Some characterizations for Legendre curves in the 3-Dimensional Sasakian space

Transcription

1 IJST (05) 9A4: 5-54 Iaia Joual of Sciece & Techology Some chaacteizatio fo Legede cuve i the -Dimeioal Saakia pace H Kocayigit* ad M Ode Depatmet of Mathematic, Faculty of At ad Sciece, Celal Baya Uiveity, Muadiye Campu, Muadiye, Maia 4540, Tukey hueyikocayigit@cbuedut & mehmetode@cbuedut Abtact I thi pape, we give ome chaacteizatio fo Legede pheical, Legede omal ad Legede ectifyig cuve i the -dimeioal Saakia pace Futhemoe, we how that Legede pheical cuve ae alo Legede omal cuve I paticula, we pove that the ivee of cuvatue of a Legede ectifyig cuve i a o-cotat liea fuctio of the aclegth paamete Keywod: Legede cuve; omal cuve; ectifyig cuve; Saakia pace Itoductio eceay ad ufficiet coditio fo a cuve to be a pheical cuve i Euclidea pace E have bee give by Wog (96; 97) The coepodig chaacteizatio fo pheical cuve i the Mikowki -pace have bee tudied by Petović-Togašev ad Šućuović (000; 00) Aalogue to the -dimeioal pheical cuve, the autho have give chaacteizatio fo 4- dimeioal pheical cuve i the Mikowki 4- pace (Camcı et al, 00; Öde ad Kocayiğit 007) Moeove, Camcı et al (008) have coideed the cocept of pheical cuve i the - dimeioal Saakia pace ad have give ome coditio fo the Legede pheical cuve i thi pace I the Euclidea pace E, to each egula uit : IR E, with at leat fou peed cuve cotiuou deivative, it i poible to aociate thee mutually othogoal uit vecto field T, ad B, called the uit taget, the picipal omal ad the biomal vecto field, epectively The plae paed by T,, T, B ad, B ae kow a the oculatig plae, the ectifyig plae ad the omal plae, epectively The cuve : IR E fo which the poitio vecto alway lie i it omal plae i called omal cuve (Che, 00) Theefoe, fo a omal cuve, by defiitio the poitio vecto atifie the equatio ( ) ( ) ( ) ( ) B( ) whee () ad () ae diffeetiable fuctio of aclegth paamete The chaacteizatio fo omal cuve i ome pace uch a Euclidea -pace, Mikowki -pace ad dual Mikowki -pace have bee tudied by ome autho (Che, 00; İlala, 005; Öde, 006) Simila to omal cuve, if the poitio vecto alway lie i it ectifyig plae, the the : IR E i called ectifyig cuve cuve (Che, 00) By thi defiitio, the poitio vecto of a ectifyig cuve atifie the equatio ( ) ( ) T( ) ( ) B( ) fo ome diffeetiable fuctio () ad () Oe of the mot iteetig chaacteitic of ectifyig cuve i that the atio of thei toio ad cuvatue i a ocotat liea fuctio of ac legth paamete Rectifyig cuve lyig fully i the Euclidea pace E ae detemied explicitly by Che (00) I thi pape, fit we give ome chaacteizatio fo a Legede cuve to be a pheical cuve i the - dimeioal Saakia pace Late, we defie Legede omal cuve ad give the chaacteizatio fo thi cuve We how that Legede omal cuve ae pheical cuve Moeove, we give a defiitio ad chaacteizatio of Legede ectifyig cuve I paticula, we pove that the ivee of cuvatue of a Legede ectifyig cuve i a o-cotat liea fuctio of aclegth paamete Alo, we fid ome paametizatio fo Legede ectifyig cuve i the Saakia -pace *Coepodig autho Received: 7 Septembe 04 / Accepted: 4 Jauay 05

2 IJST (05) 9A4: Pelimiaie Let M be a mooth maifold A cotact fom o M i a -fom uch that d( ) 0 o M A maifold M togethe with a cotact fom i called a cotact maifold The ditibutio D defied by the Phaffia equatio 0 i called the cotact tuctue detemied by That i, D = {X x(m) η: x(m) C (M, η), η(x) = 0} The maximum dimeio of itegal ubmaifold of D i called a Legede ubmaifold of ( M, ) The eel vecto field (killig vecto field) i defied by ( ), d(, X) 0 (Yao ad Ko, 98; Belkelfa ad et al 00) O a cotact maifold ( M, ), thee exit a edomophim field ad a metic g atifyig X ( X), g( X, Y) g( X, Y) ( X) ( Y), d( X, Y) g( X, Y) fo all vecto field X ad Y o M whee g(, ) Whe ad, the g i Riemaia ad Loetzia metic, epectively The tuctue teo (,, g) ae called the aociated almot cotact tuctue of (Belkelfa et al 00) A cotact maifold ( M, ;,, g, ) i aid to be a Saaki maifold if M atifie ( X) Y g( X, Y) ( Y) X, X, Y ( M) (Yao ad Ko, 98; Belkelfa ad et al 00) ow let M ( M,,,, g, ) be a cotact -maifold with a aociated metic g A cuve ( ): I M paameteized by aclegth paamete i aid to be a Legede cuve if i taget to cotact ditibutio D of M It i obviou that i a Legede cuve if ad oly if ( ) 0 Let be a Legede cuve i Feet fame of i give by T, B T, ( ) ad B M The the, whee The, the Feet fomulae of ae give explicitly by TT T 0 0 T T 0 T B B 0 0 B whee the fuctio i the cuvatue of amely, evey Legede cuve ha cotat toio (Baikoui ad Blai, 994; Belkelfa et al 00) I a -dimeioal Saakia pace M, the phee i defied by H S whee H PM : g( P, P) ad S PM : g( P, P) (Camci et al 008) Legede Spheical Cuve I thi ectio we give the chaacteizatio fo Legede pheical cuve i the -dimeioal Saakia pace Theoem Let be a Legede cuve with cuvatue 0 i the -dimeioal Saakia pace The i a Legede pheical cuve if ad oly if, () hold, whee i the adiu of the phee Poof: Let m be the cete of phee o which lie The, we have g( m, m) () If we deive thi equatio with epect to, the ac-legth of, the we have g( m, T) 0, () whee T i uit taget vecto of If we epeat the deivatio we get g( m, ), (4) ad the deivatio of the lat equality give u

3 55 IJST (05) 9A4: 5-54 g( m, B) (5) The, fom (4) ad (5) we ca wite m B (6) Sice we have that the adiu of the phee i g( m, m), we obtai that which complete the poof Coveely, aume that () hold If we defie the vecto m B, (7) ad coide (), we have that m 0, ie, m i a cotat Moeove (7) give that g( m, m) ad fom hypothei,, (8) g( m, m) cotat, (9) we get that the Legede cuve lie o the phee whoe cete i m ad adiu i Let ow give the itegal chaacteizatio fo Legede pheical cuve i the -dimeioal Saakia pace Theoem I -dimeioal Saakia pace, a uit peed Legede cuve () with cuvatue 0 i a pheical cuve if ad oly if thee ae cotat A, B IR uch that A( ) B( ) (0) whee ( ) i( ), ( ) co( ) if ; ad ( ) ih( ), ( ) coh( ) if Poof: Aume that the Legede cuve lie o the phee with cete m ad adiu The, fom Theoem we have If we take y i () we get () y ( y) () The we ca wite that d d y ad y Thee diffeetial equatio ca be put ito thei vaiable, o we ca wite d, y y d, () epectively If, fom the lat equalitie we get (4) d, y y d By itegatig the equatio i (4), we have 0 0 y y 0 0 d, d, (5) ad the the paticula olutio of thee equalitie ae y i, y co, (6) epectively So, the geeal olutio of diffeetial equatio () with i y c y c y c i c co, (7) whee c, c c B IR If we wite y, c A,, the itegatio of () will be i the fom

4 IJST (05) 9A4: Ai( ) Bco( ), (8) that fiihe the poof If, the fom () we have 0 0 y y 0 0 d, d (9) The we obtai the paticula olutio of thee equalitie a follow y ih, y coh (0) So, the geeal olutio of diffeetial equatio () with i y c y c y c ih c coh, () whee c, c IR If we wite y, c A, c B, the itegatio of () will be i the fom Aih( ) Bcoh( ) () Uig (8) ad () we ca wite A( ) B( ) () whee ( ) i( ), ( ) co( ) if ; ad ( ) ih( ), ( ) coh( ) if ad A, B IR Coveely, if () hold fo the Legede cuve, we have () The, fom Theoem, we ay that the cuve () i a Legede pheical cuve i - dimeioal Saakia pace 4 The Legede omal Cuve I thi ectio, we give the defiitio ad the chaacteizatio of Legede omal cuve ad how that Legede omal cuve ae alo Legede pheical cuve i the -dimeioal Saakia pace Defiitio 4 Let () be a uit peed Legede cuve i the -dimeioal Saakia pace with Feet fame T,, B ad cuvatue 0 The cuve () i called Legede omal cuve if the poitio vecto alway lie o the omal plae of the Legede cuve () By defiitio, fo a Legede omal cuve the poitio vecto atifie the equatio ( ) ( ) ( ) ( ) B( ) fo ome diffeetiable fuctio () ad () The, we ca give the followig chaacteizatio Theoem 4 Let () be a uit peed Legede omal cuve with cuvatue ( ) 0 i the - dimeioal Saakia pace The the followig tatemet hold: i) The cuvatue () atifie the followig equality d ( ) d ( ), (4) whee ( ) i( ), ( ) co( ) if ; ad ( ) ih( ), ( ) coh( ) if ad d, d IR ii) The picipal omal ad biomal compoet of the poitio vecto of the Legede cuve () ae give by g( ( ), ) d( ) d ( ), g( ( ), B) d ( ) d( ), epectively (5) Poof: Suppoe that () i a uit peed Legede omal cuve The by Defiitio 4, we have ( ) ( ) ( ) ( ) B( ) (6) Diffeetiatig (6) with epect to ad uig the Feet equatio, we fid

5 57 IJST (05) 9A4: 5-54 ( ) ( ), ( ) ( ) 0, ( ) ( ) 0 (7) Fom the fit ad ecod equatio of (7) we get, (8) Thu, B (9) Futhe, fom the thid equatio i (7) ad uig (8) we fid the followig diffeetial equatio 0 a Puttig y () (0), equatio (0) ca be witte y y 0 () The olutio of () i d ( ) d ( ), () whee ( ) i( ), ( ) co( ) if ; ad ( ) ih( ), ( ) coh( ) if ad d, d IR Thu we have poved tatemet (i) By Theoem, we ee that the Legede omal cuve () i a pheical cuve So we ca give the followig coollay Coollay 4 Evey Legede omal cuve () i alo a Legede pheical cuve i the - dimeioal Saakia pace Futhemoe, Camcı et al (008) have how that thee ae o Legede pheical cuve i the - IR ( ) The, by dimeioal Saakia pace coideig Coollay 4, we ca give the followig coollay Coollay 4 Thee ae o Legede omal cuve () i the -dimeioal Saakia pace IR ( ) Let u ow pove tatemet (ii) Subtitutig () ito (8) ad (9), we get ( ) d ( ) d ( ), () ( ) d( ) d ( ), (4) d ( ) d ( ) ( ) d( ) d ( ) B( ) (5) Theefoe, ice g( B, B), fom (5) we eaily fid that g(, ) d d, (6) g( ( ), ) d ( ) d ( ), (7) g( ( ), B) d( ) d ( ) (8) Coequetly we have poved (ii) Coveely, uppoe that tatemet (i) hold The we have d ( ) d ( ), (9) whee ( ) i( ), ( ) co( ) if ; ad ( ) ih( ), ( ) coh( ) if ad d, d IR Sice ( ) ( ) ad ( ) ( ), by diffeetiatig (9) two time with epect to we fid d( ) d ( ) () (40) The fom (9) ad (40) we have 0 Equatio (4) how that (4)

6 IJST (05) 9A4: cotat It alo mea that () i a Legede pheical cuve So, fom (6), by applyig Feet equatio we obtai d ( ) ( ) B( ) 0 d ( ) ( ) (4) Coequetly, () i coguet to a Legede omal cuve ext, aume that tatemet (ii) hold The the equatio (6) ad (7) ae atified Diffeetiatig (6) with epect to ad uig (7), we fid g(, T) 0, which mea that () i a Legede omal cuve 5 The Legede Rectifyig Cuve I thi ectio, we give the defiitio ad chaacteizatio of Legede ectifyig cuve i the -dimeioal Saakia pace Defiitio 5 Let () be a uit peed Legede cuve i the -dimeioal Saakia pace with Feet fame T,, B ad cuvatue 0 The cuve () i called Legede pheical cuve if the poitio vecto alway lie o the omal plae of the Legede cuve () By defiitio, fo a Legede omal cuve the poitio vecto atifie the equatio ( ) ( ) T( ) ( ) B( ) fo ome diffeetiable fuctio () ad () Theoem 5 Let () be a uit peed Legede ectifyig cuve i Saakia -pace with cuvatue ( ) 0 The the followig tatemet hold: (i) The ditace fuctio atifie ( ), fo ome, IR (ii) The tagetial compoet of the poitio vecto g(, T), whee of i give by IR (iii) The omal compoet of the poitio vecto of the cuve ha a cotat legth ad the ditace fuctio i o-cotat (iv) The biomal compoet of the poitio vecto of the cuve i cotat, ie, g(, B) i cotat Coveely, if () i a uit peed Legede cuve i the Saakia -pace with cuvatue ( ) 0 ad oe of the tatemet (i), (ii), (iii) ad (iv) hold, the i a ectifyig cuve Poof: Let u uppoe that () i a uit peed Legede ectifyig cuve The the poitio vecto of the cuve atifie the equatio ( ) ( ) T( ) ( ) B( ), (4) whee () ad () ae ome diffeetiable fuctio of aclegth paamete Diffeetiatig (4) with epect to ad applyig the Feet- Seet equatio give, 0, 0 (44) Theefoe, it follow that ( ), ( ), 0,, IR (45) Fom the equatio (4) ad (45), we eailiy fid ( ), ad o (i) hold Futhe, fom (4) we obtai g(, T) which togethe with (45), g(, T) IR ad (ii) hold, whee ext, fom the elatio (4) it follow that the of the poitio vecto i omal compoet give by B Theefoe 0 Thu we poved tatemet (iii) Fially, fom (4) g(, B) cotat ad o we eaily get the tatemet (iv) i poved Coveely, aume that tatemet (i) o tatemet (ii) hold The, thee hold the equatio g(, T), whee IR Diffeetiatig thi equatio with epect to, we get ( ) g ( ), ( ) 0 Sice ( ) 0, it follow that g(, ) 0 Hece i a Legede ectifyig cuve ext, uppoe that tatemet (iii) hold Let u put ( ) m( ) T( ), m( ) IR The we eaily fid that g(, ) C cotat g(, ) g(, T) Diffeetiatig thi equatio with epect to we have

7 59 IJST (05) 9A4: 5-54 g(, ) 0 (46) Sice 0, we have g(, T) 0 Moeove, ice ( ) 0 fom (46) we obtai g(, ) 0, which mea that i a Legede ectifyig cuve Fially, if tatemet (iv) hold, the by applyig Feet equatio, we eaily obtai that the cuve i a Legede ectifyig cuve I the ext theoem, we pove that the ivee of the cuvatue of a Legede ectifyig cuve i a o-cotat liea fuctio of ac legth paamete Theoem 5 Let () be a uit peed Legede ectifyig cuve with cuvatue ( ) 0 i the Saakia -pace The, up to iometie of Saakia -pace, the cuve i a Legede ectifyig cuve if ad oly if 4 hold, whee, 4 () IR Poof: Let u fit uppoe that () i Legede ectifyig cuve By the poof of Theoem 5 ad by elatio (44) ad (45), it follow that (), (47) whee, IR Coequetly, / ( ) 4 whee /, 4 / ae eal cotat Coveely, let u uppoe that / ( ) 4 ad, 4 IR The, we /, / whee Applyig () may chooe 4, IR Hece, the Feet equatio, we eaily fid that d ( ) ( ) T B 0 d, which mea that up to iometie of Saakia pace, the Legede cuve () i a ectifyig cuve Theoem 5 Let () be a uit peed Legede cuve i the Saakia -pace The i a Legede ectifyig cuve if ad oly if, up to a paametizatio, i give by co t ih t ( t) y( t), IR, if ( t) y( t), IR, if (48) whee yt () i a uit peed Legede cuve lyig o the phee H () S () Poof: Let u fit aume that () i a uit peed Legede ectifyig cuve Sice g( T, T), g( B, B), by the poof of Theoem 5, it follow that ( ),, IR IR We may chooe Alo, we may apply a talatio with epect to, uch that ext, we defie a cuve by y () () () (49) ad aume that the cuve y i lyig o the phee H () S () The we have ( ) y( ) (50) Diffeetiatig (50) with epect to, we get T( ) y( ) y( ) (5) Sice g( y, y), it follow that g( y, y) 0 Fom (5) we obtai g( T, T) g( y, y)( ) ad hece g( y, y) ( ) Fom (5), we get 0 (5) y( ) / ( ) Let t y() u du be the aclegth paamete of the cuve y The we have

8 IJST (05) 9A4: t du u 0 If, the we get, the we get ta t ad if coth Subtitutig t thee ito (50), epectively, we obtai the paametizatio give i (48) Coveely, aume that i a cuve defied by (48) whee yt () i a uit peed Legede cuve lyig o the phee H S with adiu Diffeetiatig the equatio (48) with epect to, we get ( t) y( t)i t y ( t)co t, if co t ( t) y( t)coh t y( t)ih t, if ih t By aumptio, we have g( y, y), g( y, y) ad coequetly g( y, y) 0 Theefoe, it follow that i t g(, ), g(, ), if 4 (5) co t co t coh t g(, ), g(, ), if 4 ih t ih t ad coequetly ( t) / co t, if ( t) / ih t, if Let u put ( t) m( t) ( t), whee m() t IR ad i omal compoet of the poitio vecto The we eaily fid that, ad theefoe m g(, )/ g(, ) Sice (, ) (, ) (, ) g g g g(, ) g(, ), if co t g(, ), if ih t by uig (5), the lat equatio become g(, ) cotat It follow that cotat ad ice / co t cotat, Theoem 5 implie that i a Legede ectifyig cuve 6 Cocluio I the tu of cotact maifold, Legede cuve have a impotat ole I the cotact maifold, a diffeomophim i a cotact tafomatio if ad oly if ay Legede cuve i a domai of it go to Legede cuve (Baikoui ad Blai, 994) The the tu of pecial Legede cuve i faciatig By coideig the impotace of thi, Legede pheical, Legede omal ad Legede ectifyig cuve i Loetzia Saakia pace have bee itoduced It i how that Legede omal cuve ae alo Legede pheical cuve Refeece Baikoui, C D, & Blai, E (994) O Legede cuve i cotact -maifold Geom Dedicata, 49, 5 4 Belkhelfa, M, Hiica, I E, Roca, R, & Vetlaele, L (00) O Legede cuve i Riemaia ad Loetzia Saaki pace Soochow J Math, 8(), 8 9 Camcı, Ç, İlala, K, & Šućuović, E (00) O Peudohypebolic Cuve i Mikowki pace-time Tuk J Math, 7(), 5 8 Camcı, Ç, Yaylı, Y, & Hacıalihoğlu, H H (008) O the chaacteizatio of pheical cuve i - dimeioal Saakia pace J Math Aal Appl, 4(), 5 59 Che, B Y (00) Whe doe the poitio vecto of a pace cuve alway lie i it ectifyig plae? Ame Math Mothly, 0(), 47 5 İlala, K (005) Spacelike omal cuve i Mikowki pace E Tuk J Math, 9(), 5 6 İlala, K, ešović, E, & Petović-Togašev, M (00) Some Chaacteizatio of Rectifyig Cuve i the Mikowki -Space ovi Sad J Math, (), Öde, M (006) Dual Timelike omal ad Dual Timelike Spheical Cuve i Dual Mikowki Space D SDU Joual of ciece, (-), Öde, M, & Kocayiğit, H (007) O Loetzia pheical timelike ad ull cuve i Mikowki pacetime CBU Joual of Sciece, (), 5 58 Petović-Togašev, M, & Šućuović, E (000) Some chaacteizatio of Loetzia pheical pacelike cuve with the timelike ad ull picipal omal Mathematica Moavica, 4, 8 9 Petović-Togašev, M, & Šućuović, E (000) Some chaacteizatio of cuve lyig o the peudohypebolic pace H 0 i the Mikowki pace E Kagujevac J Math,, 7 8

9 54 IJST (05) 9A4: 5-54 Petović-Togašev, M, & Šućuović, E (00) Some chaacteizatio of the Loetzia pheical timelike ad ull cuve MATEMATИУKИ BECHИK, 5, 7 Wog, Y C (96) A global fomulatio of the coditio fo a cuve to lie i a phee Moatchefte fu Mathematik, 67(4), 6 65 Wog, Y C (97) O a explicit chaacteizatio of pheical cuve Poceedig of the Ameica Math Soc, 4, 9 4 Yao, K, & Ko, M (984) Stuctue o maifold Seie i Pue Mathematic,, Wold Scietific Publihig Co, Sigapoe