INTEGRAL CHARACTERIZATIONS FOR TIMELIKE AND SPACELIKE CURVES ON THE LORENTZIAN SPHERE S *

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1 Iranian Journal of Science & echnology ranaction A Vol No A Printed in he Ilamic epublic of Iran 8 Shiraz Univerity INEGAL CHAACEIZAIONS FO IMELIKE AND SPACELIKE CUVES ON HE LOENZIAN SPHEE S * M KAZAZ ** H H UGULU AND A OZDEMI Department of Mathematic Faculty of Art and Science Univerity of Celal Bayar Muradiye Campu 57 Mania urkey mutafakazaz@bayaredutr Gazi Univerity Gazi Faculty of Education Department of Secondary Education Science and Mathematic eaching Mathematic eaching Program Ankara urkey hugurlu@gaziedutr Abtract V Dannon howed that pherical curve in E can be given by Frenet-like equation and he then gave an integral characterization for pherical curve in E In thi paper Lorentzian pherical timelike and pacelike curve in the pace time are hown to be given by Frenet-like equation of timelike and pacelike curve in the Euclidean pace E and the Minkowki -pace hu finding an integral characterization for a Lorentzian pherical -timelike and pacelike curve i identical to finding it for E curve and -timelike and pacelike curve In the cae of E curve the integral characterization coincide with Dannon Let { N B be the moving Frenet frame along the curve α () in the Minkowki pace Let α () be a unit peed C -timelike (or pacelike) curve in o that α ( ) = hen α () i a Frenet curve with curvature κ () and torion τ () if and only if there are contant vector a and b o that (i) ( ) = κ( ) { aco ξ( ) + bin ξ( ) + co [ ( ) ( )] ( ) ( ) ξ ξ δ δ κ δ dδ i timelike ξ ξ (ii) ( ) = κ( ) { ae + be + coh ( ξ( ) ξ( δ) ) ( δ) κ( δ) dδ N i timelike where ξ () = τδ ( ) dδ Keyword Lorentzian -phere timelike curve pacelike curve curvature INODUCION he differential equation characterizing a pherical curve i given by d dρ + ρ() τ() = d τ () d () where i the length of arc ρ() = κ() i the radiu of curvature and τ () i the torion of the curve [] Breuer and Gottlieb [] gave an explicit olution of thi differential equation: where a and b are arbitrary contant [] ρ() = aco τδ ( ) dδ+ bin τδ ( ) dδ eceived by the editor July 5 6 and in final revied form January 8 Correponding author

2 6 M Kazaz / et al Let α () be a unit peed C curve in E o that α ( ) = In [] V Dannon howed that α i a Frenet curve with curvature κ () and torion τ () if and only if there are contant vector a and b o that { ( ) = κ( ) aco ξ( ) + bin ξ( ) co ξ( ) ξ( δ) ( δ) κ( δ) dδ where ξ () = τδ ( ) dδ he differential equation characterizing a Lorentzian pherical curve in Minkowki -pace explicit olution of thi differential equation are given in [5 6] On the other hand it i alo hown in [5 6] that a unit peed timelike (rep pacelike) on a Lorentzian phere if and only if there i a function f C o that ( κ) fτ = f + τ (rep f τ ) κ = κ = Set ρ = κ and conider thoe equation rewritten in the form ρ = τ f f = τρ (rep f = τρ) and curve lie n hu the characteritic of a Lorentzian pherical curve in i the Frenet Pattern So the problem of getting an integral characterization to a Lorentzian pherical curve i nothing ele but the integration of Frenet equation hi mean that Lorentzian correpondence of the Breuer-Gottlieb characterization i iomorphic to the integral characterization of Frenet equation in the Lorentzian plane hen the method ued to obtain characterization of Lorentzian pherical curve can be extended to include Frenet equation in Space-time i a Euclidean pace PELIMINAIES provided with the tandard flat metric given by g = dx + dx + dx dx where ( x x x x ) i a rectangular coordinate ytem in Since g i an indefinite metric an arbitrary vector v can have one of three caual character: it can be pacelike if g( v v ) > or v = timelike if g( v v ) < and null (lightlike) if g( v v ) = and v Similarly an arbitrary curve α = α() in can locally be pacelike timelike or null (lightlike) if all of it velocity vector α ( ) are repectively pacelike timelike or null Alo recall that the norm of a vector v i given by v = g( v v) herefore v i a unit vector if g( v v ) = ± wo vector v w in are aid to be orthogonal if g( v w ) = he velocity of a curve α () i given by α ( ) he Lorentzian phere with center m= ( m m m m) and radiu r + in the pace-time i the hyperquadric { ( ) g( ) S = a = a a a a a m a m = r Let u denote the moving Frenet frame along the pacelike curve α () in the pace by { hen and are the tangent the principal normal the firt binormal and econd binormal vector field repectively A timelike or pacelike curve α () i aid to be parameterized by arc length function if g( α ( ) α ( )) = Iranian Journal of Science & echnology ran A Volume Number A Winter 8

3 Integral characterization for 7 Let α () be a curve in the pace time parameterized by arc length function hen for the curve α () the following Frenet equation are given in [7]: If i timelike and the other are pacelike then the Frenet formulae ha the form κ κ τ = τ µ µ if i timelike and the other are pacelike then Frenet equation are given by κ κ τ = τ µ µ and finally if i timelike and the other are pacelike then κ κ τ = τ µ µ S CUVES AND FENE CUVES Similar to the extenion (baed on Wong condition [8]) given by Dannon [] we give an extenion of Petrovic-orgaev and Sucurovic condition [5 6] to for expreing the connection between S curve and Frenet curve In ince i the fourth orthonormal vector to µ () = g( ) meaure the change of direction of the pace panned by hen we have the following cae: Cae : i timelike: Propoition Let α () be an 5 unit peed C timelike Frenet curve with curvature function κ() τ () µ () hen the following are equivalent: i) α () lie on a Lorentzian phere ii) κ() and there are two C function f () and g() o that ρ = τ f f = τρ + µ g ( ρ = κ ) g = µ f () Note that the equation () are the Frenet formulae of an E curve in Proof: i) ii ) Aume that α () lie on an Lorentzian phere of radiu a which we may aume to have center x at the origin By repeated differentiation of a = g( α α) and uing Frenet equation we obtain κ and f = Ff where fi = g( α i) i = ( ρ = f f = f and g = f ) Winter 8 Iranian Journal of Science & echnology ran A Volume Number A

4 8 M Kazaz / et al ii) i ) Given f = α fii = cont x i= x x = fi i= herefore Ff and define the curve hen we have γ() = α f i i () then d f i i = i= d α i= α x f Uing the orthogonality give = i i i= g( α α ) Differentiation of thi give g( α x α x) = o g( α x α x) = cont= a ie α lie on the Lorentzian -phere of radiu a about x Cae : i timelike: 5 Propoition Let α () be an unit peed C pacelike Frenet curve with curvature function κ() τ () µ () hen the following are equivalent: i) α () lie on an Lorentzian phere ii) κ() and there are two C function f () and g() o that ρ = τ f f = τ ρ + µ g ( ρ = κ ) g = µ f and f + g > ρ () Note that the equation () are the Frenet formulae of an Proof: he proof i imilar to that of Propoition timelike curve Cae : i timelike: Propoition Let α () be an 5 unit peed C pacelike Frenet curve with curvature function κ() τ () µ () hen the following are equivalent: i) α () lie on an Lorentzian phere ii) κ() and there are two C function f () and g() o that ρ = τ f f = τ ρ + µ g ( ρ = κ ) g = µ f and ρ + g > f () Proof: he proof i imilar to that of Propoition HE INEGAL CHAACEIZAIONS OF FENE CUVES In ection we ee that timelike or pacelike pherical curve equation have the tructure of timelike or pacelike Frenet curve Conequently finding an integral characterization for an Lorentzian pherical curve i identical to finding it for an curve hu to obtain characterization of Lorentzian pherical curve we extend the method given in [] Cae : i timelike: he two bottom E Frenet equation are of the form g ( ) = λ f + µ h h( ) = µ g Auming that µ i non vanihing (the concluion are free from thi aumption) then we get g µ = ( λ f µ ) + h Differentiation then ubtitution of h and eventually application of the change of variable ξ () = µδ ( ) dδ reduce thi equation to g + g= f i ( λ µ ) Iranian Journal of Science & echnology ran A Volume Number A Winter 8

5 Integral characterization for 9 where the variable i ξ A particular olution for g( ξ ) i hen we have: [ ] co ξ ( ) ξδ ( ) λδ ( ) f ( δ ) dδ heorem Let α () be a C curve in E parameterized by it arc length o that α ( ) = hen the following are equivalent: i) α () ha a Frenet ytem N B curvature κ () and torion τ () that atify the Frenet equation ii) here are contant vector a and b o that where ξ () = τδ ( ) dδ ( ) = κn N( ) = κ + τb B( ) = τn { ( ) = κ( ) acoξ + binξ co ξ( ) ξ( δ) ( δ) κ( δ) dδ Proof : he characterization coincide with Dannon Cae : i timelike: he two bottom timelike Frenet equation are of the form g ( ) = λ f + µ h h( ) = µ g Auming that µ i non vanihing (the concluion are free from thi aumption) then we get g µ = ( λ f µ ) + h Differentiation then ubtitution of h and eventually application of the change of variable ξ () = µδ ( ) dδ reduce thi equation to g + g= f i ( λ µ ) where the variable i ξ A particular olution for g( ξ ) i hen we have: [ ] co ξ ( ) ξδ ( ) λδ ( ) f ( δ ) dδ heorem Let α () be a C timelike curve in parameterized by it arc length o that α ( ) = hen the following are equivalent: i) α () ha a Frenet ytem N B curvature κ () and torion τ () that atify the Frenet equation ii) here are contant vector a and b o that ( ) = κn N( ) = κ + τb B( ) = τn Winter 8 Iranian Journal of Science & echnology ran A Volume Number A

6 where ξ () = τδ ( ) dδ M Kazaz / et al { ( ) = κ( ) acoξ + binξ + co ξ( ) ξ( δ) ( δ) κ( δ) dδ Proof: ii) i ) Suppoe the condition hold Put and N () = co ξ () ξδ ( ) ( δκδ ) ( ) dδ+ acoξ+ binξ B () = in ξ ( δ) ξ() ( δ) κ( δ) dδ ainξ + bco ξ hen N and B atify the Frenet equation i) ii ) If the equation hold the above N and B olve the coupled ytem he lat equation κ N = i our condition N = κ + τb B = τn Cae : i timelike: he two bottom pacelike Frenet equation are of the form g ( ) = λ f + µ h h( ) = µ g Auming that µ i non vanihing (the concluion are free from thi aumption) then we get g µ = ( λ f µ ) + h Differentiation then ubtitution of h and eventually application of the change of variable ξ () = µδ ( ) dδ reduce thi equation to g g= f i ( λ µ ) where the variable i ξ A particular olution for g( ξ ) i hen we have: [ ] coh ξ ( ) ξδ ( ) λδ ( ) f ( δ ) dδ heorem Let α () be a C pacelike curve in parameterized by it arc length o that α ( ) = hen the following are equivalent: i) α () ha a Frenet ytem N B curvature κ () and torion τ () that atify the Frenet equation ii) here are contant vector a and b o that ( ) = κn N( ) = κ + τb B( ) = τn ξ ξ { ( ) ( ) = κ( ) ae + be + coh ξ( ) ξ( δ) ( δ) κ( δ) dδ Iranian Journal of Science & echnology ran A Volume Number A Winter 8

7 Integral characterization for where ξ () = τδ ( ) dδ Proof: ii) i ) Suppoe the condition hold Put and ξ N () = coh ξ() ξ( δ) ( δ) κ( δ) dδ + ae+ be ξ ξ B () = inh ξδ ( ) ξ() ( δκδ ) ( ) dδ+ ae be hen N and B atify the Frenet equation i) ii ) If the equation hold the above N and B olve the coupled ytem N = κ + τb B = τn he lat equation κ N = i our condition n he method extend to timelike Frenet curve through ucceive application of tranformation of the form A ( δ ) dδ by uing the terminology of [9] EFEENCES Kreyzig E (959) Differential geometry oronto Univ of oronto Pre Breuer S & Gottlieb D (97) Explicit characterization of pherical curve Proc Amer Math Soc Wong Y C (97) On an explicit characterization of pherical curve Proc Amer Math Soc 9- Dannon V (98) Integral Characterization and the heory of Curve Proc Amer Math Soc 8() Petrović-orgašev M & Šućurović E () Some characterization of Lorentzian pherical pace-like curve with the timelike and the null principal normal Mathematica Moravica Petrović-orgašev M & Šućurović E () Some characterization of the Lorentzian pherical timelike and null curve Matematiqki Venik Walrave J (995) Curve and urface in Minkowki pace Doctoral hei K U Leuven Faculty of Science Leuven 8 Wong Y C (96) A global formulation of the condition for a curve to lie in a phere Monath Math Klingenberg W (978) A coure in differential geometry Berlin and New York Springer-Verlag ξ Winter 8 Iranian Journal of Science & echnology ran A Volume Number A

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