Infinitesimal Automorphisms in the Tangent Bundle of a Riemannian Manifold with Horizontal Lift of Affine Connection

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1 Cag a J Sc 2007; 34(2) 5 Cag a J Sc 2007; 34(2) : 5-59 wwwscececmuact/joural-scece/josctml Cotrbuted Paper Iftesmal Automorpsms te Taget Budle of a Remaa afold wt orzotal Lft of Affe Coecto Ayd Gezer *, ad Atatürk Üverstes, Fe-Edebyat Fakültes, atematk Bölümü, Erzurum, Turkey * Autor for Correspodece, e-mal : agezer@atauedutr Receved: ay 2006 Accepted: 9 Aprl 2007 ABSTRACT Te ma purpose of te preset paper s to study codtos for a vertcal ftesmal affe trasformato te taget budle of a Remaa mafold wt respect to te orzotal lft of affe coecto ad te to apply te results obtaed to te study of fbre-preservg ftesmal affe trasformato ad also to vestgate ftesmal sometry ts settg Keywords: lft, taget budle, ftesmal affe trasformato, fbre-preservg trasformato, ftesmal sometry INTRODUCTION Let be a Remaa mafold wt metrc g wose compoets a coordate egborood U are g j ad deote by Γ j te Crstoffel symbols formed wt g j If, te egborood π ( U ) of te taget budle T( ) over, U beg a egborood of, te g as compoets gve by t t Γ j gt +Γ gjt g j g = g j 0 wt respect to ( x, y ) duced coordates T( ) ad Γ j, = y Γ j Γ jbeg compoets of te affe coecto Let be a pseudo-remaa metrc, te te orzotal lft g of g wt respect to s a pseudo-remaa metrc T( ) Sce C g s defed by g = g γ ( g), were γ ( g) s a tesor feld of type, wc as te compoets of te form s y sg 0 γ ( g) = 0 0, we ave g ad C g cocde f ad oly f g = 0 [, p05] 2 j If we wrteds = g jdx dx te pseudo- Remaa metrc gve by g, te te pseudo-remaa metrc T( ) gve by te g of g to T( ) wt respect to a affe coecto s, () were ad are compoets of te coecto defed by

2 52 Cag a J Sc 2007; 34(2), Y, T0 ( ), [, p67 ] We sall ow defe te orzotal lft of affe coecto to T( ) by te codtos V Y = V 0, 0 V Y =, (2) Y = ( Y ), ( ) Y = Y V V for Y, I 0( ) From (2), te orzotal lft of as compoets Γ K suc JI tat Γ =Γ, Γ k j= Γ k j= Γ k j= Γ k j= 0, Γ k s k s k j = y sγj y Rsj, Γ k j= Γ k j= Γ k j, (3) k k j j wt respect to te duced coordates T( ), were k Γj are compoets of Let g ad be, respectvely, a pseudo-remaa metrc ad a affe coecto suc tat g = 0 Te g = 0, were g s a pseudo-remaa metrc Te coecto as otrval torso eve for te Remaa coecto determed by g, uless g s locally flat [, p] Let tere be gve a affe coecto ad a vector feld I ( ) 0 Te te Le dervatve L wt respect to s, by defto, a elemet of I ( ) 2 suc tat ( L )( Y, Z) = L( YZ) Y( LZ) Z = L, Z Y Z, (4) [ Y, ] [ ] [ Y, ] for ay Y, Z I 0( ) I a mafold wt affe coecto, a ftesmal affe trasformato ' x = x + ( x,, x ) tdefed by a vector feld I ( ) 0 s called a ftesmal affe trasformato f L = 0, [, p67 ] Te ma purpose of te preset paper s to study te ftesmal affe trasformato ad ftesmal sometry T( ) wt affe coecto 2 VERTICAL INFINITESIAL AFFINE TRANSFORATIONS IN A TANGENT BUNDLE WIT α From (4) we see tat, terms of compoets Γ γβ of, s a ftesmal affe trasformato - dmesoal mafold f ad oly f, α λ α λ α α λ α λ + Γ Γ +Γ +Γ = 0, αβ,, =,, (5) γ β λ γβ γβ λ λβ γ γλ β Let tere be gve wt a affe coecto wt Crstoffel symbols k Γ j

3 Cag a J Sc 2007; 34(2) 53 Let, were =, = =, x y x = +,, 2 be a vector feld T( ) Te, takg accout of (3), we ca easly see from (5) tat s a ftesmal affe trasformatos T( ) wt f ad oly f te followg codtos (6)-(3) old:, (6), (7), (8), (9) (0), (), (2), (3) Let be a vertcal ftesmal affe trasformato T( ) Te as compoets wt respect to te duced coordates Tus, from (3), we ave =0, e, were C ad C C dx = ad, te, (4) D deped oly o varables Sce s a vector feld ( ) T, D= D are defed elemets of ( ) I ad I ( ) 0, respectvely Teorem If s a vertcal ftesmal affe trasformato of T( ) wt (a) L + C ( D R) = 0, D D x =, ( ) k I0 ad C ( D R) = D R kj D

4 54 Cag a J Sc 2007; 34(2) (b) C s parallel wt respect to, e, C = 0 (c) C( T( Y, Z)) = T( CY, Z) = T( Y, CZ), for ay Y, Z I ( ) 0, were T deotes te torso tesor of, e T s pure tesor wt respect to C (d) C( ZT)( Y, W) = ( CZT)( Y, W), for ay Y, Z, W I ( ) 0 (e) Coversely, f C ad D satsfy te codtos (a), (b), (c) ad (d), te vector feld s a ftesmal affe trasformato of T( ) wt coecto, were γ C s a vertcal vector feld, wc as compoets of te form γ C 0 = yc Proof: (a) Substtutg (4) ad = 0 (0), we ave k k k k k k k j Cs + Cs Γ k j Γj kcs Γ s jck +Γ k jcs +Γjk Cs CsRkj + RsjCk = 0, (5) ad k k k k k D + D Γ Γ D +Γ D +Γ D D R = 0, (6) j k j j k k j jk kj wc meas tat L + C ( D R) = 0 (b) Substtutg (4) D = 0 ad (2), we obta, k k C Γ C +Γ C = 0, (7) j j k k j Substtutg (4) ad = 0 (), we obta, k k C Γ C +Γ C = 0, (8) j j k jk wc meas C s parallel (c) Itercagg ad j (8), we ave, k k Cj Γ jck +Γ kcj = 0, ad subtractg te resultg equato from (7), we ave, k k TC j k = TC k j, (9) tat s, C( T( Y, Z)) = T( CY, Z) (20) for ay Y, Z I ( ) 0 From (9), we obta TYCZ (, )) = TCZ (, ) = CTZY ( (, )) = CTY ( (, Z )) ad ece C( T( Y, Z)) = T( CY, Z) = T( Y, CZ) wc s te formula (c)

5 Cag a J Sc 2007; 34(2) 55 (d) Usg (7) ad (8), we elmate all partal dervatves of k k k j l k l j C j from (5) Te we obta, C T = T C, e T s φ - tesor wt respect to C [3] (2) (e) If we assume tat te codtos (a), (b), (c) ad (d) are establsed, te we see tat, gve (e), s a ftesmal affe trasformato Cosequetly, Teorem s completely proved Teorem 2 Let C be as Teorem If s a ftesmal affe trasformato of wt affe coectos ad RYZξ (,, ; ) s pure wt respect to ad ξ, so s C 3 FIBRE-PRESERVING INFINITESIAL AFFINE TRANSFORATION WIT A trasformato of T( ) s sad to be fbre-preservg f t seds eac fbre of T( ) to a fbre A ftesmal trasformato of T( ) s sad to be fbre-preservg f t geerates a local -parameter group of fbre-preservg trasformatos A ftesmal trasformato wt compoets s fbre-preservg f ad oly f deped oly o te varables T( ) From x,, x wt respect to te duced coordates ( x, y ) we see tat a fbre-preservg ftesmal trasformato wt compoets duces a ftesmal trasformato wt compoets te base space Sce = 0 ad, te from (6) we ave: Teorem 3 If a fbre-preservg ftesmal trasformato of T( ) wt orzotal lft of a affe coecto to T( ), te te ftesmal trasformato duced o from s also affe wt respect to Teorem 4 Let be a affe coecto Te, C C C C C ( LC )( Y, Z) = ( L )( Y, Z) + γ ( L R)(, Y, Z), for ay I ( ) 0

6 56 Cag a J Sc 2007; 34(2) Proof Our proposto follows from te followg computatos: C C C C C ( L )( Y, Z) = L ( Z) ( L Z) C Z C C C C C C Y Y [, Y] = L [ ( Z) γ ( R(, Y) Z)] ( L Z) Z C C C C C C Y Y [, Y] C C C C = [, Y] [, γ( R(, Y) Z)] ( ( L Z)) + γ( R(, Y) L Z) Y C ( Z [, ] ) +γ R Y ([, Y ] Z )] = ( L Y) ( ( L Z)) ( Z) γ ( L R(, Y) Z) C C C Y [, Y] + γ( R(, Y) L Z) + γ( R(, L Y) Z) = C ( L )( C Y, C Z) + γ ( L R(, Y) Z + R(, Y) L Z + R(, L Y) Z) C C C = ( L )( Y, Z) γ ( L R)(, Y, Z) were R(, ) Y deotes a tesor feld W of type (,) suc tat W( Z) = R( Z, ) Y for ay Z I ( ) 0 Let ad be as Teorem 3 From Teorem 4 we see tat, t s also kow tat f s ftesmal affe trasformato, L R= 0 [] 5, te c s a ftesmal affe trasformato of T( ) wt Sce c as te compoets, t follows tat - c s a vertcal ftesmal affe trasformato T( ) wt Tus we ave, Teorem 5 If s a fbre-preservg ftesmal affe trasformato of T( ) wt te lft, te = c + v D+ γ C, were D ad C are tesor felds of type (, 0) ad (,), respectvely, satsfyg codtos (a), (b) ad (c) of Teorem g 4 INFINITESIAL ISOETRY WIT A vector feld I s sad to be a ftesmal sometry or a Kllg vector ( ) 0 feld of a Remaa mafold wt metrc g, f L g = 0 [] 4 I terms of compoets gj of g, s ftesmal sometry f ad oly f L g = g + g + g = + α α α j α j α j α j j j α beg compoets of, were s te Remaa coecto of te metrc g Let be vector feld T( ) ad ts compoets wt respect to duced coordates Te te covarat dervatve as compoets

7 Cag a J Sc 2007; 34(2) 57 s Γ beg gve by (3), wt respect to duced coordates J I We ow cosder a vector feld I V ( ( )) c I 0 T, complete lft I ( ( )) 0 T ad orzotal lft ave respectvely compoets of te form V 0 = x, c x = x, wt respect to te duced coordates T( ), were (22) ( ) 0, te ts vertcal lft x = Γ x Γ x = y Γ x s s I T 0 ( ( )) We ow compute te Le dervatves of te metrc g wt respect to V c, ad, by meas of (3) ad (23) Te Le dervatves of g wt respect to V, c ad ave respectvely compoets (23) 0 0 V J V I LV g = ( I + J ) = j j + 0 j + j 0 c J c I Lc g = ( I + J ) = s j s j k j y s( j ) y ( Rsk Rsjk ) j j + j 0 J I L g = ( I + J ) = j j Γ Γ j + j (24) Takg accout of te fact tat k = 0 mples j k R = 0 k ad R = 0 We ave, Teorem 6 Necessary ad suffcet codtos order tat a) Vertcal V I ( ( )) 0 T b) Complete c) orzotal c I T sk 0 ( ( )) I T 0 ( ( )) lfts to T( ) wt te metrc g, of a vector feld be a Kllg vector feld T( ) are tat a) s ftesmal sometry b) s ftesmal sometry wt vasg covarat dervato c) s ftesmal sometry wt vasg covarat dervato sjk Let ad Y be vector felds If ad Y are Kllg vector felds from te defto of Kllg vector feld, te we ave,,

8 58 Cag a J Sc 2007; 34(2) L g = L ( ) ( ) 0, LYg LY Lg = (25) [ xy] e [ Y, ] s ftesmal sometry elemets of We ow deote by I ( ) 0, defed by for ay Z I ( ) 0 Te we ave, [] 3 AY te tesor feld of type (,), ad Y beg two gve [ ] [, ] ( AYZ ) = ( L )( YZ, ) = L, Z Z (26) Y Y V c V, Y [, Y c = ], c c, Y [, Y c = ], [ ], Y =, Y γ ( A Y) (27) 0 were γ ( AY ) I 0( ), wc as te compoets of te form s y ( AY) s A ftesmal trasformato defed by vector feld I ( ) 0 s sad to be ftesmal trasformato wt affe coecto, f L = 0 Te, from (26) ad (27) [ ] c, Y =, Y (28) We compute te Le dervatves of te metrc V c g wt respect to [ Y, ] ad [ Y, ] L g = L g = L ( L g ) L ( L g ) V [ Y, ], V c V c c V Y Y Y L c c c c ( c ) c ( c ) [, ] g = L g = L L g L L g Y, Y Y Y (29) from (24) ad (29), we get, Teorem 7 Suffcet codtos order tat te vertcal, complete lfts of a vector feld [ Y, ] ad Y s a ftesmal sometry to T( ) be a ftesmal sometry wt metrc g are tat wt vasg ter covarat dervatos Let be ftesmal affe trasformato From (25) ad (29), we ave L c c ( ) ( c ) [, ] g = L g = L L g L L g Y, Y Y Y (30) from (24) ad (30), we get, [ Y, ] Teorem 8 Suffcet codtos order tat te orzotal lft of a vector feld to T( ) be a ftesmal sometry wt metrc g are tat ad Y are ftesmal sometry wt vasg ter covarat dervatos ACKNOWLEDGEENT We are grateful to Professor AA Salmov for s valuable suggestos

9 Cag a J Sc 2007; 34(2) 59 REFERENCES [] Yao K ad Isara S Taget ad cotaget budles, arcel Dekker, New York, 973 [2] agde A ad Salmov A A, orzotal lfts of tesor felds to sectos of te taget budle (Russa), Izv Vyss Uceb Zaved at; 200, 3; [3] Kobayas S ad Nomzu K Foudatos of Dfferetal Geometry, A Wley-Iter Scece Publcato, New York, 963 [4] ayers S B ad Steerod NE, Te group of Isometres of a Remaa mafold, Aals of at, 939; 40 : [5] Nomzu K ad Yao K, O ftesmal trasformatos preservg te curvature tesor feld ad ts covarat dfferetals, Aales de l sttut Fourer, tome 4, 964; 2 : [6] Yao K, Te Teory of Le Dervatves ad Its Applcatos, Amsterdam, 957