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Infinitesimal Affine Transformations in the Tangent Bundle of a Riemannian Manifold with Respect to the Horizontal Lift of an Affine Connection

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Infinitesimal Affine Transformations in the Tangent Bundle of a Riemannian Manifold with Respect to the Horizontal Lift of an Affine Connection Hacettepe Journal of Mathematics and Statistics Volume 35 (2) (2006), 155 { 159 INFINITESIMAL AFFINE TRANSFORMATIONS IN THE TANGENT BUNDLE OF A RIEMANNIAN MANIFOLD WITH RESPECT TO THE HORIZONTAL LIFT OF AN AFFINE CONNECTION A. Gezer¤y and K. Akbulut¤ Received 28 : 02 : 2006 : Accepted 04 : 12 : 2006 Abstract The main purpose of the present paper is to study properties of vertical in¯nitesimal a±ne transformation in the tangent bundle of a Riemann- ian manifold with respect to the horizontal lift of an a±ne connection, and to apply the results obtained to the study of ¯bre-preserving in¯n- itesimal a±ne transformations in this setting. Keywords: Lift, Tangent bundle, In¯nitesimal a±ne transformation, Fibre-preserving transformation. 2000 AMS Classi¯cation: 53 B 05, 53 C 07, 53 A 45 1. Introduction Let Mn be a Riemannian manifold with metric g whose components in a coordinate h neighborhood U are gji, and denote by ¡ji the Christo®el symbols formed with gji. If, in ¡1 the neighborhood ¼ (U) of the tangent bundle T (Mn) over Mn, U being a neighborhood H of Mn, then g has components given by ¡t g + ¡tg g H g = j ti i jt ji µ gji 0 ¶ i i h j h h with respect to the induced coordinates (x ; y ) in T (Mn), where ¡i = y ¡ji, ¡ji being the components of the a±ne connection in Mn. Let g be a pseudo-Riemannian metric. Then the horizontal lift H g of g with respect H H C to r is a pseudo-Riemannian metric in T (Mn). Since g is de¯ned by g = g ¡ γ(rg), ¤Department of Mathematics, Faculty of Art and Sciences, AtaturkÄ University, 25240, Erzu- rum, Turkey. E-mail: (A. Gezer) [email protected] (K. Akbulut) [email protected] yThis paper is supported by TUBITAK grant number 105T551/TBAG-HD-112 156 A. Gezer, K. Akbulut where γ(rg) is a tensor ¯eld of type (0; 2), which has components of the form γ(rg) = ysr g 0 s , we have that H g and C g coincide if and only if rg = 0 [1, p.105]. µ 0 0¶ 2 j i If we write ds = gjidx dx the pseudo-Riemannian metric in Mn given by g, then H the pseudo-Riemannian metric in T (Mn) given by the g of g to T (Mn) with respect to an a±ne connection r in Mn is 2 j i (1) ds = 2gji±y dx ; ~ j j ~j l k ~h h ~ where ±y = dy +e¡lky dx and ¡ji = ¡ij are components of the connection r de¯ned ~ 1 by rX Y = rY X + [X; Y ] ; 8 X; Y 2 T0 (Mn), [1, p.67 ]. H We shall now de¯ne the horizontal lift r of the a±ne connection r in Mn to T (Mn) by the conditions H V H H rV X Y = 0; rV X Y = 0; (2) H V V H H H rH X Y = (rX Y ) ; rH X Y = (rX Y ) ; 1 H H K for X; Y 2 =0(Mn). From (2), the horizontal lift r of r has components ¡JI such that H k k H k H k H k H k¹ ¡ij = ¡ij ; ¡i¹j = ¡¹{j = ¡¹{¹j = ¡¹{¹j = 0; (3) H k¹ s k s k H k¹ H k¹ k ¡ij = y @s¡ij ¡ y Rsij ; ¡¹{j = ¡i¹j = ¡ij k with respect to the induced coordinates in T (Mn), where ¡ij are the components of r in Mn. Let g and r be, respectively, a pseudo-Riemannian metric and an a±ne connection such that rg = 0. Then H rH g = 0, where H g is a pseudo-Riemannian metric. The connection H r has nontrivial torsion even for the Riemannian connection r determined by g, unless g is locally flat [1, p.111]. 1 Let there be given an a±ne connection r and a vector ¯eld X 2 =0(Mn). Then the 1 Lie derivative LX r with respect to X is, by de¯nition, an element of =2(Mn) such that (4) (LX r)(Y; Z) = LX (rY Z) ¡ rY (LX Z) ¡ r[X;Y ]Z 1 for any Y; Z 2 =0(Mn). In a manifold Mn with a±ne connection r, an in¯nitesimal a±ne transformation 0 h h h 1 n 1 x = x + X (x ; : : : ; x )¢t de¯ned by a vector ¯eld X 2 =0(Mn) is called an in¯ni- tesimal a±ne transformation if LX r = 0, [1, p.67 ]. The main purpose of the present paper is to study the in¯nitesimal a±ne transforma- H tion in T (Mn) with a±ne connection r. 2. Vertical in¯nitesimal a±ne transformations in a tangent bundle with H r ® From (4) we see that, in terms of the components ¡γ¯ of r, X is an in¯nitesimal a±ne transformation in the m-dimensional manifold Mn if and only if, ® ¸ ® ¸ ® ® ¸ ® ¸ (5) @γ @¯ X + X @¸¡γ¯ ¡ ¡γ¯ @¸X + ¡¸¯ @γ X + ¡γ¸@¯ X = 0; ®; ¯; : : : = 1; : : : ; m: Let there be given in Mn with metric g an a±ne connection r with Christo®el symbols k ~ ~ i ~ ¹{ @ @ @ ¡ij . Let X = X @i + X @¹{, where @i = @xi , @¹{ = @yi = @x¹{ , ¹{ = n + 1; : : : ; 2n be a vector ¯eld in T (Mn). Then, taking account of (3), we can easily see from (5) that X~ In¯nitesimal A±ne Transformations 157 H is an in¯nitesimal a±ne transformations in T (Mn) with r if and only if the following conditions (6){(13) hold: ~ h ~ k h k ~ h k ~ h h ~ k h ~ k @j @iX + X @k¡ji ¡ (¡ji@kX + @¡ji@k¹X ) + ¡ki@j X + ¡jk@iX (6) s k ~ h + y Rsji@k¹X = 0; ~ h k ~ h h ~ k (7) @j @¹{X ¡ ¡ji@k¹X + ¡jk@¹{X = 0; ~ h k ~ h h ~ k (8) @¹j @iX ¡ ¡ji@k¹X + ¡ki@¹j X = 0; ~ h (9) @¹j @¹{X = 0; ~ h¹ ~ k h ~ k¹ h k ~ h¹ k ~ h¹ h ~ k @j @iX + (X @k@¡ji + X @k¡ji) ¡ (¡ji@kX + @¡ji@k¹X ) + (@¡ki@j X h k¹ h k h k¹ k¹ h s k h (10) + ¡ki@j X~ ) + (@¡jk@iX~ + ¡jk@iX~ ) ¡ X~ Rkji ¡ y X~ @kRsji s k ~ h¹ s h ~ k s h ~ k + y Rsji@k¹X ¡ y Rski@j X ¡ y Rsjk@iX = 0; ~ h¹ ~ k h k ~ h¹ h ~ k h ~ k h ~ k¹ @j @¹{X + X @k¡ji ¡ ¡ji@k¹X + ¡ki@j X + (@¡jk@¹{X + ¡jk@¹{X ) (11) s h k ¡ y Rsjk@¹{X~ = 0; ~ h¹ ~ k h k ~ h¹ h ~ k h ~ k¹ h ~ k @¹j @iX + X @k¡ji ¡ ¡ji@k¹X + (@¡ki@¹j X + ¡ki@¹j X ) + ¡jk@iX (12) s h ~ k ¡ y Rski@¹j X = 0; ~ h¹ h ~ k h ~ k (13) @¹j @¹{X ¡ ¡ki@¹j X + ¡jk@¹{X = 0: Let X~ be a vertical in¯nitesimal a±ne transformation in T (Mn). Then X~ has components 0 h¹ ¹ with respect to the induced coordinates. Thus, from (13), we have @¹@¹{X~ = 0, µX~ h¶ j i.e., h¹ h i h (14) X~ = Ci y + D ; h h h where Ci and D depend only on the variables x . Since X~ is a vector ¯eld in T (Mn), h i h 1 1 C = Ci @h­dx and D = D @h are de¯ned elements of =1(Mn) and =0(Mn), respectively. H 2.1. Theorem. If X~ is a vertical in¯nitesimal a±ne transformation of T (Mn) with r, then r ­ h @ 2 =1 ­ k h (a) LD + C(D R) = 0, D = @ @xh , D 0(Mn) and C(D R) = D Rkji. (b) C is parallel with respect to r, i.e., rC = 0. 1 (c) C(T (Y; Z)) = T (CY; Z) = T (Y; CZ), for any Y; Z 2 =0(Mn), where T denotes the torsion tensor of r, i.e. T is a pure tensor with respect to C. 1 (d) C(rZ T )(Y; W ) = (rCZ T )(Y; W ), for any Y; Z; W 2 =0(Mn). (e) Conversely, if C and D satisfy the conditions (a), (b), (c) and (d) then the vector ¯eld @ X~ = (Chyi + Dh) = γC +v D i @yh H is an in¯nitesimal a±ne transformation of T (Mn) with connection r, where 0 γC is a vertical vector ¯eld which has components of the form γC = i h . µy Ci ¶ Proof. (a). Substituting (14) and X~ h = 0 in (10), we have h k h k h k h h k h k k h k h (15) @j @iCs +Cs @k¡ji ¡¡ji@kCs ¡@s¡jiCk +¡ki@j Cs +¡jk@iCs ¡Cs Rkji +RsjiCk = 0; and h k h k h h k h k k h (16) @j @iD + D @k¡ji ¡ ¡ji@kD + ¡ki@j D + ¡jk@iD ¡ D Rkji = 0; 158 A. Gezer, K. Akbulut which means that LDr + C(D ­ R) = 0. (b). Substituting (14) and X~ h = 0 in (12), we obtain h k h h k (17) @iCj ¡ ¡jiCk + ¡kiCj = 0: Substituting (14) and X~ h = 0 in (11), we obtain h k h h k (18) @j Ci ¡ ¡jiCk + ¡jkCi = 0; which means that C is parallel in Mn. (c). Interchanging i and j in (18), we have h k h h k @iCj ¡ ¡ij Ck + ¡ikCj = 0; and subtracting the resulting equation from (17), we have k h h k (19) TjiCk = TkiCj ; that is, (20) C(T (Y; Z)) = T (CY; Z) 1 for any Y; Z 2 =0(Mn).
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