifold establishing an area preserving representation, Tensor N.S. 47 (1988) 128–139. [7] A. Turgut, H.H. Hacisalihoglu, Time-like ruled surfaces in the Minkowski 3-space, Far East J. Math. Sci. 5 (1) (1997) 83–90.
www.elsevier.com/locate/amc
Erratum
Erratum to: ‘‘Ruled surfaces with time like rulings’’ [Appl. Math. Comput. 147 (2004) 241–248] _ Emin Kasap *, Ismail Aydemir, Nuri Kuruog˘lu Ondokuz Mayis University, Faculty of Arts and Sciences, Department of Mathematics, 55139 Kurupelit-Samsun, Turkey
Abstract In this article, a new type of ruled surfaces in a Lorentz 3-space R31 is obtained by a strictly connected time-like oriented line moving with FrenetÕs frame along a space-like curve. These surfaces are classified into time-like and space-like surfaces. The wellknown theorems due to Bonnet and Chasles in the 3-dimensional Euclidean space are proved for a time-like ruled surface. Ó 2005 Published by Elsevier Inc.
1. Introduction Definition 1. Let R31 be a 3-dimensional Lorentzian space with the pseudo-met~; ~ ~ and ~ ~ and ~ ric ds2 = dx2 dy2 + dz2. If hX Y i ¼ 0 for all X Y , the vectors X Y are called perpendicular in the sense of Lorentz, where h , i is the induced inner product in R31 [1]. *
DOI of original article: 10.1016/S0096-3003(02)00664-1. Corresponding author. E-mail address: [email protected] (E. Kasap).
0096-3003/$ - see front matter Ó 2005 Published by Elsevier Inc. doi:10.1016/j.amc.2005.07.004
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~ 2 R3 is denoted by kX ~k and defined as Definition 2. The norm of X 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~k ¼ jhX ~; X ~ij. kX ~ 2 R3 is called a space-like, time-like and null (light-like) vector if The vector X 1 ~ ~ ~ ¼ 0, hX ~; X ~i < 0, and hX ~; X ~i ¼ 0 for X ~ 6¼ 0, respectively [5]. hX ; X i > 0 or X Definition 3. A regular curve aðsÞ : I ! R31 , I R in R31 is said to be a spacelike, time-like and null curve if the velocity vector ~ a0 ðsÞ ¼ da=ds is a space-like, time-like or null vector respectively [4]. Definition 4. A surface in a 3-dimensional Lorentz space is called a time-like surface if the induced metric on the surface is a Lorentz metric, i.e., the normal on the surface is a space-like vector [7]. As revealed from the foregoing definitions, one can prove the following: Lemma 1. In the Lorentz space R31 , the following properties are satisfied: (i) Two time-like vectors are never orthogonal. (ii) Two null vectors are orthogonal if and only if they are linearly dependent. (iii) A time-like vector is never orthogonal to a null (light-like) vector [3]. Let ~ a ¼~ aðsÞ be a unit speed space-like curve in R31 ; by j(s), s(s) we denote the natural curvature and torsion of ~ aðsÞ respectively. Consider the Frenet frame f~ e1 ;~ e2 ;~ e3 g attached to the space-like curve ~ a ¼~ aðsÞ such that ~ e2 ¼ ~ e2 ðsÞ is the principal normal vector field of type time-like and ~ e3 ¼ ~ e3 ðsÞ is the binormal vector field of type space-like and ~ e1 ¼ ~ e1 ðsÞ is the unit tangent vector field with the normalization [4]. h~ e1 ;~ e1 i ¼ h~ e2 ;~ e2 i ¼ h~ e3 ;~ e3 i ¼ 1; h~ ei ;~ ej i ¼ 0
ð1:1Þ
for i 6¼ j; detð~ e1 ;~ e2 ;~ e3 Þ ¼ 1.
The infinitesimal displacements of the frame are given as ~ e01 ðsÞ ¼ jðsÞ~ e2 ;
~ e02 ðsÞ ¼ jðsÞ~ e1 þ sðsÞ~ e3 ;
~ e03 ðsÞ ¼ sðsÞ~ e2 .
ð1:2Þ
One can easily see the following: Lemma 2. Assume that ~ aðsÞ is a unit speed space-like curve in R31 and f~ e1 ;~ e2 ;~ e3 g be its FrenetÕs frame field. Then ~ e2 ¼ ~ e3 ; e1 ^~
~ e1 ^~ e3 ¼ ~ e2 ;
~ e2 ^~ e3 ¼ ~ e1 .
ð1:3Þ
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2. Ruled surfaces with time-like generators in R31 A time-like straight line ~ L in R31 such that it is strictly connected to FrenetÕs frame of the space-like curve ~ a ¼~ aðsÞ is represented, uniquely with respect to this frame, in the form ~ LðsÞ ¼
3 X
‘i ðsÞ~ ei ðsÞ;
h~ LðsÞ; ~ LðsÞi > 0;
ð2:1Þ
i¼1
where the components ‘i = ‘i(s) (i = 1, 2, 3) are scalar functions of the arc length parameter of the base curve ~ a ¼~ aðsÞ. Without loss of generality the family of ruled surfaces under investigation are characterized by the regular parametrization. M : Uðs; mÞ ¼ ~ aðsÞ þ m~ LðsÞ; ‘21 ‘22 þ ‘23 ¼ 1;
) ðIÞ
0 ~ L ðsÞ 6¼ 0.
Definition 5. If there exists a common perpendicular to two constructive rulings in the skew surface, then the foot of the common perpendicular on the main ruling is called a central point. The locus of the central point is called the striction curve [7]. Using (1.2) and (2.1), it is easy to see that the parametrization of the striction curve on a ruled surface of type (I) is given by ~ bðsÞ ¼ ~ aðsÞ /ðsÞ~ LðsÞ;
where /ðsÞ ¼
‘01 þ ‘2 j . 0 0 h~ L ðsÞ; ~ L ðsÞi
ð2:2Þ
The unit normal vector ~ n on the ruled surface of type (I) is given by 0 ~ LðsÞ þ m~ L ðsÞ ^ ~ LðsÞ a0 ðsÞ ^ ~ ~ n¼ 0 . 0 ~ ~ ~ k~ a ðsÞ ^ LðsÞ þ mL ðsÞ ^ LðsÞk
ð2:3Þ
Thus, from (2.3) and (1.2) the unit normal vector to the surface M at the point (s, o) is ‘2~ e3 þ ‘3~ e2 ~ nðs; oÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ‘22 ‘23
ð2:4Þ
From (2.4), it follows that the base curve of the surface (I) to be a geodesic curve (~ nðs; oÞ ¼ ~ e2 ) if ‘2 = 0, ‘3 5 0. From (2.2) and (2.4) it follows that the BonnetÕs theorem for the ruled surface (I) can be formulated as
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Theorem 1. If a space-like curve on a ruled surface (I) in R31 has two of the following properties, it has the third also (i) It is a geodesic (‘2 = 0 or j = 0). (ii) It cut the rulings at a constant angle (‘1 = const.). (iii) It is a striction curve (/(s) = 0). The induced metric on the ruled surfaces (I) is given by [2] 0 0 g ¼ ð1 þ ‘21 Þ ð2m/ðsÞ þ m2 Þh~ L ;~ L i.
Therefore, one can give a classification to the ruled surfaces (I) into time-like or space-like ruled surfaces as follows: The class of ruled surfaces (I) is of type time-like (g < 0) if one of the following conditions: (i) m and /(s) have the same sign, (ii) /(s) = 0, (iii) m and /(s) have opposite sign and 0 0 ð1 þ ‘21 Þ þ ðm2 2m/ðsÞÞh~ L ðsÞ; ~ L ðsÞi > 0
is satisfied. The ruled surfaces (I) are space-like ruled surfaces if and only if m and /(s) have opposite sign and 0 0 ð1 þ ‘21 Þ þ ðm2 2m/ðsÞÞh~ L ðsÞ; ~ L ðsÞi < 0.
Theorem 2. The class of time-like ruled surfaces is divided into three subclasses according to the conditions (i), (ii) and (iii) respectively. 2.1. Time-like ruled surface in R31 Here, we study a subclass of time-like ruled surfaces which is characterized by the condition (ii). For this type, the striction curve is the base curve of a ruled surface of type (I), i.e., M t : Uðs; mÞ ¼ ~ aðsÞ þ m~ LðsÞ; ‘21 ‘22 þ ‘23 ¼ 1;
0 k~ L ðsÞk 6¼ 0;
‘01 þ ‘2 j ¼ 0.
) ðIIÞ
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The distribution parameter k(s) of the time-like ruled surface Mt is defined as kðsÞ ¼
0 ½~ a0 ðsÞ; ~ LðsÞ; ~ L ðsÞ 0 0 h~ L ðsÞ; ~ L ðsÞi
and from (1.2) and (2.1) we have kðsÞ ¼
‘2 ‘03 ‘3 ‘02 þ ð1 þ ‘21 Þs ‘1 ‘3 j . 0 0 h~ L ðsÞ; ~ L ðsÞi
ð2:5Þ
From (2.3) and (II) it is easy to see that, the unit normal vector to the time-like ruled surface Mt at (s, m) and (s, o) are ~ nðs; mÞ ¼
k~ lðsÞ þ m~ lðsÞ ^ ~ LðsÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; 2 2 k þm
~ lðsÞ ¼ no ¼ ~
0 ~ L ðsÞ . 0 k~ L ðsÞk
ð2:6Þ
From Definition 4 and Lemma 1, one can see that ~ n and ~ no are unit space-like vectors. The angle h of rotation from the normal ~ no to the normal ~ n is given from m sin h ¼ k~ no ^ ~ nk ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 2 k þ m2 Hence Chasles theorem is valid for a time-like ruled surface Mt in R31 . As an immediate result we have the following: Corollary 1. The tangent plane turns evidently through 180° along a ruling in a non-developable (k 5 0) time-like ruled surface Mt. The Gaussian curvature K of the surface in R31 is given by K = (ehab)/gab, where e ¼ h~ n;~ ni and gab, hab are the first and second fundamental quantities, respectively [6]. After simple calculation, one can see that the Gaussian curvature of the time-like ruled surface Mt is K¼
k2 2
ðk2 þ m2 Þ
.
ð2:7Þ
Thus we have the following: Theorem 3. The Gaussian curvature K of a time-like ruled surface Mt in R31 is non negative and K is equal to zero only along the ruling which meet the striction curve at a singular point (k = 0, m 5 0). Eq. (2.7) allows us to give a geometric interpretation of the (regular) central points of a time-like ruled surface. Indeed, the points of a ruling except perhaps the central point, are regular points of the surface if k 5 0. The function K(s, m) is a continuous function on the ruling (s = const.) and the central point is characterized by the fact that K(s, m) has a maximum there.
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2.2. Time-like ruled surfaces with constant parameter of distribution From (2.5), one can see that a time-like ruled surface Mt with a constant distribution parameter satisfies the following differential equation: 0 0 L ðsÞ; ~ L ðsÞi; ‘2 ‘03 ‘3 ‘02 þ ð1 þ ‘21 Þs ‘1 ‘3 j ¼ Ch~
where C is constant. Using (II), one can see that R 9 ‘1 ¼ ‘2 j ds þ c1 ; > > > = R Ch~L0 ðsÞ;~L0 ðsÞi ‘3 ¼ ‘2 s ds þ c ; ‘ ¼ 6 ‘ ; 2 2 3 2 1þl1 > > > ; 2 2 2 ‘1 ‘2 þ ‘3 ¼ 1.
/ðsÞ ¼ 0;
ð2:8Þ
The second equation of the triple (2.8) is an integral equation for the unknown ‘3 = ‘3(S). Therefore, if we given ‘2 = ‘2(s) we get ‘1 = ‘1(s) and ‘3 = ‘3(s). Thus, we have the existence theorem: Theorem 4. The range of existence of a one parametric time-like ruled surfaces {Mt} with constant parameter of distribution comprises within one arbitrary function of one variable. Developable time-like ruled surface are special class of the ruled surfaces described by Theorem 4 (k = c = 0). If a point P be displaced orthogonally along ~ L to a neighbouring point Po, we have an orthogonal trajectory. The condition that the point P be displaced orthogonally to the ruling is hd~ c=ds, ~ Li ¼ 0. One can see that m¼
Z ‘1 ds.
ð2:9Þ
From (2.9) and Theorem 1, for the time-like ruled surface Mt, one can give an interpretation to the parameters m, ‘1, as Theorem 5. For a time-like ruled surface Mt in R31 . If the base curve is geodesic, then the distance m along a ruling from the base curve ~ a ¼~ aðsÞ to an orthogonal trajectory is proportional to the arc length of the base curve. 2.3. The geodesic and normal curvature Let ~ c ¼~ cðsÞ be a curve on the time-like ruled surface Mt, then it can be represented in the form ~ cðsÞ ¼ ~ aðsÞ þ mðsÞ~ LðsÞ.
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Using (1.2), it is easy to see that the unit tangent vector along the curve ~ c ¼~ cðsÞ is ~ c0 ðsÞ ¼
n1~ e1 þ n2~ e2 þ n3~ e3 pffiffiffi ; R
where n1 ¼ 1 þ m0 ‘1 ;
n2 ¼ mu1 þ m0 ‘2 ;
u1 ¼ ‘1 j þ ‘02 þ ‘3 s;
n3 ¼ mu2 þ m0 ‘3 ;
u2 ¼ ‘2 s þ ‘03 ;
R ¼ jn21 n22 þ n23 j. Therefore, one can see that ~ e1 þ ðn02 þ n1 j þ n3 sÞ~ e2 þ ðn03 þ n2 sÞ~ e3 R c00 ðsÞ ¼ R3=2 ð½ðn01 þ n2 jÞ~
R0 ðn ~ e1 þ n2~ e2 þ n3~ eÞÞ. 2 1
Using (2.6), the unit normal vector field on the time-like ruled surface Mt along the curve (m = m(s)) is mð‘3 u1 ‘2 u2 Þ~ e1 þ ð‘3 m‘1 u2 Þ~ e2 þ ð‘2 m‘1 u1 Þ~ e3 ffi. ~ nðs; mðsÞÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 m2 ð‘3 u1 ‘2 u2 Þ þ ð‘3 m‘1 u2 Þ þ ð‘2 m‘1 u1 Þ The geodesic curvature of the curve ~ c ¼~ cðsÞ is given by n1 n2 n3 1 kg ¼ n02 þ n1 j þ n3 s n03 þ n2 s . n01 þ n2 j Rk~ nðs; mðsÞÞk mð‘3 u ‘2 u Þ ‘3 m‘1 u2 ‘2 m‘1 u1 1 2 ð2:10Þ Then, the differential equation of the geodesic curves on the time-like ruled surface Mt is given by 2
f ðsÞm00 þ hðsÞðm0 Þ þ f ðsÞm0 þ rðsÞm þ ‘2 j ¼ 0; where f ðsÞ ¼ 1 þ ‘21 þ m2 ðu22 u21 Þ; hðsÞ ¼ m½u21 u22 þ ‘02 u1 ‘03 u2 ‘2 su2 þ ð‘1 j þ ‘3 sÞu1 ; gðsÞ ¼ m‘1 ½u22 u21 2u1 ð‘1 j þ ‘3 sÞ þ ‘03 u2 ‘3 u02 þ 2‘2 su2 ‘02 u1 þ m2 ðu1 u01 u2 u02 Þ þ ‘1 ‘2 j; rðsÞ ¼ ‘2 u01 ‘3 u02 ‘1 ju1 þ ‘2 su2 ‘3 su1 þ m½‘1 ðu2 u02 u1 u01 Þ þ ‘2 jðu22 u21 Þ þ m2 ½ðu01 u2 u1 u02 Þð‘2 u2 ‘3 u1 Þ þ ðu31 u1 u22 Þð‘1 j þ ‘3 sÞ.
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From (2.10), the geodesic curvature of the base curve (m = 0) is ‘2 j ðk g Þb ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ‘22 ‘23 Then, the normal curvature of a curve ~ c ¼~ cðsÞ is 0 1 R n1 ð‘3 u1 ‘2 u2 Þ m n01 þ n2 j k n ¼ 1=2 2R nðs; mðsÞÞk R k~ 0 R n02 þ n1 j þ n3 s n2 ð‘3 m‘1 u2 Þ 2R R0 0 n ð‘2 m‘1 u1 Þ . þ n3 þ n2 s 2R 3
ð2:10aÞ
ð2:11Þ
Then, the normal curvature of the base curve (m = 0) is ‘3 j ðk n Þb ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ‘22 ‘23
ð2:11aÞ
Therefore, from (2.10a), (2.11a) and (I) we have Corollary 2. The geodesic curvature kg and the normal curvature kn of a spacelike base curve on the time-like ruled surface Mt in R31 space satisfy (kg)2 (kn)2 = j2.
References [1] J.K. Beem, P.E. Ehrlich, Global Lorentzian Geometry, Marcel Dekker Inc., New York, 1981. [2] K.L. Duggal, A. Bejancu, Lightlike Submanifolds of Semi Riemannian Manifolds and Applications, Kluwer Academic Publishers, 1996. [3] W. Greub, Linear Algebra, fourth ed., Springer-Verlag, New York, 1975. [4] T. Ikaw, On curves and submanifolds in indefinite Riemannian manifold, Tsukuba J. Math. 9 (2) (1985) 353–371. [5] B. OÕNeill, Semi-Riemannian Geometry, Academic Press, New York, 1983. [6] G. Tsagas, B. Papanyoniou, On the rectilinear congruences of Lorentz manifold establishing an area preserving representation, Tensor N.S. 47 (1988) 128–139. [7] A. Turgut, H.H. Hacisalihoglu, Time-like ruled surfaces in the Minkowski 3-space, Far East J. Math. Sci. 5 (1) (1997) 83–90.