SmarandacheCurvesandSphericalIndicatricesintheGalilean3-SpaceH.S.Abdel-Aziz andM.KhalifaSaadDept. ofMath.,FacultyofScience,SohagUniv.,82524Sohag,EgyptAbstract. Inthe present paper, Smarandache curves for some special curves inthe three-dimensional GalileanspaceG3areinvestigated. Moreover, spherical indicatricesforthehelixaswellascircularhelixareintroduced. Furthermore,somepropertiesforthesecurvesaregiven. Finally,inthelightofthisstudy,somerelatedexamplesofthesecurvesareprovided.Keywords. Smarandachecurves,Sphericalindicatrices,Frenetframe,Galileanspace.MSC(2010). 53A35,53B30.1 IntroductionDiscovering Galilean space-time is probably one of the major achievements of non relativistic physics.NowadaysGalileanspaceisbecomingincreasinglypopularasevidencedfromtheconnectionof thefundamental conceptssuchasvelocity, momentum, kineticenergy, etc. andprinciplesasindicatedin [7]. In recent years,researchers have begun to investigate curves and surfaces in the Galilean spaceandthereafterpseudo-Galileanspace. AregularcurveinEuclideanspacewhosepositionvectoriscomposedbyFrenetframevectorsonanotherregularcurveiscalledaSmarandachecurve. Smaran-dachecurveshavebeeninvestigatedbysomedierential geometers[1, 9]. M. TurgutandS. Yilmazhavedenedaspecial caseof suchcurvesandcall itSmarandacheTB2curvesinthespaceE41[9].TheyhavedealedwithaspecialSmarandachecurveswhichisdenedbythetangentandsecondbi-normal vector elds. Additionally, they have computed formulas of this kind curves. In [1], the authorhas introduced some special Smarandache curves in the Euclidean space. He has studied Frenet-Serretinvariants of a special case. In this paper, we investigate Smarandache curves for the position vector ofanarbitrarycurveintermsofthecurvatureandthetorsionwithrespecttostandardframe. Besides,special Smarandache curves such as TN, TBand TNBaccording to Frenet frame in Galilean 3-spacearestudied. Furthermore, wenddierential geometricproperties of thesespecial curves andwecalculatecurvatures(natural curvatures)of thesecurves. OurmainresultinthisworkistostudySmarandachecurvesforsomespecialcurvesintheGalilean3-spaceG3.E-mailaddress:mohamedkhalifa77@science.sohag.edu.eg1arXiv:1501.05245v1 [math.DG] 21 Jan 20152 PreliminariesLet us recall thebasicfacts about thethree-dimensional GalileangeometryG3. ThegeometryoftheGalileanspacehasbeenrstlyexplainedin[11]. ThecurvesandsomespecialsurfacesinG3areconsideredin[3]. TheGalileangeometryisareal Cayley-Kleingeometrywithprojectivesignature(0, 0, +, +) accordingto[5]. Theabsoluteof theGalileangeometryis anorderedtriple(w, f, I )wherewis theideal (absolute) plane(x0=0), f is alineinw(x0=x1=0) andI is elliptic((0: 0: x2: x3) (0: 0: x3: x2))involutionof thepointsof f . IntheGalileanspacetherearejusttwotypesofvectors, non-isotropicx(x, y, z)(forwhichholdsx =0). Otherwise, itiscalledisotropic. Wedonot distinguishclasses of vectors amongisotropicvectors inG3. Aplaneof theformx=const. intheGalileanspaceiscalledEuclidean, sinceitsinducedgeometryisEuclidean.Otherwiseitiscalledisotropicplane. Inanecoordinates, theGalileaninnerproductbetweentwovectorsP= (p1, p2, p3)andQ = (q1, q2, q3)isdenedby[4]:P, QG3=_p1q1ifp1 = 0 q1 = 0,p2q2 + p3q3ifp1= 0 q1= 0.(2.1)AndthecrossproductinthesenseofGalileanspaceisgivenby:(P Q)G3=___0 e2e3p1p2p3q1q2q3; ifp1 = 0 q1 = 0,e1e2e3p1p2p3q1q2q3; ifp1= 0 q1= 0.(2.2)TheGalileansphereofradiusdandcentercofthespaceG3isdenedbyS2G(c, d) = {X c G3: X c, X cG3= d2}.A curve (t) = (x(t), y(t), z(t)) is admissible in G3if it has no inection points ( (t) (t) = 0) . AnadmissiblecurveinG3isananalogueofaregularcurveinEuclideanspace.Foranadmissiblecurve:I G3, I Rparameterizedbythearclengthswithdierentialformdt = ds,givenby(s) = (s, y(s), z(s)). (2.3)Thecurvature(s)andtorsion(s)ofaredenedby(s) =___

(s)___ =_y

(s)2+ z

(s)2,(s) =y

(s)z

(s) + y

(s)z

(s)2(s). (2.4)Notethatanadmissiblecurvehasnon-zerocurvature.2TheassociatedtrihedronisgivenbyT(s) =

(s) = (1, y

(s), z

(s)),N(s) =

(s)(s)=(0, y

(s), z

(s))(s),B(s) =(0, z

(s), y

(s))(s). (2.5)For derivatives of the tangent T, normal N and binormal B vector eld, the following Frenet formulasintheGalileanspacehold[11]__TNB__

=__0 00 0 0 0____TNB__. (2.6)From(2.5)and(2.6),wederiveanimportantrelation

(s) =

(s)N(s) + (s)(s)B(s).Letusconsiderthefollowingdenitions:Denition2.1Asinthethree-dimensionalEuclideanspace,anadmissiblecurveinG3,whose posi-tionvectoriscomposedbyFrenetframevectorsonanotheradmissiblecurveiscalledaSmarandachecurve[9].Inthelight of theabovedenition, weadapt it toadmissiblecurves intheGalileanspaceasfollows:Denition2.2let = (s)beanadmissiblecurveinG3and {T, N, B}beitsmovingFrenetframe.SmarandacheTN, TBandTNBcurvesarerespectively,denedbyTN=T+NT+N,TB=T+BT+B,TNB=T+N+BT+N+B. (2.7)Denition2.3Ahelix is a geometric curve with non-vanishing constant curvature and non-vanishingconstanttorsion [10]Denition2.4A family of curves with constant curvature but non-constant torsion is called Salkowskicurves and a family of curves with constant torsion but non-constant curvature is called Anti-Salkowskicurves[6].Denition2.5As in Euclidean 3-space ,let be an admissible curve in Galilean 3-space with FrenetvectorsT, NandB. TheunittangentvectorsalongthecurvegenerateacurveTonaGalilean3sphereof radius 1about theorigin. ThecurveTis calledthespherical indicatrixof Tor morecommonly, Tiscalledtangentindicatrixof thecurve. If =(s)isanatural representationof, thenT(sT)=T(s)will bearepresentationof T. Similarlyoneconsiderstheprincipal normalindicatrixN(sN) = N(s)andbinormal indicatrixB(sB) = B(s)[8].3 SmarandachecurvesofspecialcurvesInthissection,weconsiderthepositionvectorofanarbitrarycurvewithcurvature(s)andtorsion(s)intheGalileanspaceG3whichcomputedfromthenaturalrepresentationformasfollows[2]r (s) =_s,_ __(s) cos__(s)ds_ds_ds,_ __(s) sin__(s)ds_ds_ds_. (3.1)ItsmovingframeisT(s) =_1,__(s) cos__(s)ds_ds_,__(s) sin__(s)ds_ds__,N(s) =_0, cos__(s)ds_, sin__(s)ds__,B(s) =_0, sin__(s)ds_, cos__(s)ds__. (3.2)LetusinvestigateFrenetinvariantsofSmarandachecurvesaccordingtodenition2.2.3.1 TN-SmarandachecurveLetr1(s1)beaTN-Smarandachecurveofr(s)writtenasr1(s1) =_1,_(s) cos__(s)dsds + cos__(s)ds,_(s) sin__(s)dsds + sin__(s)ds_. (3.3)Bydierentiating(3.3)withrespecttos,wegetT1=12+ 2_0, (s) cos__(s)ds(s) sin__(s)ds,(s) sin__(s)ds+ (s) cos__(s)ds_, (3.4)whereds1ds=_2+ 2. (3.5)Also,dierentiating(3.4),weobtain1N1=1(2+ 2)2(0, 1, 2) , (3.6)then,thecurvatureandprincipalnormalvectoreldofr1(s1)arerespectively,1=1(2+ 2)2_21 + 22, N1=1_21 + 22(0, 1, 2) , (3.7)where41=_3 2 3 + _sin__(s)ds_+_22+ 2 4 _cos__(s)ds_, (3.8)2=_3+ 2 + 3 _cos__(s)ds_+_222 + 4+ _sin__(s)ds_. (3.9)Ontheotherhand,thebinormalvectorofthiscurveisexpressedasB1=1_(2+ 2)_21 + 22__2_(s) cos__(s)ds(s) sin__(s)ds_1_(s) sin__(s)ds+ (s) cos__(s)ds_, 0, 0_, (3.10)itfollowsthat1= 0. (3.11)3.2 TB-SmarandachecurveLetr2(s2)beaTB-Smarandachecurvegivenbyr2(s2) =_1,_(s) cos__(s)dssin__(s)ds,_(s) sin__(s)ds+ cos__(s)ds_. (3.12)DierentiatingthisequationwithrespecttosyieldsT2=_0, cos__(s)ds_, sin__(s)ds__, (3.13)whereds2ds= ( ) . (3.14)Again,dierentiating(3.13)gives2=( ), (3.15)N2=_0, sin__(s)ds_, cos__(s)ds__. (3.16)Fromtheabovementioned,onecanobtainB2= (1, 0, 0) . (3.17)Fromwhich2= 0. (3.18)53.3 TNB-SmarandachecurveBydenition2.2,theTNB-Smarandachecurveisgivenbyr3(s3) =_1,_(s) cos__(s)ds+ cos__(s)dssin__(s)ds,_(s) sin__(s)ds+ sin__(s)ds+ cos__(s)ds_. (3.19)Similarly,itstangent,theprincipalnormalandthebinormalvectorsarerespectively,T3=1_( )2+ 2_0, ( ) cos__(s)ds(s) sin__(s)ds,( ) sin__(s)ds+ (s) cos__(s)ds_, (3.20)inwhichds3ds=_( )2+ 2, (3.21)N3=_0, 1 cos__(s)ds2 sin__(s)ds,2 cos__(s)ds+ 1 sin__(s)ds__21 + 22, (3.22)B3=(2 ( ) 1)_( )2+ 2_21 + 22(1, 0, 0) , (3.23)where1(s) =__( )2+ 2__ 2_( ) (( ) ( ) + )_,2(s) =__( )2+ 2__ 2+ _ (( ) ( ) + )_. (3.24)Thecurvatureandthetorsionofthiscurvearerespectively,givenby3=_21 + 22_( )2+ 2_2, 3= 0. (3.25)Theorem3.1Letr(s)givenby(3.1)beahelix(s)inG3(/ = m = const.)whichcanbewrittenas(s) =_s,1m_sin_m_(s)ds_ds, 1m_cos_m_(s)ds_ds_, (3.26)thenTN, TBandTNB-Smarandachecurvesofareplanecurveswithconstantcurvatures.Proof. WeprovethistheoremforTN-Smarandachecurveonly. Thestraightforwardcomputationsongiverespectively,thetangent,theprincipalnormalandthebinormalvectorsofasfollowsT(s) =_1,1m sin_m_(s)ds_, 1m cos_m_(s)ds__,N(s) =_0, cos_m_(s)ds_, sin_m_(s)ds__,6B(s) =_0, sin_m_(s)ds_, cos_m_(s)ds__. (3.27)TheTN-Smarandachecurveofcanbeexpressedas1(s1) =_1,1m sin_m_(s)ds+ cos_m_(s)ds,1m cos_m_(s)ds+ sin_m_(s)ds_. (3.28)IthasthemovingFrenetframeT1=11 + m2_0, cos_m_(s)dsmsin_m_(s)ds,sin_m_(s)ds+ mcos_m_(s)ds_,N1=1(1 + m2)_0, sin_m_(s)dsmcos_m_(s)ds,cos_m_(s)dsmsin_m_(s)ds_,B1=_11 + m2, 0, 0_. (3.29)Fromtheabovedata,thecurvatureandtorsionof1are1= m, 1= 0. (3.30)Thecurvatureisconstantandtorionisvanished. ThesameresultsforTBandTNB-Smarandachecurvesofarevalid. Hencetheproofiscompleted.Remark3.1If r(s) is a circular helix ( = const. , = const.), then TN, TB and TNB-Smarandachecurvesarealsoplanecurveswithconstantcurvatures.Theorem3.2Ifr(s)isafamilyofSalkowskicurves(s)inG3(= (s), = a = const.)whichcanbewrittenas(s) =_s, a_ __cos__(s)ds_ds_ds, a_ __sin__(s)ds_ds_ds_, (3.31)thenTN, TBandTNB-Smarandachecurvesofareplanecurveswithnon-constantcurvatures.Proof. Aftersomecalculationson, thetangent, theprincipalnormalandthebinormalvectorsofare,respectivelyT (s) =_1, a_ _cos__(s)ds__ds, a_ _sin__(s)ds__ds_,N (s) =_0, cos__(s)ds_, sin__(s)ds__,B (s) =_0, sin__(s)ds_, cos__(s)ds__. (3.32)Here,TN-Smarandachecurveofis1(s1) =_1, a_ _cos__(s)ds_ds + cos__(s)ds,a_ _sin__(s)ds_ds + sin__(s)ds_, (3.33)7withtheFrenetvectorsT1=1_a22(s)_0, a cos(_(s)ds) (s) sin(_(s)ds),a sin(_(s)ds) + (s) cos(_(s)ds)_,N1=1_0, 1 (s) sin(_(s)ds) + 2 (s) cos(_(s)ds),2 (s) sin(_(s)ds) + 1 (s) cos(_(s)ds)_; =_21 + 22,B1=1_(a1 2)_a22(s), 0, 0_. (3.34)Fromtheaforementionedcalculations,thecurvaturesof1(s1)are1=(a22(s))2, 1= 0, (3.35)where1 (s) =_(a)_a22_+ 2_, 2 (s) =_a 2_a22__. (3.36)As above, we can do similar calculations for TB and TNB-Smarandache curves of . In all cases, thecurvaturesarenon-constants(theydependonthetorsionoftheSalkowskicurve)andthetorionsarevanished. Thus,thiscompletestheproof.Remark3.2Ifr(s)isafamilyofAnti-Salkowskicurves(s)inG3(= b = const., = (s)),whichcanbewrittenas(s) =_s,_ __(s) cos(bs)ds_ds,_ __(s) sin(bs)ds_ds_, (3.37)then TN, TB and TNB-Smarandache curves of are also plane curves withnon-constant curvatures.4 SphericalindicatricesofageneralhelixinG3Inwhatfollows,weinvestigatethespherical indicatrixofthetangentofthehelix(s).Bydier-entiating(3.26)withrespecttos,wehavethetangentsphericalcurveasfollows:T= T(sT) =_1,1m sin(m_(s)ds), 1mcos(m_((s)ds)_. (4.1)TheFrenetvectorsofT(sT)areTT(sT) =_0, cos(m_(s)ds), sin(m_(s)ds)_,NT(sT) =_0, sin(m_(s)ds), cos(m_(s)ds)_,BT(sT) = (1, 0, 0), (4.2)anditscurvaturesaregivenbyT(sT) = m, T(sT) = 0. (4.3)8Theorem4.1Let= (s)beahelixintheGalileanspaceG3with (s) = 0. Thecurvaturesofthetangentspherical curveT(sT)offoreachsT I RsatisfythefollowingequalitiesTT, T

G3= 0,NT, T

G3= 1T,BT, T

G3=

T2TT.Proof. ByassumptionwehaveT, T

G3= d2, (4.4)foreverysT I R anddistheradiusof theGalileansphereS2G. Bydierentiating(4.4)withrespecttos,wehave_dTdsTdsTds, T_G3= 0,wheresTisthearclengthofthetangentsphericalcurveinG3. So,wegetdsTds= (s),then(s) TT, T

G3= 0,TT, T

G3= 0. (4.5)Byanewdierentiationof(4.5),wendthatT NT, T

G3+ TT, TT

G3= 0.FromwhichNT, T

G3= 1T. (4.6)Moredierentiationof(4.6)givesT BT, T

G3+ NT, TT

G3= T2T,sinceNT, TT

G3= 0, dNTdsT= TBT,thenBT, T

G3=TT2T.Thus,theproofiscompleted.Fromaforementionedinformation,wehavethefollowingproposition.Proposition4.1TheGalileanspherical imagesof ageneral helix(orcircularhelix)inthethree-dimensional Galileanspaceareplanecurveswithconstantcurvatures.Proposition4.2Wecanalsodoinsimilarwaythecalculationsfortheotherspherical imagesofTN, TBandTNB-Smarandache curves, we ndthat theyare plane curves withnon-constantcurvatures.95 ExamplesExample5.1Let (s)beageneral helixinthree-dimensional Galilieanspacewith= =12s,parameterizedby(s) =_s, 2_sins s coss, 2_coss +s sins_TheTN-Smarandachecurveof(s)isgivenby1 (s1) =_1, sin_s_+ cos_s_, cos_s_+ sin_s__.Then,theFrenetvectorsareT1(s1) =_0, cos (s) sin (s)2, cos (s) + sin (s)2_,N1(s1) =_0, (cos (s) + sin (s))2, cos (s) sin (s)2_,B1(s1) =_cos2_s_+ sin2_s_, 0, 0_.Thederivativesofthesevectorsgive1 (s1) =12, 1 (s1) = 0.TheTBandTNB-Smarandachecurvesof (s)canbecalculatedasabove.Thetangentspherical imageof(s)isdenedbyT=_1, sins, coss_.Fromwhich,onecanobtainTT=_0, coss, sins_,NT=_0, sins, coss_,BT=__sins_2+_coss_2, 0, 0_.ItfollowsthatT= 1, T= 0.Inananalogousway,onecanobtainthenormalandbinormalsphericalimagesofandtheircurva-tures. Thegeneral helix(s)anditsTN-Smarandachecurveandtangentspherical imageareshowninFigures1,2,3.10Figure1: Thegeneralhelix(s).Figure2: TheTN-Smarandachecurveof(s).Figure3: Thetangentsphericalimageof(s).Example5.2Consider a circular helix (s) in G3with non-zero constant curvature and torsion, whichisgivenby(s) = (s, coss, sins) ,therepresentationofitstangentspherical curveisT= (1, sins, coss) .11Then,simplecalculationsleadtoTT= (0, coss, sins) ,NT= (0, sins, coss) ,BT=_(sins)2+ (coss)2, 0, 0_.Bytheaidofthederivativesoftheaboveformulas,weobtainthecurvaturesofTT= 1, T= 0.Thecurveanditstangentspherical imageareshowninFigures4,5.Figure4: Acircularhelix(s).Figure5: Thetangentsphericalimageof(s).6 ConclusionInthethree-dimensional Galileanspace, Smarandachecurvesof somespecial curvessuchashelix,circularhelix,SalkowskiandAnt-Salkowskicurvesarestudied. Furthermore,thesphericalimagesofthehelixaregiven. Someinterestingresultsarepresented. Finallyasanapplicationforthiswork,someexamplesaregivenandplotted.12References[1] A. T. Ali, Special SmarandachecurvesintheEuclideanspace, International Journal ofMathe-maticalCombinatorics,2(2010),30-36.[2] A.T.Ali,PositionvectorsofcurvesintheGalileanspaceG3,MatematickiVesnik,64(3)(2012),200-210.[3] B. DivjakandZ. MilinSipus, Mindings isometries of ruledsurfaces inGalileanandpseudo-Galileanspace.J.Geom.,77(2003),35-47.[4] B.J.PavkovicandI.Kamenarovic,TheequiformdierentialgeometryofcurvesintheGalileanspaceG3,GlasnikMatematicki,22(42)(1987),449-457.[5] E. Molnar, The projective interpretationof the eight 3-dimensional homogeneous geometries.BeitrageAlgebraGeom.,38(2)(1997),261-288.[6] E.Salkowski,Zurtransformationvonraumkurven,Math.Ann.,66(1909),517-557.[7] I. M. Yaglom, Asimplenon-Euclideangeometryandits physical basis, Springer-Verlag, NewYork,1979.[8] M. Ergut and A. O. Ogrenmis, Some characterizations of a spherical curves in Galilean space G3,JournalofAdvancedResearchinPureMathematics,1(2)(2009),18-26.[9] M. TurgutandS. Yilmaz, SmarandachecurvesinMinkowski space-time, International JournalofMathematicalCombinatorics,3(2008),51-55.[10] M. P. DoCarmo, Dierential geometryofcurvesandsurfaces, PrenticeHall, EnglewoodClis,NJ,1976.[11] O.Roschel,DieGeometriedesGalileischenRaumes.Habilitationsschrift,Institutf urMath.undAngew.Geometrie,Leoben,1984.13