Helix Hypersurfaces and Special Curves

Transcription

1 It J Cotemp Math Scieces, Vol 7,, o 5, 45 Helix Hypersurfaces ad Special Curves Evre ZIPLAR Departmet of Mathematics, Faculty of Sciece Uiversity of Aara, Tadoğa, Aara, Turey evreziplar@yahoocom Ali ŞENOL Caırı Karatei Uiversity, Faculty of Sciece Departmet of Mathematics, Caırı, Turey Yusuf YAYLI Departmet of Mathematics, Faculty of Sciece Uiversity of Aara, Tadoğa, Aara, Turey Abstract I this paper, we study special curves o helix hypersurfaces whose taet space maes a costat ale with a fixed directio i Euclidea -space E Besides, we ivestiate special curves o r -helix hypersurfaces whose taet space maes a costat ale with r liearly idepedet directios Also, we ive the relatios betwee helix hypersurfaces ad the Gauss trasformatios of these surfaces i Euclidea -space Mathematics Subject Classificatio: 5A4, 5B5, 5C4, 5C5 Keywords: r -helix hypersurface, Gauss trasformatio, lie of curvature, eodesic curve, asymptotic curve, slat helix Itroductio I differetial eometry of surfaces, a helix hypersurface i E is defied

2 4 E Ziplar, A Şeol ad Y Yayli by the property that taet plaes mae a costat ale with a fixed directio i [] Di Scala ad Ruiz- Herádez have itroduced the cocept of these surfaces i [] Moreover, they have studied r -helices submaifolds [] That is to say submaifolds such that its taet space maes a costat ale with r liearly idepedet directios Helix hypersurfaces has bee wored i oflat ambiet spaces i [,4] Cermelli ad Di Scala have also studied helix hypersurfaces i liquid cristals i [8] Nowadays, M Ghomi wored out the shadow problem ive by HWete Ad, He metioed the shadow boudary i [7] Ruiz- Herádez ivestiated that shadow bouderies are related to helix submaifolds whose taet space maes costat ale with a fixed directio i [5] AI Nistor has also itroduced certai costat ale surfaces costructed o curves i E i [] Özaldi ad Yaylı ive some characterizatio for a curve lyi o a surface for which the uit ormal maes a costat ale with a fixed directio i [9] Oe of the mai purposes of this wor is to observe the relatios betwee helix hypersurfaces ad special curves i Euclidea -space E Aother purpose of this study is to ive the relatios betwee helix hypersurfaces ad the Gauss trasformatios of these surfaces i Euclidea -space E Prelimiaries Defiitio Let α : Ι ΙR E be a arbitrary curve i E Recall that the curve α is said to be of uit speed ( or parametrized by the arc-leth fuctio s ) if α ( s), α ( s) =, where, is the stadart scalar product i the Euclidea space E ive by for each X, Y = i= x y i, X = ( x, x,, x ), Y = ( y, y,, y ) E ( s), V ( s),, V ( s) V be the movi frame alo α, where the vectors V i are mutually orthooal vectors satisfyi V, = The Freet equatios for Let { } α are ive by V V V = V V i i V i V V V V V

3 Helix hypersurfaces ad special curves 5 Recall that the fuctios i (s) are called the i -th curvatures of α [6] Defiitio Let α : Ι ΙR E be a uit speed curve with ozero curvatures i ( i =,,, ) i E ad let { V, V, V } deote the Freet frame of the curve α We call α a V -slat helix, if the -th uit vector field V maes a costat ale ϕ with a fixed directio Χ, that is, π V, Χ = cos( ϕ), ϕ, ϕ = costat alo the curve, where Χ is uit vector field i E [6] Defiitio Let α : Ι ΙR M be a uit speed curve o a hypersurface M i Euclidea -space E Tae the frame { T, Y = T N, N} alo the curve α, where T is the taet vector field of α ad N is a ormal vector field o M The, the frame { T, Y, N} is called Darboux frame of α with respect to M ad the vector fields T, Y, N are called Darboux vector fields of α with respect to M The Darboux equatios for this frame are Here T = Y N, (*) + Y = T + τ N, N = T τ Y is called the ormal curvature of α o M, is called the eodesic curvature of α o M ad τ is called the eodesic torsio of α o M [9] Defiitio 4 Give a hypersurface M ΙR ad a uitary vector d i Ι R, we say that M is a helix with respect to the fixed directio d if for each q M the ale betwee d ad Tq M is costat Note that the above defiitio is equivalet to the fact that d, ξ is costat fuctio alo M, where ξ is a ormal vector field o M [] Theorem Let H ΙR be a orietable hypersurface i η be a uitary ormal vector field of H The, θ ΙR ( si( θ ) η( x) + cos( θ ) d ), f : H ΙR ΙR, costat f ( x, s) = x + s θ = θ ad let is a helix with respect to the fixed directio d i Ι R, where x H ad s ΙR Here d is the vector (,,,) ΙR such that d is orthooal to η ad H Besides, ξ ( x ) = cos( θ ) η( x) + si( θ ) d is a uitary ormal vector field of f θ, where x H [] Defiitio 5 A submaifold M ΙR is a r helix if there is exist a liear subspace H ΙR of dimesio r = dim(h ) such that M is a helix with

4 6 E Ziplar, A Şeol ad Y Yayli respect to ay directio directios [] d H The subspace H is called the subspace of helix MAIN THEOREMS Theorem Let M be a helix hypersurface with the directio d i E ad let α : Ι ΙR M be a uit speed curve o M If d Sp{ N, Y} alo the curve α, the α is a lie of curvature o M ad = ta(θ ) = costat, where { Y} N, are Darboux vector fields of α ad θ is the costat ale betwee d ad N Coversely, ıf α is a lie of curvature o M, the d Sp{ N, Y} Proof: Sice M is a helix hypersurface with the directio d ad d Sp{ N, Y}, we ca write as d = cos( θ ) N + si( θ) Y () alo the curve α, where < N, d >= cos( θ ) = costat Tai the derivative i each part of the equatio (), we obtai: = cos( θ ) N + si( θ ) Y Usi the Darboux equatios i (* ), we have cos( θ ) si( θ ) T + τ cos( θ ) Y + τ si( θ ) N = () ( ) ( ) Sice the system { T Y, N} that, is liear idepedet, we deduce from the equatio () cos( θ ) + si( θ ) = () τ cos( θ ) = (4) τ si( θ ) = (5) So, from the equatios (4) ad (5), we et τ = It follows that the curve α is a lie of curvature o M Moreover, from the equatio (), we have = ta(θ ) =costat Coversely, we assume that α is a lie of curvature o M The, ıt follows that τ = (6) O the other had, < N, d >= costat (7) By tai the derivative of the equatio (7), we obtai

5 Helix hypersurfaces ad special curves 7 < N, d >= (8) where N = T τ Y Ad, so we have N = T by usi (6) Cosequetly, if we also cosider (8), we obtai < T, d >= It follows that d Sp{ N, Y} This completes the proof Theorem Let M be a helix hypersurface with the directio d i E ad let α : Ι ΙR M be a uit speed curve o M If d Sp{ N, T} alo the τ curve α, the α is a asymptotic curve o M ad = ta(θ ) = costat, where { N, T} are Darboux vector fields of α ad θ is the costat ale betwee d ad N Coversely, ıf α is a asymptotic curve o M, the d Sp{ N, T} Proof: Sice M is a helix hypersurface with the directio d ad d Sp{ N, T}, we ca write as d = cos( θ ) N + si( θ) T (9) alo the curve α, where < N, d >= cos( θ ) = costat Tai the derivative i each part of the equatio (9), we obtai: = cos( θ ) N + si( θ) T Usi the Darboux equatios i ( *), we have cos( θ ) T + τ cos( θ ) + si( θ ) Y + si( θ ) N = () ( ) ( ) ( ) Sice the system { T Y, N} () that, is liear idepedet, we deduce from the equatio τ cos( θ ) + si( θ ) = () cos( θ ) = () si( θ ) = () So, from the equatios () ad (), we et = It follows that the curve α is a asymptotic curve o M Moreover, from the equatio (), we have τ = ta( θ ) = cst Coversely, we assume that α is a asymptotic curve o M The, ıt follows that = (4) O the other had, < N, d >= costat (5)

6 8 E Ziplar, A Şeol ad Y Yayli By tai the derivative of the equatio (5), we obtai < N, d >= (6) where N = T τ Y Ad, so we have N = τ Y by usi (4) Cosequetly, if we also cosider (6), we obtai < Y, d >= It follows that d Sp{ N, T} This completes the proof Theorem Let M be a r helix hypersurface i E ad let H be the subspace of helix directios of M If α : Ι ΙR M be a uit speed eodesic curve o M, the the curve α is a V -slat helix with respect to ay directio d H i E Proof: Let ξ be a ormal vector field o M Sice M is r helix hypersurface, < d, ξ >= costat, where d H is ay directio That is, the ale betwee d ad ξ is costat o every poit of the surface M Ad, α (s) = λξ alo the curve α α (s) sice α is a eodesic curve o M Moreover, by usi the Freet equatio α ( s ) = V = V, we obtai λξ = α ( s ) V, where is first curvature of α Thus, from the last equatio, by tai orms o both sides, we obtai ξ = V or ξ = -V So, d, V is costatalo the curve α sice d, ξ = costat I other words, the ale betwee d ad V is costat alo the curve α Cosequetly, the curve α is a V -slat helix with respect to ay directio d H i E This latter Theorem has the followi corollary Corollary Let M be a helix hypersurface with the directio d i E ad let α : Ι ΙR M be a uit speed eodesic curve o M The, the curve α is a V -slat helix with the directio d i E [] Theorem 4 Let M be a r helix hypersurface i E ad let H be the subspace of helix directios of M If α : Ι ΙR M a uit speed curve o M ad the -th uit vector field V of α equals to ξ or - ξ, the α is a V -slat helix with respect to ay directio vector field o M d H i E, where ξ is a ormal Proof: Let ξ be a ormal vector field o M Sice M is r helix hypersurface, < d, ξ >= costat, where d H is ay directio That is, the ale betwee d ad ξ is costat o every poit of the surface M Let the -th uit vector field V of α be equals to ξ or - ξ The d, V is costat alo the curve α sice d, ξ = costat That is, the ale betwee d ad V is costat alo the curve α Fially, the curve α is a V -slat helix with respect to ay directio d H i E

7 Helix hypersurfaces ad special curves 9 The Theorem 4 has the followi corollary Corollary Let M be a helix hypersurface i E ad let α : Ι ΙR M be a uit speed curve o M If the -th uit vector field V of α equals to ξ or - ξ, the α is a V -slat helix i E, where ξ is a ormal vector field o M [] Theorem 5 Let M be r helix hypersurface i E ad let H be the subspace of helix directios of M If α : Ι ΙR M ( α ( t) M, t Ι ) is a curve o the surface M ad α is a lie of curvature o M, the d Sp{} T alo the curve α, where T is taet vector field of α ad d H is ay directio Proof: Sice M is r helix hypersurface, < N, d >= costat ( d H is ay directio) alo the curve α, where N is the ormal vector field of M If we are tai the derivative i each part of the equality with respect to t, we obtai : ( N ), d = alo the curve α Sice α is a lie of curvature o M, ( N ) = S( T ) = λt alo the curve α, where S is the shape operator of the surface M So, we have T, d = Fially, { T} d Sp alo the curve α, where d H is ay directio The latter Theorem has the followi corollary Corollary Let M be a helix hypersurface with the directio d i E ad let α : Ι ΙR M ( α ( t) M, t Ι ) be a curve o the surface M If α is a lie of curvature o M, the d Sp{ T} alo the curve α, where T is taet vector field of α [] Theorem 6 Let M be a hypersurface i Euclidea -space E ad let N be the uit ormal vector field of M If M is a -helix i E, the the positio vectors o all poits of the surface η ( M ) = { X S X = N( P), P M } mae a costat ale with a fixed directio i E Here, η : M S E r P η( P) = N( P) S is Gauss trasformatio of M, where is the uit hypersphere i E Proof: Let d ΙR be the helix directio of M Sice M is a -helix i E,

8 4 E Ziplar, A Şeol ad Y Yayli < N ( P), d >= cos( θ ) = costat for every P M So, we obtai < X, d >= cos( θ ) = costat, for every X = N r r ( P) η( M ) At the same time, sice X = N(P) is the positio vector of η (M ) o the uit hypersphere S, η (M ) mae a costat ale with a fixed directio i E This completes the proof The followi examples ad are related to the Theorem 6 Example We cosider the hypersurface M ive by v v v ϕ : M E, ϕ(u,v) = cos ( )cos,si ( )cos, u u u u Let us determie the ormal to the surface To do this, we compute the partial derivatives of ϕ with respect to u ad v v v ϕ = u siu + ( )siu,cosu + ( )siu,, ϕ v = cosu, cosu, Usi by ϕ u ad ϕ v, the ormal to the surface is ive by N = ϕ ϕ ϕ ϕ u v = u v ( cosu, siu, ) Ad, < N, d >= = costat, where d = (,,) E That is, ϕ is a -helix with respect to d So, the rae of η : η ( M ) = ( cosu, siu, ), where η : M S E is the Gauss trasformatio of M The -helix M is show i Fiure ad the rae of the Gauss trasformatio of M is show i Fiure

9 Helix hypersurfaces ad special curves Fiure The -helix M i η o E Fiure The surface (M ) S E Example I Theorem, let us tae the hypersurface φ : H E, φ( u v) = ( u, v, u v) H E as Let us determie the ormal to the surface H To do this, we compute the partial derivatives of φ with respect to u ad v φ u = (,, v), φ v = (,, u) Usi by φ u ad φ v, the ormal to the surface H is ive by φu φv η = = ( v, u,) φ u φv v + u + So, accordi to Theorem, the hypersurface M = fθ ive by 4 ϕ : M E, ϕ( u, v, s) = ( u, v, u v,) + s ( v, u,,) + (,,,) v + u + is a -helix i E 4 4 with respect to d = (,,,) E, where θ = π 4 ad s ΙR Also, accordi to Theorem, the ormal vector field of M : ξ = ( v, u,,) + (,,,) v + u + Hece, the rae of μ :

10 4 E Ziplar, A Şeol ad Y Yayli v u μ ( M ) = (,,,), v + u + v + u + v + u + 4 where μ : M S E is the Gauss trasformatio of M Fially, μ (M ) maes a costat ale with the fixed directio 4 S E if we cosider the Theorem 6 4 d E o Theorem 7 Let M be a -helix hypersurface with the directio d i Euclidea -space E ad let d TpM be for every p M If α : Ι ΙR M ( α ( t ) Ι, t Ι) is a eodesic curve o the surface M, the the curve α t) = X S X = N, α( t) M ( ) { } η ( α ( t ) ca ot be a eodesic curve o the uit hypersphere the uit ormal vector field of M alo the curve α ad η : M S E r α( t) η α( t) = N ( ) ( ) α t S E, where N α ( t) Gauss trasformatio of M o the curve α Proof: Sice α : Ι ΙR M is a eodesic curve o the surface M, we ca write as α t) = λn ( α ( t) alo the curve α We assume that β = η( α(t) ) trasformatio of M So, we obtai sice α ( t) = λn α ( t) had, β = α, λ ad ( α t) ) N ( ) ( α t, where η Gauss η = alo the curve α O the other < ( ), d >= cos( θ ) = costat N α t alo the curve α sice M is a -helix surface with the directio d Thus, we have < N ( ), α t d >=< α, d > λ =< β, d > = cos( θ ) = costat If we tae the derivative i each part of the equality < β, d >= cos( θ ) twice, we obtai: < β, d >= (7)

11 Helix hypersurfaces ad special curves 4 Now, we suppose that the curve β is a eodesic curve o β = N α ( t), where N α ( t) is also the ormal vector field of et < N α ( t ), d >= S The S Hece, we by usi the equatio (7) It follows that d Tα ( t) M for every poit α ( t) M But, accordi to the hypothesis i this Theorem, d TpM for every p M So, it is a cotradictio As a result, β = η( α(t) ) ca ot be a eodesic curve o the uit hypersphere S E Theorem 8 Let M be a -helix hypersurface with the directio d i Euclidea -space E ad let α : Ι ΙR M be a curve o M If the system { T,T} is liear depedet alo the curve α, where T the taet compoet of d ad T the taet to the curve α, the α is a asymptotic curve o M Proof: Sice M is a -helix hypersurface with the directio d, < N, d >= cos( θ ) = costat, where N is the uit ormal vector field of M alo the curve α Ad, we ca decompose the vector d i its ormal ad taet compoets: d = cos( θ ) N + si( θ ) T, (8) where θ is costat If we tae derivative i each part of equality < N, d >= cos( θ ), we obtai < N, d >= By usi the equatio (8), we ca write : =< N, d >=< N,cos( θ ) N + si( θ ) T > = cos( θ ) < N, N > + si( θ ) < N, T > O the other had, < N, N >= sice N is the uit ormal vector field Therefore, we have : < N, T >=, where depedet, we ca write θ Accordi to the this Theorem, sice the system { T,T} T = λt Hece, we obtai: < N, T >= is liear Cosequetly, α is a asymptotic curve o M Theorem 9 Let M be a -helix hypersurface with the directio d i Euclidea -space E ad let α : Ι ΙR M be a curve o M If α is a d Sp T T,, T, N, where lie of curvature o M, the {, } Sp{ T, T, T,, T, N} = χ(e ) ad (, T,, T ) T vector fields i χ (E ) Note that T the uit taet vector field of α ad N the uit ormal vector field of M

12 44 E Ziplar, A Şeol ad Y Yayli Proof: Sice M is a -helix hypersurface with the directio d, < N, d >= cos( θ ) = costat, where N is the uit ormal vector field of M alo the curve α If we tae derivative i each part of equality < N, d >= cos( θ ), we obtai < N, d >= Moreover, N = λt sice α is a lie curvature Hece, we have: < T, d >=, This completes the proof Theorem Let M be a -helix hypersurface with the directio d i Euclidea -space E ad let α : Ι ΙR M be a curve o M If the system { T,T} is liear depedet, where T the derivative of the taet compoet of d ad T the taet to the curve α, the α is a lie of curvature o M Proof: Sice M is a -helix hypersurface with the directio d, we ca decompose the vector d i its ormal ad taet compoets: It follows that d Sp{ T T,, T, N} d = cos( θ ) N + si( θ ) T, (9) where < N, d >= cos( θ ) = costat ad N the uit ormal vector field of M If we tae derivative i each part of the equatio (9), we obtai: = si( θ ) T + cos( ) N θ () is liear depedet Hece, we ca write T = λt By usi the equatio (), we have the system { N,T } is liear depedet It follows that α is a lie of curvature o M This completes the proof Accordi to the hypothesis, the system { T,T} REFERENCES [] AI Nistor, Certai costat ale surfaces cosructed o curves, arxiv: 94475v [mathdg], (9) [] AJ Di Scala ad G Ruiz-Herádez, Helix submaifolds of euclidea spaces, Moatsh Math 57, (9) 5-5 [] F Dille, J Fasteaels, J Va der Vere ad L Vrace, Costat ale surfaces i S ΙR, Moatsh Math 5, (7) [4] F Dille ad MI Muteau, Costat ale surfaces i H ΙR, arxiv: 75744, (7) [5] G Ruiz-Herádez, Helix, shadow boudary ad miimal submaifolds arxiv: 7654 (7) [6] I Gö, Ç Camcı ad HH Hacısalihoğlu, V -slat helices i Euclidea -space E, Math Commu, Vol 4, No, (9) pp 7-9 [7] M Ghomi, Shadows ad covexity of surfaces, A Math () 55 (), () 8-9

13 Helix hypersurfaces ad special curves 45 [8] P Cermelli ad AJ Di Scala, Costat ale surfaces i liquid crystals, Philos Ma 87, (7) [9] S Özaldı ad Y Yayli, Costat ale surfaces ad curves i E, Iteratioal electroic joural of eometry, Vol 4, No, () pp 7-78 [] Y Yaylı ad E Zıplar, Costat ale ruled surfaces i Euclidea spaces, (), Submitted Received: November,