Semi-Riemann Metric on. the Tangent Bundle and its Index

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1 t J Cotem Math Sceces ol 5 o Sem-Rema Metrc o the Taet Budle ad ts dex smet Ayha Pamuale Uversty Educato Faculty Dezl Turey ayha@auedutr Erol asar Mers Uversty Art ad Scece Faculty Mers Turey yerol@mersedutr Abstract ths aer t s examed relatos betwee the causal character of lfted taet vector o the taet budle wth sem-rema metrc of a sem- Rema mafold ad the causal character of the taet vector o the sem- Rema mafold wth sem-rema metrc Moreover t s roved that whch s obtaed term of the horzotal lft of a sem-rema metrc wth - dex o a dfferetable mafold s a sem-remaa metrc wth -dex Mathematcs Subject Classfcato: 53C5 53C7 Keywords: the sem-rema metrc ad secal metrc o taet budle troducto The study about metrcs o taet budle of a Rema mafold was troduced at the ed of the 95 s [ao ad shhara 97] The metrcs o M were lfted TM by vertcal comlete ad horzotal lfts [ao ad shhara 97] [Orou ad Paahuc 998] examed the dfferetal eometry of TM by ths metrc Moreover they defed that obtaed metrc o TM by the horzotal lft of Rema or sem-rema metrc o M s sem-rema metrc wthout roof addto They also defed that lfted metrc o TM by horzotal lft s a sem-rema metrc wth ostve ad eatve ss wthout roof

2 34 Ayha ad E asar ths aer t s roved that TM s a sem-rema mafold The t s showed that the taet vector M ca resectvely sacele ull ad tmele f the taet vector TM whch has bee lfted a taet vector M s sacele ull ad tmele Fally t s obtaed that obtaed metrc o TM by the horzotal lft of a Rema or a sem-rema metrc o M s sem-rema metrc wth ostve ad eatve ss A dfferetable mafold ad ts taet budle Let M be a dfferetable -dmesoal mafold ad TM be ts taet budle Suose that x { x x } s a system of local coordates defed the ehborhood U M Sce the caocal rojecto π :TM M obtas / the equalty π π U U s oe ehborhood of the ot π { } TM x y x x [ x ] [ x ] Therefore the ma xy whch s defed wth the equalty s a local ma U / TM ad the system of x y ; s duced local coordate system TM TTM the taet budle of TM has subvector budle TM Çe τ M * whch s called vertcal dstrbuto o TM ad TM whch s called horzotal dstrbuto o TM addto t ca be exressed TTM as drect sum of subvector budles TM ad TM TTM TM TM { δ ; } s adated local frame TM where δ s local frame TM δ j j j δ N N j y Γ 3 x δx x y ad s local frame TM 4 x y Furthermore { δy dx ; } s adated local dual frame TM where j δ y dy N j dx N j y Γj 5 Orou ve Paahuc 998 s defed vector felds o TM as bellow for ay χ M ad χ TM 6

3 Sem-Rema metrc 35 the horzotal lft of the metrc tesor wth comoets j o M has comoets j j jdx δ y jδy dx 7 wth resect to adated local frame TM or j j j j jγ j y Γj y dx dx jdx dy jdy dx 8 wth resect to duced local coordates TM Moreover the matrx reresetato of s j jγ j y Γj y j 9 j ao ad shhara 97 3 Sem-Rema mafold Sem-Rema eometry volves a artcular d of tesor o taet saces To study these eeral let be a fte dmesoal real vector sace A blear form o s a R-blear fucto b : R ad let b be a symmetrc Defto 3 A symmetrc blear form b o s called ostve [eatve] defte rovded v mles b v v > [ < ] ad s called odeeerate rovded b v w for all w mles v f b s a symmetrc blear form o the for ay subsace W of the restrcto b W W deoted merely by b W s aa symmetrc ad blear Defto 3 The dex of a symmetrc blear form b o s the larest teer that s the dmeso of a subsace W o whch b W s eatve defte Defto 33 A scalar roduct o a vector sace s odeeerate symmetrc blear form o Defto 34 f smoothly asss to each ot of M a scalar roduct o the taet sace T M ad the dex of s the same for all a smooth mafold M furshed wth a metrc tesor s called a sem-rema mafold A sem-rema mafold s deoted a ordered ar M Defto 35 A taet vector v a sem-rema mafold M s sacele vector f v v > or v ull vector f v v ad v tmele vector f v v <

4 36 Ayha ad E asar The cateory to whch a ve taet vector falls s called ts causal character To reveto clumsy ad to mae comutato easer we wll use ormal coordate system whle comut the dex of a sem-rema mafold Theorem 3 Let M be a sem-rema mafold wth dex f x x s a ormal coordate system at M t s j j j j j j ε ε δ Γ j for j O Nell The metrc TM Theorem 4 Let M be a dfferetable mafold ad be a Rema or sem- Rema metrc o M f TM χ s a set of vector felds TM ad R TM C s a r of dfferetable fucto whose rae set s real umber s sem- Rema metrc TM where s R TM C TM TM : χ χ Proof: Let be a vector feld M ad be a vector feld TM All vector felds TM are exressed wth drect sum of vector felds ad due to roerty as below 3 We et ad for ay M χ TM χ ad R Thus we obta that s blear By the equalty 3 we et Thus we obta that s symmetrc By the defte odeeeracy of a metrc we et for ad TM χ We fd that ad by smlary oerato we et

5 Sem-Rema metrc 37 We also fd that Thus we et for χ TM Namely s odeeerate cocluso sce t s rovded o deeerate symmetrc ad blear roertes of s a sem-rema metrc TM Theorem 4 Let M be a sem-rema mafold ad sem-rema metrc TM Let T M be the taet sace at a ot M ad T TM be the taet sace at a ot TM whch rovde equalty π f are sacele or tmele vectors M are ull vectors T TM T both ad Proof: By the equato 6 ad defto 35 t ca be see roof of the clam strahtforward Theorem 43 Ay vector whch s defed a sem-rema mafold TM s ull vector f ad oly f the vector whch s defed o TM les vertcal or horzotal vector subsace taet sace of TM or rojected vector T M of the vector whch s defed T TM s ull vector Proof: By the equato 6 t s clear By the equalty 3 t s wrte where TTM ad TM We et addto we et Thus s also ull vector Theorem 44 f ay vector whch s defed o sem-rema mafold TM s sacele vector the vector whch s rojected o sem-rema mafold M s sacele vector Proof: By the art of reced theorem t s wrte where TTM ad TM By the defto 35 we et > or

6 38 Ayha ad E asar Thus s sacele vector Theorem 45 f ay vector whch s defed o sem-rema mafold TM s tmele vector the vector whch s rojected o sem-rema mafold M s tmele vector Proof: By the art of theorem 43 t s wrte where TTM ad TM By the defto 35 we et < Thus f s tmele vector s tmele vector 5 The dex of the metrc Theorem 5 f M s a Rema mafold mafold wth -dex TM s a sem-rema Proof: Let be a Rema metrc o M We tae a ormal coordate system M terms of the equaltes t s obtaed that j j Γj j j δ 4 for The equaltes 4 ta to accout 9 the matrx reresetato of The eevalue of ths obtaed metrc x x ca be see by the equalty s x x x 5 x x The equalty 5 ca be roved by the methot of ducto as below t s true that x for We suose that follow equalty s true x

7 Sem-Rema metrc 39 The equalty ca be roved as follow for x det x Sce the dex of s deedet chose of ma o TM [see the defto 34] has ostve ad eatve ee value whose equal to ad - The ee vectors whch s corresoded the ee value are ad the ee vectors whch s corresoded the ee value - are The matrx whch are obtaed terms of eevectors ad ts verse

8 4 Ayha ad E asar are ad Thus P x x x x P x P P the daoal matrx of x x x x x s exressed as follow Fally terms of the matrx reresetato of we ca see that sem-rema metrc wth ostf ad eatf s strahtforward s a Theorem 5 f M s a Rema mafold wth - dex sem-rema mafold wth -dex TM s a Proof: Let M be a sem-rema mafold wth - dex f we tae ormal coordate system M t s ; j j ; j j where s daoal square matrx The frst elemets whch are o the daoal of the ths matrx are - ad rest elemets are Moreover t s Γ Thus the reresetato of s that j The eevalue of ths obtaed metrc ca be see by the equalty x x x The equalty 6 ca be roved that t s cosdered two cases as follow Case : Suose that -chaeable ad o chaeable we wll the methot of ducto so that we rove case t s true that 4 x4 for 6

9 Sem-Rema metrc 4 We suose that follow equalty s true for x We ca rove the true of the equalty of 6 for as below x x det Case : t s -o chaeable ad chaeable x 3 5 t s true that ad 3 6 x6 for 3 ad We suose that follow equalty s true for 3 6 x6 for 3 ad ad x We ca rove that the true of the equalty for ad as follow

10 4 Ayha ad E asar x x x x x x x Thus the value of the detemat x s deedet the choose of both ad Furthermore has ostve ad eatve eevalue whch equal to ad - The ee vector whch s corresoded the ee value ad the eevector whch s corresoded the eevalue -

11 Sem-Rema metrc 43 The matrx whch are obtaed terms of eevectors ad ts verse are P ad P Thus P P the daoal matrx of s exressed as follow Fally terms of the matrx reresetato of we ca see that s a sem-rema metrc wth ostve ad eatve s strahtforward Refereces [] Ayha Cöe A Caşar E C Sem-Rema Metrc O The Taet Budle ad ts dex CBU Joural of Scece [] Dombrows P O the eometry of the taet budle J Ree Aew Math [3] O ell B Sem-Remaa Geometry Academc Pres c Lodo 983 [4] Orou Paahuc N O the eometry of taet budle of a seudo-remaa mafold A Stt Uv Al Cuza as SerNoua Mat 36 No

12 44 Ayha ad E asar [5] ao K shhara S Taet ad Cotaet Budles Marcel Decer c New or Receved: Arl 9