THE DIFFERENTIAL GEOMETRY OF REGULAR CURVES ON A REGULAR TIME-LIKE SURFACE

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1 Dynamic Systems and Applications TH DIFFRNTIAL OMTRY OF RULAR CURVS ON A RULAR TIM-LIK SURFAC MIN OZYILMAZ AND YUSUF YAYLI Department of Mathematics ge Uniersity Bornoa Izmir Trkey Department of Mathematics Ankara Uniersity Dogol cad. Ankara Trkey ABSTRACT. In this stdy we consider time-like reglar srface in Minkowski space as y y and inestigate Darbox ectors of the time-like cres on time-like srface as c c 1 and c 2 which are not intersect perpendiclarly. Moreoer we gie a relation between the Darbox ectors of these Darbox frames. By this relation we obtain general Lioille formla and general form ler and O. Bonnet. AMS MOS Sbject Classification. 53A0453B INTRODUCTION It is well-known that if a cre differentiable on an open interal at each point a set of mtally orthogonal nit ectors can be constrcted. And these ectors are called Frenet frame or moing frame ectors. The set whose elements are frame ectors and cratres of a cre is called Frenet apparats of the cres. In recent years the theory of degenerate sbmanifol has been treated by researchers and some classical differential geometry topics hae been extended to Lorentz manifol. For instance in [6] the athors extended and stdied spacelike inolte-eolte cres in Minkowski space. Classical differential geometry of the cres may be srronded by the topics which are general helices inolte-eolte cre coples spherical cres and Bertrand cres. Sch special cres are inestigated and sed in some of real world problems like mechanical design or robotics by well-known Frenet-Serret eqations. Becase we think of cres as the path of a moing particle in the clidean space. At the beginning of the twentieth centry instein s theory opened a door to new geometries sch as Minkowski space-time which is simltaneosly the geometry of special relatiity and the geometry indced on each fixed tangent space of an arbitrary Lorentzian manifold. Some athors hae aimed to determine Frenet Serret inariants in higher dimensions. There exists a ast literatre on this sbject for instance [ ]. In the light of the aailable literatre in [4] the athor extended spherical Receied April $15.00 c Dynamic Pblishers Inc.

2 350. OZYILMAZ AND Y. YAYLI images of cres to a for-dimensional Lorentzian space and stdied sch cres in the case where the base cre is a space-like cre according to the signatre By sing the Darbox ector arios well-known formlas of differential geometry had been prodced by [5]. Then in [1] athors had been gien these formlas in Minkowski 3-space. In this work we stdy to inestigate the formlae between the Darbox ectors of the cre c the parameter cres c 1 and c 2 which are not intersecting perpendiclarly. Ths we will find an opportnity to inestigate reglar time-like srface by taking the parameter cres which are intersect nder the angle θ. 2. PRLIMINARIS To meet the reqirements in the next sections here the basic elements of the theory of cres in the space R1 3 are briefly presented. A more complete elementary treatment can be fond in [1]. The Minkowski 3-space proided with the standard flat metric gien by 2.1 dx dx 2 2 dx 2 3 where x 1 x 2 x 3 is rectanglar coordinate system in R 3 1. Recall that the norm of an arbitrary ector a R1 3 is gien by a a a. Let Φ Φs be a reglar cre in R1 3. Φ is called an nit speed cre if the elocity ector of Φ satisfies 1. For the ectors w R1 3 it is said to be orthogonal if and only if w 0. On the other hand the ector w is called anglar elocity ector of motion. If we consider any orthogonal trihedron as e 1 e 2 e 3 } we can write their deriatie formlas as follows: 2.2 d e i w e i i where is Lorentzian ectorial prodct [1]. Let s take a time-like srface as y y. Denote by t N B} the moing Frenet-Serret frame along the time-like cre c on y y. Another orthogonal frame on y y is the Darbox trihedron as t g N }. For an arbitrary time-like cre c on time-like srface the orientation of the Darbox trihedron is written as 2.3 N t g t g N g N t and the Darbox ector of this trihedron is written as 2.4 w t + g R n N R g where t. t 1 g. g 1 N. N 1 and 1 1 R n and 1 R g are geodesic torsion normal cratre and geodesic cratre respectiely. Alsothe Darbox deriatie formlae

3 RULAR CURVS ON A RULAR TIM-LIK SURFAC 351 can be written as follows: 2.5 [1]. d t w t d g w g d N w N 3. TH DARBOUX VCTOR FOR TH DARBOUX TRIHDRON OF A TIM-LIK CURV Let s express the parameter cres const. as c 1 and const. as c 2 which are constant on a time-like srface y y. Bt these cres are intersect nder the angle θ not perpendiclar. Let any time-like cre that is passing throgh a point P on the srface be c. Let s take time-like cres which are passing throgh the same point P as c 1 and c 2. Let the nit tangent ectors of cres c c 1 and c 2 and at the point P be t t 1 and t 2 respectiely. From [1] the edges of the Darbox trihedrons of parameter cres are 3.1 N t 1 g 1 t 1 g 1 N g 1 N t 1. Here three Darbox trihedrons are written as below: t g N} t 1 g 1 N} t 2 g 2 N}. Let s s 1 and s 2 be the arc-elements of the cres c c 1 and c 2 respectiely. Ths we can write 3.2 t 1 r r r t 2 r r r d t r + r d Moreoer becase of the parameter cres are intersect nder the angle θ we hae 3.3 t 1. t 2 chθ Then the normal ector of time-like srface is 3.4 N t 1 t 2 t 1 t 2 t 1 t 2 Other than considering the first two formlae of 3.2 in the third term d 3.5 t r + r d d t 1 + d t 2 is written [1]. On the other hand let s consider the hyperbolic angle between t and t 1 as α and if we take inner prodct both sides of 3.5 with t 1 and t 2 then 3.6 t. t 1 chα d chθ d 3.7 t. t 2 chθ α chθ d d

4 352. OZYILMAZ AND Y. YAYLI are obtained. Ths from 3.6 and d shα d are written. Finally if we pt 3.8 into 3.5 we hae the following eqation between the tangent ectors of the cres c c 1 and c 2 as 3.9 t t 1 + shα t 2 Here we shall denote the arc elements 1 and 2 of the parameter cres which are belongs to time-like srface y y and then we express as follows: d 2 + 2Fdd + d d2 2 2 d 2 Ths considering 3.8 and 3.10 we hae 3.11 d 1 shα d 2 Corollary 3.1. The third elements g g 1 and g 2 of the Darbox trihedrons t g N t 1 g 1 N} and t 2 g 2 N} are linear dependent. Proof. If we sbstitte the eqation 3.9 in the first eqality of 2.3 and consider the Darbox trihedrons of c 1 and c 2 we hae 3.12 g g 1 + shα g 2 Ths we get the expression. Theorem 3.1. The Darbox trihedrons t 1 g 1 N } } and t 2 g 2 N } of the parameters cres c 1 and c 2 of the time-like srface are written by Darbox instantaneos ectors as follows: 3.13 t i s j w i t i g i s j w i t i Proof. If we consider the Darbox trihedrons t 1 g 1 N } N s j w i N i j 1 2 the normal ector N is coincide. Then considering 3.4 g 1 t 1 N t 1 t 2 t 1 t 2 t 2 1 t 1 t 1. t 2 and t 2 g 2 N } we see that

5 RULAR CURVS ON A RULAR TIM-LIK SURFAC g 1 t 1 chθ t 2 g 2 t 2 N t 1 t 2 t 2 t 2 t 1. t 2 t 1 t g 2 t 2 chθ + t 1 are obtained. From 2.2 we write 3.16 t 1 w 1 t 1 g 1 w 1 g 1 N w 1 N 3.17 s 2 w 2 t 2 g 2 s 2 w 2 g 2 N s 2 w 2 N If 3.14 is sbstitted in the third eqality 3.16 we get [ ] t 1 chθ t 2 g 1 1 t 1 chθ t [ w 1 t 1 chθ ] t g 1 w 1 g 1 w 1 Then from 3.18 and 3.19 we hae 3.20 [ ] t 1 chθ t 2 w 1 t 2. 1 [ w 1 t 1 chθ w 1 t 2. Ths the deriatie of t 2 with respect to s 1 is written by the Lorentzian ectorial prodct of w 1 and t 2. Similarly it is easy to see that the other ectors can be written by the same method. Corollary 3.2. By sing the ectors t 1 t 2 and N we can express w w 1 and w 2 as follows: 3.21 w t + g R n N t 1 R g [ w 1 t 1 [ w 2 ] chθ α + [ t 2 shα chα ] N R n R n R g chθ R n1 ] t 2 R n1 N R g1 1 1 t 1 + chθ N t 2 + R n2 2 R n2 R g2

6 354. OZYILMAZ AND Y. YAYLI where 1 1 R n and 1 R g respectiely. are geodesic torsion normal cratre and geodesic cratre From 2.4 we can write the Darbox ectors of the t 2 g 2 N} as 3.24 w t + g R n N R g t w g 1 N 1 R n1 R g1 t w g N. 2 R n2 R g2 t g N} t 1 g 1 N } and Then if we consider the eqations and 3.15 according to the ectors t 1 and t 2 and sbstitte in 3.24 we get and Theorem 3.2. If we consider the tangent ectors t 1 and t 2 of the parameter cres c 1 and c 2 on the time-like srface then we obtain the following relations: t i t 1 t ii t 2 t 1 s 2 t 1 s 2. Proof. i From 3.2 t 1 r and t 2 r are written. And also we know that 3.27 t 1 t 2 chθ r r chθ r 2 r r r 2 r r. By taking differential from t 1 r and t 2 r we obtain Ths we write t t 2 r r t 1 r r r r. r

7 3.31 t 1 r r On the other hand we hae RULAR CURVS ON A RULAR TIM-LIK SURFAC 355 r t 1 t 1 1 t 1 2 d 1 s t 1 1 t 2 1 d 1. Ths taking inner prodct of 3.32 and 3.33 by the ector t 2 and the ector t 1 and considering 3.30 and 3.31 we hae 3.26 and The other cases can be seen easily. get Reslt 3.1. If we take differential from t 1 t 2 chθ with respect to and we 3.34 t 1 t 2 t t 1 Ths we hae 3.36 t 1 +. t 1. Theorem 3.3. The following geodesic cratre eqalities are satisfied for the parameter cres c 1 and c R g1 chθ R g2 +.

8 356. OZYILMAZ AND Y. YAYLI Proof. i From 3.25 and 3.20 we write t 1 From here we hae w 1 t 2. t 1 w 1 t 2 w 1 t 2 t 1 w 1. Then from 3.24 if we take inner prodct both of side w 1 with N we obtain Ths N w 1 N 2 R g 1 N w 1 1 R g 1. 1 R g 1 1 is obtained. Similarly ii can be proofed. chθ + Theorem 3.4. Let s consider any cre c on the time-like srface and the arc elements of cres c c 1 and c 2 as s s 1 and s 2 respectiely. Let the Darbox instantaneos rotation ectors of c 1 and c 2 be w 1 and w 2 and if the hyperbolic angle between the tangent t of cre c and t 1 is α then 3.39 w 1 + shα w 2 t 1 A t are satisfied. d t 1 A t 1 d t 2 A t 2 d N A N Proof. If we consider 3.11 and 3.13 then d t 1 t 1 1 s + t 1 2 s 2 s w 1 + shα w 2 is obtained. Similarly the others are satisfied. Reslt 3.2. The following eqality w1 t 1 + shα t 1 A t t 2 d t 1 t 1 d t 2 A N is alid. w2 t 1

9 RULAR CURVS ON A RULAR TIM-LIK SURFAC 357 Theorem 3.5. Let s consider the cres c c 1 and c 2 which are intersect a point P on time-like srface. Let the Darbox instantaneos rotation ectors of these cres at the point P be w w 1 and w 2 respectiely. Then the following eqality is satisfied 3.42 w Proof. From t w 1 + shα w 2 + N dα. t 1 + shα t 2 can be written. Then by taking deriaties with respect to s from eqation 3.43 we obtain 3.44 d t d t 1 + shα d t 2 chθ α t 1 chα dα t 2. } On the other hand considering the Darbox trihedrons t 1 g 1 N we write 3.45 t 1 g 1 N t 2 g 2 N. From 3.14 and 3.15 if g 1 and g 2 are sbstitted in 3.45 we obtain 3.46 t 1 1 chθ t 1 t 2 N t 2 1 chθ t 2 + t 1 N. And then sbstitting the eqations 3.46 in 3.44 we hae d t d t 1 + shα d t 2 chθ α chθ t sh 2 1 t 2 N θ chα t sh 2 1 t 2 chθ N θ dα According to the Theorem 3.4 d t 1 A d t 2 t 1 A t 2 A w 1 + shα w 2 are known. And by sing the trigonometric expression we find d t A t 1 + shα A t 1 + shα t 2 A t + N dα t A + N dα t. and t 2 g 2 N A t 2 dα t 1 + shα t 2 N dα t 1 + shα t 2 N }

10 358. OZYILMAZ AND Y. YAYLI Ths 3.47 d t b t 3.48 b A + N dα are written. After that 3.49 A b N dα is obtained. By writing 3.49 in the third expression of 3.40 we obtain dn 3.50 A N b N dα N b N. Since w is Darbox ector we hae 3.51 d t w t d N w N. Then considering and 3.51 d t w t b t b t w t b w t 3.52 b w λ t 3.53 dn w N b N b N w N b w N are written. At the end if we make eqal 3.52 to 3.53 we hae b w λ t µ N λ µ 0. Finally b w 0 can be written. Ths we get the theorem. Corollary 3.3 eneral Form of ler Formla. Taking dot prodct both of the 3.42 with g we hae following eqation among the timelike cres c c 1 and c 2 : R n shα [ 1 1 ] + 1 [ chα 1 2 R n1 ] shαchθ α R n2 where 1 i 1 R ni i 1 2 and 1 R n are geodesic torsion normal cratres of the parameter cres and the normal cre respectiely. Corollary 3.4 eneral Form of O. Bonnet. Taking dot prodct both of the 3.42 with t we hae following eqation among the timelike cres c c 1 and c 2 : [ 1 chα shα ] + shα [ ] chθ α 1 R n2 2 R n2

11 RULAR CURVS ON A RULAR TIM-LIK SURFAC 359 where 1 1 i R ni i 1 2 and 1 are geodesic torsions normal cratres of the parameter cres and the normal torsion of normal cre respectiely. Corollary 3.5 Lioille Formla. Taking dot prodct both of the 3.42 with N we hae following eqation among the timelike cres c c 1 and c 2 : shα R g R g1 R g2 + dα where 1 R gi and 1 R g i 1 2 are geodesic cratres of the parameter cres and the normal cre respectiely. Now we gie some special cases of the formlae 3.54 and Corollary 3.6. If we take i.e. the parameter cres are cratre lines in 3.55 we hae [ 1 shα ] R n1 R n2 where 1 and 1 R ni i 1 2 are geodesic torsion of normal cre and normal cratre of parameter cres respectiely. Reslt 3.3: If we take i.e. the parameter cres are cratre lines in 3.54 we hae R n 1 where R ni i 1 2 and 1 R n cre respectiely. [ chα R n1 ] shαchθ α R n2 are normal cratres of parameter cres and normal Reslt 3.4: If we take 1 R n1 1 R n2 0 i.e. the parameter cres are asymptotic in 3.54 we hae 3.59 [ 1 shα 1 1 ] R n 1 2 where 1 i i 1 2 and 1 R n are geodesic torsions of parameter cres and normal cratre of normal cre respectiely. RFRNCS [1] Ugrl H. and Çalışkan A. The Space-like and Time-like Srface eometry by the Darbox Vector The press of the Celal Bayar Uni [2] Yılmaz S.; Ozyilmaz. and Trgt M. On The Differential eometry of the Cres in Minkowski Space-Time II Int. Jor. of Compt Math Sci 3: [3] Yaylı Y. and Hacısalihoğl H.H. Closed Cres in the Minkowski 3-space Hadronic J No: 3 [4] Yılmaz S. Spherical Indictors of Cres and Characterizations of Some Special Cres in For Dimensional Lorentzian Space Ph.D. Dissertation Dept. Math. Dokz ylül Uni. İzmir Trkey 2001.

12 360. OZYILMAZ AND Y. YAYLI [5] Akblt F. Bir Yüzey Üzerindeki ğrilerin Darbox Vektörleri. Ü. Faclty of Science İzmir [6] Kızıltğ S. and Yaylı Y. Space-Like Cre on Space-like Parallel Srface in Minkowski 3-space Int. J. Math. Compt No [7] Ali A. T.; Lopez R. and Trgt M. k-type Partially Nll and Nsedo Nll Slant Helices in Minkowski 4-space Math. Commn No [8] Trgt M. and Yilmaz S. On The Frenet Frame and A Characterization of Space-Like Inolte-olte Cre Cople in Minkowski Space-Time Int. Math. Form Vol.3 No