Harmonic Curvatures in Lorentzian Space

Transcription

1 BULLETIN of the Bull Malaya Math Sc Soc Secod See 7-79 MALAYSIAN MATEMATICAL SCIENCES SOCIETY amoc Cuvatue Loetza Space NEJAT EKMEKÇI ILMI ACISALIOĞLU AND KĀZIM İLARSLAN Aaa Uvety Faculty of Scece Depatmet of Mathematc 6 Tadoğa Aaa Tuey e-mal: emec@ceceaaaedut Abtact I the Loetza pace a egula cuve called tme-le pace-le ad ull cuve accodg to velocty vecto tatu [] [6] I th pape we obtaed the amoc cuvatue fo a egula cuve the Loetza pace We alo obtaed the elatohp betwee the amoc cuvatue ad the ow devato Itoducto I -dmeoal Eucldea Space a egula cuve decbed by t cuvatue If all cuvatue of a cuve ae detcally zeo the the cuve a geodec If oly the ft cuvatue a o-zeo cotat ad othe ae all detcally zeo the the cuve called a ccle If the ft ad ecod cuvatue ae o-zeo cotat ad othe ae all detcally zeo the cuve called a helx A egula cuve called geeal helx f t ft ad ecod cuvatue ae ot cotat but cotat [5] The ato called Ft amoc Cuvatue of the cuve ad how ghe hamoc cuvatue of a egula cuve ae defed a follow : < { [ ] } whee Feet-fame ad hghe cuvatue of a egula cuve aumed a zeo amoc cuvatue have mpotat ole chaactezato of Geeal elce The mot mpotatly the ubect ha ot bee vetgated Loetza Space The am of the peet wo to defe thoe amoc Cuvatue Loetza Space

2 7 N Emeç et al Pelmae Symmetc blea fom Let be a eal vecto pace A blea fom o a R-blea fucto : R ad we code oly the ymmetc cae v w w v fo all v w A ymmetc blea fom o [6]; a potve [egatve] defte povded v mple v v > [ < ] b potve [egatve] em defte povded v v [ ] c odegeeate povded v w fo all w mple v If a ymmetc blea fom o the fo ay ubpace W of the etcto W W deoted meely by aga ymmetc ad blea If W [em-] defte o W The dex ν of a ymmetc blea fom o the laget tege that the dmeo of a ubpace W o whch egatve defte W Thu ν dm ad ν f ad oly f potve emdefte [6] A cala poduct o a vecto pace a odegeeate ymmetc blea fom o If a ymmetc blea fom o odegeeate the t called a cala poduct o A cala poduct pace v ha a othoomal ba ad e δ whee e e ± e Fo ay othoomal ba e e L e fo the umbe of egatve g the gatue L the dex ν of If ha dex ν ν fo two vecto v p ad w p we ca wte v p w p ν v w ν v w The eultg Sem-Eucldea Space A Loetz vecto pace [6] R ν educe to R f ν L to be a cala poduct pace of dex ad dmeo

3 amoc Cuvatue Loetza Space 75 Fx the otato; fo ν fo ν Cuve A cuve a Loetza pace L a mooth mappg γ : I L whee I a ope teval the eal le R The teval I ha a coodate ytem cotg of the detty map u of I The velocty vecto of γ at t I γ t dγ u du t A cuve γ ad to be egula f γ t fo all t I A cuve γ a Loetza pace L ad to be pace-le f t velocty vecto γ ae pace-le fo all t I mlaly fo tme-le ad ull If γ a pace-le o tme-le cuvewe ca epaametze t uch that γ t γ t whee f γ pace-le ad f γ tme-le epectvely I th cae γ ad to be ut peed o t ha ac legth paametzato [] [6] Defto Let γ be a cuve L paametzed by t ow ac legth Deotg the Feet vecto feld of th cuve L Aumg; d d L the fucto defed by the equalty d < > d ae called the hghe odeed cuvatue of the cuve γ Whee []

4 N Emeç et al 76 Theoem [] [] Let L γ be a egula cuve coodate eghbohood γ I ad { } L be the Feet -fame at γ wth I The; a b c We ca wte matx epeetato a follow fo the matx epeetato we get I patcula whe the cuve tme-le we get theefoe we obta

5 amoc cuvatue amoc Cuvatue Loetza Space 77 Defto Let γ be a tme-le cuve L ad be the ft Feet vecto feld of γ X χ L beg a cotat ut vecto feld f X coh ϕ cotat the γ called a geeal helx cled cuve the pace Sp {X} called lope ax L ϕ called lope agle ad Defto Let γ : I L be a geeal helx paametzed by t ac legth Let X be a ut ad cotat vecto feld of L ad let { L } be Feet -fame at the pot of γ of γ If we code the agle betwee γ ad X a ϕ ; : I R X coh ϕ the the value of the fucto at the pot of γ called a the -th hamoc cuvatue accodg to X at the pot of γ of γ Theoem Letγ be a geeal helx cled cuve L paametzed by t ac legth all be the hghe odeed cuvatue Whe the -th hamoc cuvatue at γ of γ The Poof If we tae the devatve of the equato X coh ϕ cotat we get X

6 78 o N Emeç et al X X fom th aga tae the devatve X fom th eplacg the ' value we get o X o X X coh ϕ coh ϕ thu we have Fom Defto we ca wte X coh ϕ If we tae devato of X ug the value of aga fom th We get coh ϕ X coh ϕ o o X X coh ϕ coh ϕ coh ϕ coh ϕ

7 amoc Cuvatue Loetza Space 79 Thu we have th complete the poof of theoem Fally we obta the elatohp betwee ad value the matx fom 5 5 M fo the matx epeetato we get 5 5 Refeece N Emeç K İlala ghe cuvatue of a egula cuve Loetza pace Joual of Ittute of Mathematc ad Compute Scece Gluc ghe cuvatue of cuve Eucldea pace Ame Math Moth T Iawa O cuve ubmafold a Idefte-Remaa mafold Tuuba J Math JD Jee Gau-Boet Fomula fo Geeal Loetza Suface Geometac De dcad R Mülle Kemat Dele Aaa Üvete Fe Faülte Yayılaı Mat B O Nell Sem-Remaa Geomety wth Applcato to Relatvty Academc Pe New Yo 98 7 A Sabucuoğlu acıalhoğlu O hghe cuvatue of a cuve Commucato de la Fac Sc U AaaTome A N Taıöve Betad cuve -dmeoal Eucldea pace Joual of Kaadez T Uvety Faculty of At ad Scece ee of Mathematc-Phyc IX