THE NATURAL LIFT CURVE OF THE SPHERICAL INDICATRIX OF A TIMELIKE CURVE IN MINKOWSKI 4-SPACE

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1 Journal of Science Arts Year 5, o (, pp 5-, 5 ORIGIAL PAPER HE AURAL LIF CURVE OF HE SPHERICAL IDICARIX OF A IMELIKE CURVE I MIKOWSKI -SPACE EVRE ERGÜ Manuscript received: 65; Accepted paper: 55; Published online: 65 Abstract In this study,some interesting results about the original timelike curve were obtained depending on the assumption that the natural lift curves should be the integral curve H of the geodesic spray on the tangent bundle ( S ( Keywor: atural Lift, Geodesic Sprays IRODUCIO AD PRELIMIARIES In differential geometry, there are many important consequences properties of curve [,, ] horpe gave the concepts of the natural lift curve geodesic spray in [9] horpe provied the natural lift of the curve is an integralcurve of the geodesic spray iff is an geodesic on M in [9]Çalışkan, Sivridağ Hacısalihoğlu studied the natural lift curves of the spherical indicatries of tangent, principal normal, binormal vectors fixed centrode of a curve in []hey 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 spray on the tangent bundle ( S in [] M -integral curve of Z M - geodesic spray are defined by Sivridağ Çalışkan hey gave the main theorem: he natural lift of the curve (in M is an M -integral curve of the geodesic spray Z iff is an M -geodesic in [] Bilici, Çalışkan Aydemir studied (, being the pair of evolute-involute curves,the natural lift curve of the spherical indicatries of tangent, principal normal, binormal vectors of the involute curve hey 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 spray in Minkowski -space in Minkowski -space in [5,6 ]he anologue of the theorem of horpe was given in Minkowski -space in Minkowski -space by Ergün Çalışkan in [,5 ] Çalışkan Ergün defined M -vector field Z, M -geodesic spray, M -integral curve of Z, M -geodesic in []he anologue of the theorem of Sivridağ Çalışkan was given in Minkowski -space by Ergün Çalışkan in [] At each point of a differentiable curve a tetrad of mutually orthogonal unit vectors (called tangent, normal,binormal, trinormal was defined constructed he rates of changes of these vectors along the curve Ondokuz Mayıs University Çarşamba Chamber of Commerce Vocational School, Çarşamba, Samsun, urkey eergun@omuedutr ISS: 8 958

2 6 he natural lift curve of define the curvatures of the curve in the space E Spherical images (indicatrices are a wellknown concept in classical differential geometry of curves [6] o meet the requirements in the next sections, here the basic elements of the theory of curves in the space E are briefly presented (A more complete elementary treatment can be found in [] Let Minkowski -space inner product g given by IR be the vector space (, g X X x x x x IR equipped with the Lorentzian where X ( x, x, x, x IR A non-zero vector X ( x, x, x, x IR timelike if g ( X, X <, spacelike if g ( X, X > lightlike (or null if ( he norm of a vector X is defined by is said to be g X, X X IL ( g X, X [] We denote by { (, (, (, ( } t t B t B t the moving Frenet frame along the curve Let be a unit speed timelike space curve he functions, are called the,, B, B is first, second third curvature of Let Frenet vector fiel of be { } spacelike vector field In this trihedron, we assume that B are spacelike vector fiel B is a timelike vector field hen, Frenet formulas are given by, + B, B + B, B B, [] Let be any timelike curve with ( t ypei General Case: he helix is not confined to any - flat in space-time [7] ypeii Degenerate: he helix is lies in a -flat [7] Subtype II a : > Subtype II b : Subtype II c : < ypeiii Degenerate: he helix is lies in a -flat : It is a pseudocircle, a curve of hyperbolic type,[7] wwwjosaro

3 he natural lift curve of 7 ypeiv Degenerate: he helix is a straight line, [7] Let M be a hypersurface in integral curve of X if IR let : I M be a parametrized curve is called an d dt ( ( t X ( ( t ( for all t I where X is a smooth tangent vector field on M, [] We have where P P M P ( M M χ M M is the tangent space of M at P ( M χ is the space of vector fiel of M Definition For any parametrized curve : I M, : I M given by ( t ( t, ( t ( t ( t is called the natural lift of on M,[6] hus, we can write where D is the Levi-Civita connection on d d ( t ( t D t dt dt ( t IR Let be a unit speed timelike space curve hen the natural lift of is a spacelike space curve, [6] Definition A X χ ( M where g (, ε, [6] ( is called a geodesic spray if for V M ( ε ( (, X V g S V V heorem he natural lift of the curve is an integral curve of geodesic spray X if only if is a geodesic on M, [6] ISS: 8 958

4 8 he natural lift curve of HE AURAL LIF CURVE OF HE SPHERICAL IDICARIX OF A IMELIKE CURVE I MIKOWSKI -SPACE Let D, D D be connections in normal vector field of where ε g ( ξ, ξ S IR, S H respectively ξ be a unit H hen Gauss Equations are given by the followings X DXY D Y + ε g ( S ( X, Y ξ, DXY D X Y + ε g ( S ( X, Y ξ, S is the shape operator of ( D ξ S X X S H Let be a unit speed timelike space curve, is timelike vector field,, B B are spacelike vector fiel he atural Lift of the Spherical Indicatrix of angent Vector of of the curve have that is Let be the spherical indicatrix of tangent vectors of be the natural lift If is an integral curve of thegeodesic spray, then from heorem we D D D + ε g S, ξ D ε g S, ( ( ( D D + + B, ( ( D + + B ε g ( ξ, ξ g (,, g S, wwwjosaro

5 he natural lift curve of 9 Using these in the Gauss equation, we immediately have From the Eq ( we get ( ( D + + B + + B Since,, B, B are linearly independent we obtain So from the Eq (7 we can give the following proposition Corollary If the natural lift of is an integral curve of the geodesic on the tangent bundle ( S,then can be helices of ype III he atural Lift of the Spherical Indicatrix of Principal ormal Vector of of the curve we have that is Let be the spherical indicatrix of tangent vectors of be the natural lift If is an integral curve of the geodesic spray, then because of heorem D D D + ε g S, ξ D ε g S, ISS: 8 958

6 he natural lift curve of ( D D + B ( ( B ( B ( +, ( ( ( D + + B + + B ε g ( ξ, ξ g (,, g S, ( + Using these in the Gauss equation, we immediately have ( ( D + + B + B ( + + ( + From the Eq (8 we get ( ( + ( + B B + + ( + ( + Since,, B, B are linearly independent we obtain, ( ( ( + ( + ( + wwwjosaro

7 he natural lift curve of So from the Eq ( we can give the following proposition Corollary If the natural lift of is an integral curve of the geodesic on the tangent bundle ( S,then can be helices of ype II ( Subtype II a or Subtype II c he atural Lift of the Spherical Indicatrix of the Firsth Binormal Vectors of Let be the spherical indicatrix of tangent vectors of B be the natural lift B of the curve If B is an integral curve of the geodesic spray, then by using heorem we have that is B B D B B ( (, D D + ε g S ξ B B B B B B ( (, D g S B ε B B B B B B B ( D D + B ( ( B B ( B ( B, B B ( ( D B B + + ( ( ( B ( + ( ( B ε g ξ, ξ g B, B, g S, + Using these in the Gauss equation, we immediately have B B ( ( D + B + B ( + + B ( + From the Eq ( we get ISS: 8 958

8 he natural lift curve of + ( + ( + + ( B B ( ( Because,, B, B are linearly independent, we have, ( + ( ( + ( + ( + So from the Eq (7 we can give the following proposition Corollary If the natural lift B of is an integral curve of the geodesic on the tangent bundle ( S,then can be helices of ype II or helices of ype III B he atural Lift of the Spherical Indicatrix of the Second Binormal Vectors of Let be the spherical indicatrix of tangent vectors of B be the natural lift B of the curve If B is an integral curve of the geodesic spray, then by using heorem we have that is B D B B ( D D ε g S, + ξ B B B B B B ( D ε g S, B B B B B wwwjosaro

9 he natural lift curve of B B B B ( D D B ( B ( B + B, B B ( ( D B + B ( ( ( B ε g ξ, ξ g B, B, g S, B Using these in the Gauss equation, we immediately have D B + B B B B From the Eq (8 we get ( + B + B Because,, B, B are linearly independent, we have, So from the Eq ( we can give the following proposition Corollary If the natural lift B of B is an integral curve of the geodesic on the tangent bundle ( satisfying this condition H then we have herefore there is no such curve ISS: 8 958

10 he natural lift curve of REFERECES [] Sivridağ, Aİ, Çalışkan, M, ErcUni Fen Bil Derg, 7(, 8, 99 [] O'eill, B, Elementery Differential Geometry, Academic Press, ew York London, 967 [] O'eill, B, Semi-Riemannian Geometry, with applications to relativity, Academic Press, ew York, 98 [] Bonnor, W B, ensor,, 9, 969 [5] Ergün, E, Çalışkan, M, International Journal of Contemp Math Sciences, 6(9, 99, [6] Ergün, E, Çalışkan, M, International Journal of Contemporary Mathematical Sciences, 7(, 575, [7] Ergün, E, Çalışkan, M, International Mathematical Forum, 7(5, 77, [8] Ergün, E, Bilici, M, Çalışkan, M, Journal of Science Arts, (, 9, 5 [9] horpe, JA, Elementary opics In Differential Geometry, Springer-Verlag, ew York, Heidelberg-Berlin, 979 [] Walrave, J, Curves Surfaces in Minkowski Space, K U Leuven Faculteit Der Wetenschappen, 995 [] Bilici, M, Çalışkan, M, Aydemir, İ, Journal of Applied Mathematics, (, 5, [] Çalışkan, M, Sivridağ, Aİ, Hacısalihoğlu, HH, Some Characterizations for the natural lift curves the geodesic spray, Communications, Fac SciUnivAnkara Ser AMath,, 98 [] Çalışkan, Ergün, E, Int J of Contemp Math Sciences, 6(9, 95, [] Petroviç orgasev, M, Sucuroviç, E, ovi Sad J Math, (, 55, [5] Do Carmo, MP, Differential Geometry of Curve Surface, Prentice-Hall Inc Englewood Cliffs, ew Jersey, 976 [6] Milman R S, Parker G D, Elements of Differential Geometry, Prentice-Hall Inc, Englewood Cliffs, ew Jersey, 977 [7] Synge, J L, Proc Roy Irish Academy, A65, 7, 967 wwwjosaro

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

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