TRANSVERSAL SURFACES OF TIMELIKE RULED SURFACES IN MINKOWSKI 3-SPACE IR

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1 IJRRAS () November 0 wwwarpapresscom/volumes/volissue/ijrras 08pf TRANSVERSAL SURFACES OF TIMELIKE RULED SURFACES IN MINKOWSKI -SPACE Mehmet Öner Celal Bayar University, Faculty of Science an Arts, Department of Mathematics, Muraiye Campus, 45047, Muraiye, Manisa, Turey mehmetoner@cbueutr ABSTRACT In this stuy, we give efinitions an characterizations of transversal surfaces of timelie rule surfaces We stuy some special cases such as the striction curve is a geoesic, an asymptotic line or a line of curvature Moreover, we obtain the evelopable conitions for transversal surfaces of a timelie rule surface Key wors: Minowsi -space; timelie rule surface; transversal surfaces MSC: 5B5, 5C50 INTRODUCTION The notion of transversal surface was given by Sachs in the -imensional Eucliean space E [4] Sachs stuie -transversal surface an new transversal surfaces calle - an -transversal surfaces of rule surfaces in the same space Taoutsoglou consiere the notion of transversal surfaces for the rule surfaces of the most general type in simply isotropic space I [6] Moreover, Sipus an Divja have escribe the transversal surfaces of rule surfaces in the pseuo-galilean space Let G [5] be the Minowsi -space with stanar Lorentzian flat metric given by, x x x, where ( x, x, x) is a stanar rectangular coorinate system of An arbitrary vector v ( v, v, v) of is sai to be spacelie if vv, 0 or v 0, timelie if vv, 0 an null (lightlie) if vv, 0 an v 0 Similarly, an arbitrary curve () s is sai to be spacelie, timelie or null (lightlie), if all of its velocity vectors () s are spacelie, timelie or null (lightlie), respectively [] We say that a timelie vector is future pointing or past pointing if the first compoun of the vector is positive or negative, respectively The norm of v ( v, v, v ) the vector x ( x, x, x ) For any vectors by (See [0]) Definition ([9]) v v is given by v, y ( y, y, y ) an in, Lorentzian vector prouct of x an y is efine e e e x y x x x ( x y x y, x y x y, x y x y ) y y y i) Hyperbolic angle: Let x an y be future pointing (or past pointing) timelie vectors in unique real number 0 such that x, y x y cosh between the vectors x an y Then there is a This number is calle the hyperbolic angle 44

2 IJRRAS () November 0 Öner Transversal Surfaces of Timelie Rule Surfaces ii) Central angle: Let x an y be spacelie vectors in that span a timelie vector subspace Then there is a unique real number 0 such that x, y x y cosh This number is calle the central angle between the vectors x an y iii) Spacelie angle: Let x an y be spacelie vectors in a unique real number 0 such that x, y x y cos the vectors x an y that span a spacelie vector subspace Then there is This number is calle the spacelie angle between iv) Lorentzian timelie angle: Let x be a spacelie vector an y be a timelie vector in unique real number 0 such that x, y x y sinh angle between the vectors x an y Then there is a This number is calle the Lorentzian timelie Definition ([]) A surface in the Minowsi -space is calle a timelie surface if the inuce metric on the surface is a Lorentz metric an is calle a spacelie surface if the inuce metric on the surface is a positive efinite Riemannian metric, ie, the normal vector on the spacelie (timelie) surface is a timelie (spacelie) vector TIMELIKE RULED SURFACES IN MINKOWSKI -SPACE Let I be an open interval in the real line Let f f ( u) unit irection vector of an oriente line in be a curve in efine on I an q q( u) Then we have following parametrization for a rule surface N r( u, v) f ( u) vq( u) be a () Assume that the surface normal is spacelie Then by Definition, N is a timelie rule surface The curve f f ( u) is calle base curve or generating curve of the surface an various positions of the generating lines q q( u) are calle rulings In particular, if the irection of q is constant, then the rule surface is sai to be cylinrical, an non-cylinrical otherwise The istribution parameter (or rall) of the timelie rule surface in () is given by f, q, q, () qq, f q q (see [,7]) If f, q, q 0, then normal vectors are collinear at all points of same u u where f, ruling an at nonsingular points of the surface N, the tangent planes are ientical We then say that tangent plane contacts the surface along a ruling Such a ruling is calle a torsal ruling If f, q, q 0, then the tangent planes of the surface N are istinct at all points of same ruling which is calle nontorsal [8] Definition ([8]) A timelie rule surface whose all rulings are torsal is calle a evelopable timelie rule surface The remaining timelie rule surfaces are calle sew timelie rule surfaces Then, from () it is clear that a timelie rule surface is evelopable if an only if at all its points the istribution parameter 0 For the unit normal vector m of a sew timelie rule surface we have 444

3 IJRRAS () November 0 Öner Transversal Surfaces of Timelie Rule Surfaces r r ( f vq ) q u v m r u r v f, q q, q f vq, f vq () The unit normal of the surface along a ruling irection is calle the asymptotic normal irection an from () efine by u u approaches a limiting irection as v infinitely ecreases This q q a lim m( u, v) v q The point at which the unit normal of N is perpenicular to a is calle the striction point (or central point) C an the set of striction points of all rulings is calle striction curve of the surface The parametrization of the on a timelie rule surface is given by striction curve c c( u) v where 0 q, f qq, is calle strictional istance q, f c( u) f ( u) v0q( u) f q qq,, (4) The vector h efine by h a q is calle central normal which is the surface normal along the striction curve Then the orthonormal system C; q, h, a is calle Frenet frame of the rule surfaces N where C is the central point of ruling of timelie rule surface N an q, h a q, a are unit vectors of ruling, central normal an central tangent, respectively Let now consier rule surface N with non-null Frenet vectors an their non-null erivatives Accoring to the Lorentzian character of ruling, we can give the following classifications of the timelie rule surface N : i) If q is timelie, then timelie rule surface N is sai to be of type N ii) If q is spacelie, then timelie rule surface N is sai to be of type N In these classifications we use subscript + an - to show the Lorentzian casual character of ruling By using these classifications, parametrization of timelie rule surface N can be given as follows, r( u, v) f ( u) vq( u) where q, q ( ), h, h, a, a For the erivatives of vectors of Frenet frame length s of striction curve we have, C; q, h, a of timelie rule surface N with respect to the arc q / s 0 0q h / s 0 h a / s 0 0, (5) a 445

4 IJRRAS () November 0 Öner Transversal Surfaces of Timelie Rule Surfaces s s where, an s, s are the arc lengths of the spherical curves circumscribe by the boun s s vectors q an a, respectively Timelie rule surfaces satisfying 0, 0 are calle timelie conois (For etails see [8]) In this stuy, we introuce the efinitions an characterizations of transversal surfaces of timelie rule surfaces For a reference rule surface, we consier a timelie rule surface of the type N with timelie striction curve So, we first give some special cases for the striction curve of a timelie rule surface of the type N Of course the obtaine results of the following sections can be easily transferre to other cases such as the surface is of the type N or is a spacelie rule surface SOME SPECIAL CASES FOR THE STRICTION CURVE OF A TIMELIKE RULED SURFACE Let assume that timelie rule surface N be of the type N an let the generating curve of the surface be its striction curve an the Frenet frame of the surface be C; q, h, a Moreover, assume that the striction curve is timelie Then for the parametrization of the surface N we write r ( s, v) c ( s) vq ( s) where q, q, h, h, a, a an s is the arc length parameter of striction curve cs () unit tangent vector of the striction curve we have, (6) For the c c( s) t coshq sinha, (7) s where is the hyperbolic angle between timelie vectors t an q Then from () an (7) the istribution parameter of the surface is obtaine as sinh (8) Let now investigate some special cases for the striction curve cs () First assume that striction curve cs () is an asymptotic line on N Then central normal h an irection vector c of principal normal vector satisfy hc, 0 After a simple computation we have tanh / Then we have the following theorem Theorem Let timelie rule surface N be of the type N Then timelie striction curve cs () tanh / hols is an asymptotic line on N if an only if of the surface If the striction curve cs () is a geoesic on N, then central normal h an irection vector c of principal normal vector are linearly epene ie, we have h c where () s is a scalar function The last equality gives us h ( sinh ) q ( cosh sinh ) h ( cosh ) a, (9) an from (9) we have that is constant an we give the following theorem Theorem Let timelie rule surface N be of the type N Then timelie striction curve cs () is a geoesic on N if an only if is constant of the surface 446

5 IJRRAS () November 0 Öner Transversal Surfaces of Timelie Rule Surfaces Finally, assume that the striction curve cs () an tangent vector t of striction curve cs () scalar function Then from (5) an (7) we have is a line of curvature on N Then the erivative of central normal h are linearly epene ie, we have h t where () s tanh / which gives us following theorem Theorem Let timelie rule surface N be of the type N Then timelie striction curve cs () tanh / hols is a of the surface Now we can introuce transversal surfaces of a timelie rule surface of the type N In the following section, when we tal about timelie rule surface N an striction curve cs () an striction curve cs (), we mean that surface is of the type N is timelie an for short we on t write the Lorentzian characters of the surface an curve 4 -TRANSVERSAL SURFACES OF TIMELIKE RULED SURFACES In this section, we give the efinition an characterizations of -transversal surfaces of a timelie rule surface First, we give the following efinition Definition 4 Let N be a timelie rule surface An -transversal surface N whose rulings are straight lines through a striction point cs () etermine by ruling q ( ) q ( ) h cosh, q is timelie sinh, q is timelie ( ) ( ) sinh, q is spacelie cosh, q is spacelie an to obtaine the non-trivial cases (ruling is not q or h ) we assume ( ) 0, ( ) 0 From Definition 4 the parametrization of -transversal surface N r ( s, v) c ( s) vq ( s) an from (0) we have q is, q The strictional istance v of the -transversal surface N is obtaine as c, q cosh ( ) sinh v q, q ( ) where / s Then we obtain the following theorem: Theorem 4 The striction curve c hols of N is a rule surface in where (0), () () on every N tanh From Theorem we now that the striction curve cs () tanh / hols In this special case, Theorem 4 gives us 0 theorem: Theorem 4 Let the striction curve c asymptotic line on N if an only if is constant, () if an only if, (4) is an asymptotic line on N if an only if on every N Then we can give the following is an 447

6 IJRRAS () November 0 Öner Transversal Surfaces of Timelie Rule Surfaces We now that striction curve cs () of the surface is a geoesic on N if an only if is constant Then from (4) we have the following theorem: Theorem 4 Let the striction curve c on every N geoesic on N if an only if there exists a constant x such that x From Theorem, the striction curve cs () Then from (4) we have the following theorem: Theorem 44 Let the striction curve c line of curvature on N if an only if on every N hols is a tanh / hols Let now consier the evelopable -transversal surfaces By a simple calculation from () an () the istribution parameter of N is obtaine as Then we have the following theorem: ( )sinh cosh ( ) is a (5) Theorem 45 -transversal surface N is evelopable if an only if ( ) tanh hols Moreover, from (8) an (5) we have ( ) cosh ( ) (6) where is istribution parameter of N Since we consier non-trivial cases ie, 0, if 0 then from (6) we have 0 which gives us following corollary: Corollary 46 Let timelie rule surface N be evelopable Then N timelie conoi is evelopable if an only if N is a 5 -TRANSVERSAL SURFACES OF TIMELIKE RULED SURFACES In this section, we give the efinition an characterizations of -transversal surfaces of a timelie rule surface First, we give the following efinition Definition 5 Let N be a timelie rule surface in The -transversal surface N of N is a rule surface in whose rulings are straight lines through a striction point cs () etermine by ruling q cos h sin a where is spacelie angle between q an h, an to obtaine the non-trivial cases (ruling is not h or a ) we assume n, (n ) / where n Z From this efinition the parametrization of -transversal surface N is r ( s, v) c ( s) vq ( s) (7) 448

7 IJRRAS () November 0 Öner Transversal Surfaces of Timelie Rule Surfaces By a simple calculation the strictional istance v of the -transversal surface N is obtaine as c, q cos ( )sinh cosh v, (8) q, q ( ) cos where / s Theorem 5 The striction curve c hols Then we obtain the following theorem: on every N tanh From Theorem we have that the striction curve cs () tanh / hols In this special case, Theorem 5 gives us 0 theorem: Theorem 5 Let the striction curve c asymptotic line on N if an only if is constant We now that striction curve cs () Then from (9) we have the following theorem: Theorem 5 Let the striction curve c if an only if, (9) is an asymptotic line on N if an only if on every N Then we can give the following is an of the surface is a geoesic on N if an only if is constant (Theorem ) on every N geoesic on N if an only if there exists a constant y such that y( ) From Theorem, the striction curve cs () Then from (9) we have the following theorem: Theorem 54 Let the striction curve c curvature line on N if an only if on every N hols is a tanh / hols Let now consier the evelopable -transversal surfaces By a simple calculation, from () an (7) the istribution parameter of N Then we have the following theorem: is obtaine as cos sinh ( )cosh ( ) cos is a (0) Theorem 55 -transversal surface N is evelopable if an only if tanh cos hols Moreover, from (8) an (0) we have cos ( )cosh ( ) cos, () 449

8 IJRRAS () November 0 Öner Transversal Surfaces of Timelie Rule Surfaces where is istribution parameter of reference surface N Then () gives us following corollary: Corollary 56 Let timelie rule surface N be evelopable Then N is evelopable if an only if 6 -TRANSVERSAL SURFACES OF TIMELIKE RULED SURFACES In this section, we give the efinition an characterizations of -transversal surfaces of a timelie rule surface First, we give the following efinition Definition 6 Let N be a timelie rule surface The -transversal surface N of N is a rule surface in whose rulings are straight lines through a striction point cs () etermine by ruling q ( ) q ( ) a where cosh, q is timelie sinh, q is timelie ( ) ( ) sinh, q is spacelie cosh, q is spacelie an to obtaine the non-trivial cases (ruling is not q or a ) we assume ( ) 0, ( ) 0 From Definition 6 the parametrization of -transversal surface N is r ( s, v) c ( s) vq ( s) an from () we have q q The strictional istance v of the -transversal surface N where / s Theorem 6 The striction curve c constant or hols v, is obtaine as c, q sinh cosh q, q ( ) ( ) Then from (5) we obtain the following theorem: on every N (), () (4) tanh From Theorem we now that the striction curve cs () tanh / hols Then we can give the following theorem: Theorem 6 Let the striction curve c on every N an let be non- constant is an asymptotic line on N if an only if, (5) if an only if is, (6) is an asymptotic line on N if an only if hols We now that striction curve cs () of the surface is a geoesic on N if an only if is constant Then from (6) we have the following theorem: Theorem 6 Let the striction curve c on every N an let be nonconstant Then cs () is a geoesic on N if an only if there exists a constant z such that z 450

9 IJRRAS () November 0 Öner Transversal Surfaces of Timelie Rule Surfaces From Theorem, the striction curve cs () Then from (6) we have the following theorem: Theorem 64 Let the striction curve c on every N an let be non- constant tanh / hols hols Let now consier the evelopable -transversal surfaces By a simple calculation from () an () the istribution parameter of N is obtaine as Then we have the following theorem: cosh sinh ( ) ( ) ( ) (7) Theorem 65 -transversal surface N is evelopable if an only if tanh or hol Moreover, from (8) an (7) we have cosh ( ) ( ) ( ) where is istribution parameter of N Then if N is evelopable, ie, 0 (8), (8) gives us following corollary Corollary 66 Let timelie rule surface N be evelopable Then N hols is evelopable if an only if 7 CONCLUSIONS Transversal surfaces of a timelie rule surface of the type N are efine an characterizations of these surfaces are given In this paper, we consier the striction line as a timelie curve Of course, one can obtain corresponing theorems for a timelie rule surface with a spacelie striction line, for a surface of the type N or for a spacelie rule surface REFERENCES [] Abel-All, NH; Abel-Bay, RA; Hamoon, FM; Rule surfaces with timelie rulings, App Math an Comp (004), 47, 4 5 [] Beem, JK; Ehrlich, PE; Global Lorentzian Geometry, Marcel Deer, New Yor, 98 [] O Neill, B; Semi-Riemannian Geometry with Applications to Relativity Acaemic Press, Lonon, 98 [4] Sachs, H; Uber Transversalflachen von Regelflachen, Sitzungsber Aa Wiss Wien, (978), 86, [5] Sipus, ZM; Divja, B; Transversal Surfaces of Rule Surfaces in the Pseuo-Galilean Space, Sitzungsber Abt II, (004),, [6] Taoutsoglou, A; Transversalflachen von Regelflachen im einfach isotropen Raum, Sitzungsber Aa Wiss Wien, (994), 0, 7 48 [7] Turgut, A; Hacısalihoğlu, HH; Timelie rule surfaces in the Minowsi -space, Far East J Math Sci (997), 5(), 8 90 [8] Uğurlu, HH; Öner, M; Instantaneous Rotation vectors of Sew Timelie Rule Surfaces in Minowsi - space, VI Geometry Symposium, Uluağ University, Bursa, Turey 008 [9] Uğurlu, HH; Yaut, NN; Öztunç, S; The concept of angle in Minowsi -space, X Geometry Symposium, Balıesir, Turey, -6 Jul 0 [0] Walrave, J; Curves an surfaces in Minowsi space, PhD thesis, KU Leuven, Fac of Science, Leuven,