M -geodesic in [5]. The anologue of the theorem of Sivridağ and Çalışkan was given in Minkowski 3-space by Ergün

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1 Scholars Journal of Phsics Mathematics Statistics Sch. J. Phs. Math. Stat. 5; ():- Scholars Academic Scientific Publishers (SAS Publishers) (An International Publisher for Academic Scientific Resources) ISS (Print) ISS (Online) Some Particular Eamples for the atural Lift Curve in Minkowski -Space Evren ERGÜ* Mustafa ÇALIŞA Ondoku Mas Universit Çarşamba Chamber of Commerce Vocational School Samsun urke Gai Universit Facult of Sciences Department of Mathematics Ankara urke *Corresponding Author: Evren ERGÜ Abstract: In this stud we give some particular eamples for the natural lift curves of the spherical indicatries of tangent principal normal binormal vectors. ewords: atural Lift Geodesic Spra IRODUCİO horpe gave the concepts of the natural lift curve geodesic spra in []. horpe provied the natural lift of the curve is an integral curve of the geodesic spra iff is an geodesic on M in []. Çalışkan Sivridağ Hacısalihoğlu studied the natural lift curves of the spherical indicatries of tangent principal normal binormal vectors fied centrode of a curve in []. he gave some interesting results about the original curve were obtaied depending on the assumption that the natural lift curve should be the integral curve of the geodesic spra on the tangent bundle S in []. Some properties of M -vector field Z defined on a hpersurface M of M were studied b Agashe in []. M -integral curve of Z M -geodesic spra are defined b Çalışkan Sivridağ. he gave the main theorem: he natural lift of the curve (in M ) is an M -integral curve of the geodesic spra Z iff is an M -geodesic in [4]. ilici Çalışkan Ademir studied being the pair of evolute-involute curvesthe natural lift curve of the spherical indicatries of tangent principal normal binormal vectors of the involute curve. he gave some interesting results about the evolute curve were obtained depending on the assumption that the natural lift curve of the spherical indicatrices of the involute should be the integral curve on the tangent bundle S in []. Ergün Çalışkan defined the concepts of the natural lift curve geodesic spra in Minkowski -space in [6]. he anologue of the theorem of horpe was given in Minkowski -space b Ergün Çalışkan in [6]. Çalşkan Ergün defined M -vector field Z M -geodesic spra M -integral curve of Z M -geodesic in [5]. he anologue of the theorem of Sivridağ Çalışkan was given in Minkowski -space b Ergün Çalışkan in [5]. Walrave characteried the curve with constant curvature in Minkowski -space in []. Let Minkowski -space be the vector space equipped with the Lorentian inner product g given b g X X where X. A vector X is said to be timelike if spacelike if g X X lightlike (or null) if g X X. Similarl an arbitrar curve t g X X in where t is a pseudo-arclength parameter can locall be timelike spacelike or null (lightlike) if all of its velocit vectors t are respectivel timelike spacelike or null (lightlike) for ever ti. A lightlike vector X is said to be positive (resp. negative) if onl if ( resp. ) a timelike vector X is said to be Available Online:

2 positive (resp. negative) if onl if ( resp. ). he norm of a vector X is defined b X IL g X X [8]. he Lorentian sphere hperbolic sphere of radius in S X : g X X are given b H X : g X X respectivel[8]. he vectors X Y g X X [8].. ow let X Y be two vectors in X Y []. We denote b are orthogonal if onl if then the Lorentian cross product is given b t t t the moving Frenet frame along the curve. hen are the tangent the principal normal the binormal vector of the curve respectivel. Let be a unit speed timelike space curve with curvature torsion. Let Frenet vector fields of be. In this trihedron is timelike vector field are spacelike vector fields.for this vectors we can write where is the Lorentian cross product [] in space [].. hen Frenet formulas are given b he Frenet instantaneous rotation vector for the timelike curve is given b W. Let be a unit speed spacelike space curve with a spacelike binormal. In this trihedron we assume that are spacelike vector fields is a timelike vector field In this situation hen Frenet formulas are given b []. he Frenet instantaneous rotation vector for the spacelike space curve with a spacelike binormal is given b W. Let be a unit speed spacelike space curve with a timelike binormal. In this trihedron we assume that are spacelike vector fields is a timelike vector field.in this situation hen Frenet formulas are given b []. Available Online:

3 he Frenet instantaneous rotation vector for the spacelike space curve with a timelike binormal is given b W. heorem : Let be a unit speed timelike space curve. hen we have ) if onl if is a part of a timelike straight line; ) if onl if is a planar timelike curve; ) constant if onl if is a part of a orthogonal hperbola; 4) constant constant if onl if is a part of a timelike circular heli s s cos s sin s with ; 5) constant constant if onl if is a timelike hperbolic heli s sinh s s cosh s with ; 6) constant constant if onl if can be parameteried b s s 6 s s s []. 6 heorem: Let be a unit speed spacelike space curve with a spacelike binormal. hen we have ) constant if onl if is a part of a orthogonal hperbola; ) constant constant if onl if is a part of a spacelike hperbolic heli s cosh s s sinh s with []. heorem: Let be a unit speed spacelike space curve with a timelike binormal. hen we have ) constant if onl if is a part of a circle; ) constant constant if onl if is a part of a spacelike hperbolic heli s sinh s s cosh s with ; ) constant constant if onl if is a part of a spacelike circular heli s s cos s sin s with ; Available Online:

4 4) constant constant if onl if can be parameteried b s s s 6 s s []. 6 Definition4: Let M be a hpersurface in integral curve of X if d t X t for all t I dt let : I M be a parametried curve. is called an where X is a smooth tangent vector field on M [8]. We have M U M M where PM P P M is the tangent space of M at P M is the space of vector fields of M. Definition 5: For an parametried curve : I M : I M given b t t t t is called the natural lift of on. d d t t D t dt dt t M hus we can write where D is the Levi-Civita connection on [6]. Definition 6: A X M X V g S V V where g [6]. is called a geodesic spra if for V M heorem 7: he natural lift of the curve is an integral curve of geodesic spra geodesic on M [6]. X if onl if is a Some Particular Eamples for the atural Lift Curve in Minkowski -Space Let D D D be connections in of S S H. hen Gauss Equations are given b the followings H respectivel be a unit normal vector field D Y D X Y g S X Y X X X where g D Y D Y g S X Y S is the shape operator of S H. Available Online: 4

5 Let be a unit speed timelike space curve. Corollar 8: If the natural lift of then the curve is a part of an orthogonal hperbola []. Corollar 9: If the natural lift of H is an integral curve of the geodesic on the tangent bundle S then is an integral curve of the geodesic on the tangent bundle the curve can be classified as i. constant if onl if is a part of a orthogonal hperbola; with ii. constant constant if onl if is a part of a timelike circular heli s s cos s sin s ; iii. constant constant if onl if is a timelike hperbolic heli s sinh s s cosh s with []. Corollar : If the natural lift of is an integral curve of the geodesic spra on the tangent bundle S then we have. herefore there is no curve which holds this condition []. Let be a unit speed spacelike space curve with a spacelike binormal. Corollar : If the natural lift of then the curve is a part of an orthogonal hperbola []. Corollar : If the natural lift then the curve can be classified as i. of is an integral curve of the geodesic on the tangent bundle S is an integral curve of the geodesic on the tangent bundle H constant if onl if is a part of a orthogonal hperbola; ii. constant constant if onl if is a part of a spacelike hperbolic heli s cosh s s sinh s []. with Corollar : If the natural lift of is an integral curve of the geodesic spra on the tangent bundle S then we have. herefore there is no curve which holds this condition[]. Available Online: 5

6 Let be a unit speed spacelike space curve with a timelike binormal. Corollar 4: If the natural lift of then the curve is a part of a circle[]. Corollar 5: If the natural lift then the curve can be classified as i. with of is an integral curve of the geodesic on the tangent bundle S is an integral curve of the geodesic on the tangent bundle S constant if onl if is a part of a circle; ii. constant constant if onl if is a part of a spacelike hperbolic heli s sinh s s cosh s ; iii. constant constant if onl if is a part of a spacelike circular with heli s s cos s sin s []. Corollar 6: If the natural lift Available Online: 6 of is an integral curve of the geodesic spra on the tangent bundle H then we have. herefore there is no curve which holds this condition []. Eample 7: Let s s cos s sin s be a unit speed (timelike curve) timelike circular heli with; s sin s cos s s cos s sin s s sin s cos s s s s sin cos s s s cos sin s sin s cos s

7 Eample 8: Let s sinh s s cosh s hperbolic heli with; be a unit speed (timelike curve) timelike Available Online: 7

8 s cosh s sinh s s sinh s cosh s s cosh s sinh s s s s cosh sinh s s s sinh cosh cosh sinh s s s Eample 9: Let s 5 cosh 5 s 5 s 5 sinh 5s be a unit speed (spacelike curve with spacelike binormal) spacelike hperbolic heli with; Available Online: 8

9 5 5 5 s sinh 5 s cosh 5 s s cosh 5 s sinh 5 s s sinh 5 s cosh 5 s s s s sinh 5 cosh s s s cosh 5 sinh 5 sinh 5 cosh s s s Eample : Let s sinh s scosh s binormal)spacelike hperbolic heli with; be a unit speed (spacelike curve with timelike Available Online: 9

10 s cosh s sinh s s sinh s cosh s s cosh s sinh s s s s s s s cosh sinh sinh cosh s s s cosh sinh Eample : Let s s cos s sin s binormal) spacelike circular heli with; be a unit speed (spacelike curve with timelike Available Online:

11 s sin s cos s s cos s sin s s sin s cos s s s s sin cos s s s cos sin sin cos. s s s Available Online:

12 REFERECES. Agashe S; Curves associated with an M-vector field on a hpersurfacem of a Riemmanian manifold M ensor.s. 974; 8:7-.. Akutagawa ishikawa S; he Gauss Map Spacelike Surfacewith Prescribed Mean Curvature in Minkowski-Spaceöhoko Math. J. 99; 4: ilici M Çalşkan M Ademir İ; he natural lift curves the geodesic spras for the spherical indicatrices of the pair of evolute-involute curvesinternational Journal of Applied Mathematics ;(4): Çalşkan M Sivridağ Aİ Hacsalihoğlu HH; Some Characteriationsfor the natural lift curves the geodesic spra Communications Fac. Sci.Univ. Ankara Ser. A Math 984; (8): Çalşkan M Ergün E; On he -Integral Curves -Geodesic SprasIn Minkowski -Space International Journal of Contemp. Math. Sciences ; 6(9): Ergün E Çalşkan M; On Geodesic Spras In Minkowski -SpaceInternational Journal of Contemp. Math. Sciences ; 6(9): Lambert MS Mariam Susan FH; Darbou Vector. VDMPublishing House. 8. O'eill ; Semi-Riemannian Geometr with applications to relativit.academic Press ew York Ratcliffe JG; Foundations of Hperbolic Manifolds Springer-Verlagew York Inc. ew York Sivridağ Aİ Çalşkan M; On the -Integral Curves -Geodesic Spras Erc.Uni. Fen il. Derg 99; 7():8-87. horpe JA; Elementar opics In Differential GeometrSpringer-Verlagew York Heidelberg-erlin Walrave J; Curves Surfaces in Minkowski Space. U. Leuven FaculteitDer Wetenschappen Ergün E; On the Lift Curves the Geodesic Spras in Minkowski -Space Ph.D. hesisondoku Mas Universit Institute of Science Samsun. Available Online: