THE DARBOUX TRIHEDRONS OF REGULAR CURVES ON A REGULAR TIME-LIKE SURFACE. Emin Özyilmaz

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1 Mahemaical and Compaional Applicaions, Vol. 9, o., pp. 7-8, 04 THE DARBOUX TRIHEDROS OF REULAR CURVES O A REULAR TIME-LIKE SURFACE Emin Özyilmaz Deparmen of Mahemaics, Facly of Science, Ee Uniersiy, TR-500 İzmir, Trkey Absrac- In his work, we sdy Differenial eomery in he Minkowski -space R of he cres on a ime-like relar srface by sin parameer cres which are no perpendiclar o each oher. The aim of his sdy is o inesiae he formlae beween he Darbox Vecors of he ime-like cre () c, he ime-like parameer cre ( c ) and he space-like parameer cre ( c which are no inersecin perpendiclarly. Key wor- Time-like and space-like cres, ime-like srface, minkowski -space.. ITRODUCTIO Classical differenial eomery of he cres may be srronded by he opics which are eneral helices, inole-eole cre coples, spherical cres and Berrand cres. Sch special cres are inesiaed and sed in some of real world problems like mechanical desin or roboics by well-known Frene-Serre eqaions. A he beinnin of he wenieh cenry, Einsein s heory opened a door o new eomeries sch as Minkowski space-ime, which is simlaneosly he eomery of special relaiiy and he eomery indced on each fixed anen space of an arbirary Lorenzian manifold. In recen years, he heory of deenerae sbmanifol has been reaed by researchers and some of he classical differenial eomery opics hae been exended o Lorenzian manifol. Some ahors hae aimed o deermine Frene Serre inarians in hiher dimensions. There exiss a as lierare on his sbjec, for insance [-4,6,7]. In he lih of he aailable lierare, in [4] he ahor exended spherical imaes of cres o a for-dimensional Lorenzian space and sdied sch cres in he case where he base cre is a space-like cre accordin o he sinare (+,+,+,-). By sin he Darbox ecor, arios well-known formlas of differenial eomery had been prodced by [5]. Then, in [], ahors had been ien hese formlae in Minkowski -space R. In his work, we inesiae he formlae beween he Darbox Vecors of he cre () c, he parameer cres ( c ) and ( c which are no inersecin perpendiclarly. Ths, we will find an opporniy o inesiae relar ime-like srface by akin he parameer cres which are inersecin nder he anle.

2 The Darbox Trihedrons of Relar Cres on a Relar Time-Like Srface 7. PRELIMIARIES To mee he reqiremens in he nex secions, here, he basic elemens of he heory of cres in he space R are briefly presened (A more complee elemenary reamen can be fond in [] ). The Minkowski -space E proided wih he sandard fla meric is ien by, dx dx dx (.) where ( x, x, x ) is a recanlar coordinae sysem of arbirary ecor a R is ien by a a, a R. Recall ha, he norm of an. Le ϕ =ϕ (s ) be a relar cre in ϕ is called a ni speed cre if he elociy ecor of ϕ saisfies. For he ecors, w R hey are said o be orhoonal if and only if, 0. On he oher hand, if we consider any orhoonal rihedron as e, e, e wrie heir deriaie formlaes as follows: dei w ei i,, d R. we can (. where e is ime-like ecor, e and e are he space-like ecors and Darbox insananeos roaion ecor is w ae be ce Moreoer, is Lorenzian ecoral prodc,[]. Le s ake a ime-like srface as y y(, ), n, b he moin Frene-Serre frame alon he ime-like cre frame on y y(, ) cre. Denoe by is he Darbox rihedron as c on y y(, ). Anoher orhoonal,,. For an arbirary ime-like c on ime-like srface, he orienaion of he Darbox rihedron is wrien as,, (.) and he Darbox deriaie formlae can be wrien as follows: d d d w, w, w And also he Darbox ecor of his rihedron is wrien as w T R n R where,,,,, and, T R and n R normal crare and eodesic crare, respeciely,[]. (.4) (.5) are eodesic orsion,

3 74 E. Özyilmaz. THE DARBOUX VECTOR FOR THE DARBOUX TRIHEDRO OF A TIME-LIKE CURVE Le s express he parameer cres =cons. as ( c ) and =cons. as ( c which are on a ime-like srface y y(, ). B, hese cres inersec nder he anle (no perpendiclar). Le any ime-like cre ha is passin hroh a poin P on he srface be (c). Le s ake ime-like and space-like parameer cres which are passin hroh he same poin P as ( c ) and ( c. Le he ni anen ecors of cres (c), ( c ) and ( c a he poin P be, and edes of he Darbox rihedrons of parameer cres are,, Here, hree Darbox rihedrons are wrien as below:,,,,,,,,, respeciely. From, he (.) Le s, s and s be he arc-elemens of he cres (c), ( c ) and ( c, respeciely. Ths, we can wrie r r r E r r (. r where E r r, r r. d d r r Moreoer, becase of he parameer cres inersec nder he anle we hae sh (.),[8]. In he paper he sin will be aken as posiie i.e., i will be assmed ha sh. Then, he normal ecor of ime-like srface is ch (.4) Oher hen, considerin he firs wo formlae of (. in he hird erm, d d d d r r E (.5) is wrien,[].

4 The Darbox Trihedrons of Relar Cres on a Relar Time-Like Srface 75 On he oher hand, le s consider he hyperbolic anle beween and as, and if we ake inner prodc boh sides of (.5) wih and hen d d. ch E sh d d. sh( ) sh E (.6) (.7) are obained. Ths, from (.6) and (.7) ch( ) E ch sh d ch d (.8) are wrien. Finally, if we p (.8) ino (.5), we hae he followin eqaion beween he anen ecors of he cres () c, ( c ) and ( c ch( ) sh ch ch (.9) Here, we shall denoe he arc elemens, and of he parameer cres which are belons o ime-like srface y y(, ), and hen we express as follows: Ed Fdd d Ed d (.0) Ths, considerin (.8) and (.0), we hae ch( ) d E ch (.) sh d ch Corollary. : The hird elemens, and of he Darbox rihedrons,,,,,,, are linearly dependen :

5 76 E. Özyilmaz Proof: If we sbsie he eqaion (.9) in he firs eqaliy of (.) and consider he Darbox rihedrons of ( c ) and ( c we hae ch( ) sh ch ch (. Ths, we e he expression. Teorem.: The Darbox rihedrons,, and,, of he parameers cres ( c ) and ( c of he ime-like srface are wrien by Darbox insananeos ecors as follows: i i wi i, wi i, wi s j s j s j ( i, j, (.) Proof: If we consider he Darbox rihedrons,, and,, of parameer cres, we see ha he normal ecor is coincide. Then, considerin (.4) ( ) ( sh ch ch ch (.4) ( ( sh ch ch ch (.5) are obained. From (., we wrie w, w, w s s s w, w, w s s s If (.4) is sbsied in he hird eqaliy of (.6), we e sh ch sh ( w ) sh s s ch s s ch s w w sh ( w ) sh ( w s ch ch (.6) (.7) (.8) (.9) Then, from (.8) and (.9), we hae

6 The Darbox Trihedrons of Relar Cres on a Relar Time-Like Srface 77 w s (.0) Ths, he deriaie of wih respec o s is wrien by he Lorenzian cross prodc of w and.similarly, i is easy o see ha he oher ecors can be wrien by he same mehod. Corollary.: By sin he ecors follows:, and, we can express ww, and w as sh w T ch Rn ch R n R sh w ch T ch R T n R (. (. (. Proof: From (.5), we can wrie he darbox ecors of he,,,, as,,, and w T R n R w T Rn R w T R n R (.4) Then, if we consider he eqaions (.9), (.4) and (.5) accordin o he ecors and and sbsie in (.4), we e (.,(. and (.. Teorem.: If we consider he anen ecors and of he parameer cres (c ) and (c ) on he ime-like srface, hen we obain he followin relaions: i E sh ) s s E (.5)

7 78 E. Özyilmaz ii Proof: i) From (., sh E ) s s E r E and r sh rr sh E E E r E E r r are wrien. And also, we know ha r r r By akin differenial from r r and, we obain E r E E r E r r Ths, we wrie r E E r sh E r E E r r E sh r E (.6) (.7) (.8) (.9) (.0) (.) On he oher hand, we hae. ( d) s s s. ( Ed) s s E s E (. (.) Ths, akin inner prodc of (. and (.) by he ecor considerin (.0) and (.), we hae (.5) and (.6). The oher cases can be seen easily. Resl.: If we ake differenial from sh and he ecor, and wih respec o and, we e

8 The Darbox Trihedrons of Relar Cres on a Relar Time-Like Srface 79 sh E E E sh (.4) (.5) Ths, we hae E sh sh E E (.6) Theorem.: The followin eodesic crare eqaliies are saisfied for he parameer cres ( c ) and ( c sh E (.7) ( R ) ch E Proof: i) From (.5) and (.0), we wrie From here, we hae sh E (.8) ( R ) ch E E sh s E w s E sh E. w w. ch. w Then, from (.4), if we ake inner prodc boh of side w wih ch we obain

9 80 E. Özyilmaz ch ch w w ( R) ( R) Ths,. sh ( ) ( E) ( R ) ch E is obained. Similarly, (ii) can be proofed. Theorem.4: Le s consider any ime-like cre (c) on he ime-like srface y y(, ) and he arc elemens of cres c, c and c as s, s and s, respeciely. Le he Darbox insananeos roaion ecors of c and c be w, w, and if he hyperbolic anle beween he anen of cre c and is, hen ch( ) sh w w A (.9) ch ch and are saisfied. d d d A, A, A Proof: If we consider (.) and (.), hen d ch( ) sh s s ch ch ch( ) sh ch ch is obained. Similarly, he ohers are saisfied. = w w A w w (.40) Resl.: The followin eqaliy d d ch A. is alid. (.4)

10 The Darbox Trihedrons of Relar Cres on a Relar Time-Like Srface 8 Theorem.5: Le s consider he cres (c), ( c ) and ( c which inersec a poin P on ime-like srface. Le he Darbox insananeos roaion ecors of hese cres a he poin P be w, w and w, respeciely. we obain followin eqaliy beween he darbox ecors w, w and w. ch( ) sh d w w w (.4 ch ch Proof: From (.9) ch( ) sh (.4) ch ch can be wrien. Then, by akin deriaies wih respec o s from eqaion (.4), we obain d ch( ) d sh d sh( ) ch d ch ch ch ch On he oher hand, considerin he Darbox rihedrons,, wrie, From (.4) and (.5), if and are sbsied in (.45) we obain (.44) and,,, we (.45) sh ch sh ch (.46) And hen, sbsiin he eqaions (.46) in (.44), we hae d ch( ) d sh d sh( ) ch d sh sh ch ch ch ch Accordin o he heorem.4, d d ch( ) sh A, A, A w w ch ch are known. And, by sin he rionomeric expression, we find

11 8 E. Özyilmaz d ch( ) sh d ch( ) sh A A ch ch ch ch ch( ) sh d ch( ) sh A ch ch ch ch d A ( ) d A Ths, are wrien. Afer ha, d b d b A d A b (.47) (.48) (.49) is obained. By wriin (.49) in he hird expression of (.40) we obain d d A b b (.50) Since w is Darbox ecor, we hae d d w, w (.5) Then, considerin (.4), (.47), (.50) and (.5) d w b 0 b w b w b w d w b 0 b w b w (.5

12 The Darbox Trihedrons of Relar Cres on a Relar Time-Like Srface 8 b w (.5) are wrien. A he end, if we make eqal (.5 o (.5), we hae b w 0 Finally, bw0 can be wrien. Ths, we e he heorem. 4. REFERECES. H. Url, A. Çalışkan, The Space-like and Time-like Srface eomery by he Darbox Vecor, The press of he Celal Bayar Uni., 0.. S. Yılmaz, E. Ozyilmaz, M. Tr, On The Differenial eomery of he Cres in Minkowski Space-Time II, In.jor. of Comp Mah Sci (, Y. Yaylı, H.H. Hacısalihoğl, Closed Cres in he Minkowski -space, Hadronic Jornal (), S. Yılmaz, Spherical Indicaors of Cres and Characerizaions of some special 4 cres in for dimensional Lorenzian Space, L, Ph.D.disseraion, Dep.Mah.Dokz Eylül Uni, İzmir, Trkey, F. Akbl, Bir Yüzey Üzerindeki Eğrilerin Darbox Vekörleri, E.Ü.Facly of Science, İzmir, S. Kızılğ, Y. Yaylı, Space-like Cre on Space-like Parallel Srface in Minkowski -space E, In. J. Mah. Comp. 9(, 94 05, A.T. Ali, R. Lopez, M. Tr, k-ype parially nll and psedo nll slan helices in Minkowski 4-space, Mahemaical Commnicaions 7(), 9-0, J.. Racliffe, Fondaions of Hyperbolic Manifol, Spriner, 006.