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∗† 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 infinitesimal affine transformation in the tangent bundle of a Riemann- ian manifold with respect to the horizontal lift of an affine connection, and to apply the results obtained to the study of fibre-preserving infinitesimal affine transformations in this setting.

Keywords: Lift, Tangent bundle, Infinitesimal affine transformation, Fibre-preserving transformation. 2000 AMS Classification: 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 Γji the Christoffel symbols formed with gji. If, in −1 the neighborhood π (U) of the tangent bundle T (Mn) over Mn, U being a neighborhood H of Mn, then g has components given by Γt g + Γtg 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 Γi = y Γji, Γji being the components of the affine connection in Mn. Let g be a pseudo-Riemannian metric. Then the horizontal lift H g of g with respect H H C to ∇ is a pseudo-Riemannian metric in T (Mn). Since g is defined by g = g − γ(∇g),

∗Department of Mathematics, Faculty of Art and Sciences, Ataturk¨ University, 25240, Erzu- rum, Turkey. E-mail: (A. Gezer) [email protected] (K. Akbulut) [email protected] †This paper is supported by TUBITAK grant number 105T551/TBAG-HD-112 156 A. Gezer, K. Akbulut where γ(∇g) is a tensor field of type (0, 2), which has components of the form γ(∇g) = ys∇ g 0 s , we have that H g and C g coincide if and only if ∇g = 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 affine connection ∇ in Mn is 2 j i (1) ds = 2gjiδy dx , ˜ j j ˜j l k ˜h h ˜ where δy = dy +eΓlky dx and Γji = Γij are components of the connection ∇ defined ˜ 1 by ∇X Y = ∇Y X + [X, Y ] , ∀ X, Y ∈ T0 (Mn), [1, p.67 ]. H We shall now define the horizontal lift ∇ of the affine connection ∇ in Mn to T (Mn) by the conditions H V H H ∇V X Y = 0, ∇V X Y = 0, (2) H V V H H H ∇H X Y = (∇X Y ) , ∇H X Y = (∇X Y ) ,

1 H H K for X, Y ∈ =0(Mn). From (2), the horizontal lift ∇ of ∇ has components ΓJI such that H k k H k H k H k H k¯ Γij = Γij , Γi¯j = Γ¯ıj = Γ¯ı¯j = Γ¯ı¯j = 0, (3) H k¯ s k s k H k¯ H k¯ k Γij = y ∂sΓij − y Rsij , Γ¯ıj = Γi¯j = Γij

k with respect to the induced coordinates in T (Mn), where Γij are the components of ∇ in Mn. Let g and ∇ be, respectively, a pseudo-Riemannian metric and an affine connection such that ∇g = 0. Then H ∇H g = 0, where H g is a pseudo-Riemannian metric. The connection H ∇ has nontrivial torsion even for the Riemannian connection ∇ determined by g, unless g is locally flat [1, p.111]. 1 Let there be given an affine connection ∇ and a vector field X ∈ =0(Mn). Then the 1 Lie derivative LX ∇ with respect to X is, by definition, an element of =2(Mn) such that

(4) (LX ∇)(Y, Z) = LX (∇Y Z) − ∇Y (LX Z) − ∇[X,Y ]Z

1 for any Y, Z ∈ =0(Mn).

In a manifold Mn with affine connection ∇, an infinitesimal affine transformation 0 h h h 1 n 1 x = x + X (x , . . . , x )∆t defined by a vector field X ∈ =0(Mn) is called an infinitesimal affine transformation if LX ∇ = 0, [1, p.67 ]. The main purpose of the present paper is to study the infinitesimal affine transforma- H tion in T (Mn) with affine connection ∇.

2. Vertical inﬁnitesimal aﬃne transformations in a tangent bundle with H ∇

α From (4) we see that, in terms of the components Γγβ of ∇, X is an inﬁnitesimal aﬃne transformation in the m-dimensional manifold Mn if and only if, α λ α λ α α λ α λ (5) ∂γ ∂β X + X ∂λΓγβ − Γγβ ∂λX + Γλβ ∂γ X + Γγλ∂β X = 0, α, β, . . . = 1, . . . , m.

Let there be given in Mn with metric g an affine connection ∇ with Christoffel symbols k ˜ ˜ i ˜ ¯ı ∂ ∂ ∂ Γij . Let X = X ∂i + X ∂¯ı, where ∂i = ∂xi , ∂¯ı = ∂yi = ∂x¯ı , ¯ı = n + 1, . . . , 2n be a vector field in T (Mn). Then, taking account of (3), we can easily see from (5) that X˜ Infinitesimal Affine Transformations 157

H is an inﬁnitesimal aﬃne transformations in T (Mn) with ∇ if and only if the following conditions (6)–(13) hold: ˜ h ˜ k h k ˜ h k ˜ h h ˜ k h ˜ k ∂j ∂iX + X ∂kΓji − (Γji∂kX + ∂Γji∂k¯X ) + Γki∂j X + Γjk∂iX (6) s k ˜ h + y Rsji∂k¯X = 0, ˜ h k ˜ h h ˜ k (7) ∂j ∂¯ıX − Γji∂k¯X + Γjk∂¯ıX = 0, ˜ h k ˜ h h ˜ k (8) ∂¯j ∂iX − Γji∂k¯X + Γki∂¯j X = 0, ˜ h (9) ∂¯j ∂¯ıX = 0, ˜ h¯ ˜ k h ˜ k¯ h k ˜ h¯ k ˜ h¯ h ˜ k ∂j ∂iX + (X ∂k∂Γji + X ∂kΓji) − (Γji∂kX + ∂Γji∂k¯X ) + (∂Γki∂j X h k¯ h k h k¯ k¯ h s k h (10) + Γki∂j X˜ ) + (∂Γjk∂iX˜ + Γjk∂iX˜ ) − X˜ Rkji − y X˜ ∂kRsji s k ˜ h¯ s h ˜ k s h ˜ k + y Rsji∂k¯X − y Rski∂j X − y Rsjk∂iX = 0, ˜ h¯ ˜ k h k ˜ h¯ h ˜ k h ˜ k h ˜ k¯ ∂j ∂¯ıX + X ∂kΓji − Γji∂k¯X + Γki∂j X + (∂Γjk∂¯ıX + Γjk∂¯ıX ) (11) s h k − y Rsjk∂¯ıX˜ = 0, ˜ h¯ ˜ k h k ˜ h¯ h ˜ k h ˜ k¯ h ˜ k ∂¯j ∂iX + X ∂kΓji − Γji∂k¯X + (∂Γki∂¯j X + Γki∂¯j X ) + Γjk∂iX (12) s h ˜ k − y Rski∂¯j X = 0, ˜ h¯ h ˜ k h ˜ k (13) ∂¯j ∂¯ıX − Γki∂¯j X + Γjk∂¯ıX = 0.

Let X˜ be a vertical inﬁnitesimal aﬃne transformation in T (Mn). Then X˜ has components

0 h¯ ¯ with respect to the induced coordinates. Thus, from (13), we have ∂¯∂¯ıX˜ = 0, µX˜ h¶ j i.e., h¯ 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 field in T (Mn), h i h 1 1 C = Ci ∂h⊗dx and D = D ∂h are defined elements of =1(Mn) and =0(Mn), respectively. H 2.1. Theorem. If X˜ is a vertical infinitesimal affine transformation of T (Mn) with ∇, then ∇ ⊗ h ∂ ∈ =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 ∇, i.e., ∇C = 0. 1 (c) C(T (Y, Z)) = T (CY, Z) = T (Y, CZ), for any Y, Z ∈ =0(Mn), where T denotes the torsion tensor of ∇, i.e. T is a pure tensor with respect to C. 1 (d) C(∇Z T )(Y, W ) = (∇CZ T )(Y, W ), for any Y, Z, W ∈ =0(Mn). (e) Conversely, if C and D satisfy the conditions (a), (b), (c) and (d) then the vector field ∂ X˜ = (Chyi + Dh) = γC +v D i ∂yh H is an infinitesimal affine transformation of T (Mn) with connection ∇, where 0 γC is a vertical vector field 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 ∂kΓji −Γji∂kCs −∂sΓjiCk +Γki∂j Cs +Γjk∂iCs −Cs Rkji +RsjiCk = 0, and h k h k h h k h k k h (16) ∂j ∂iD + D ∂kΓji − Γji∂kD + Γki∂j D + Γjk∂iD − D Rkji = 0, 158 A. Gezer, K. Akbulut

which means that LD∇ + C(D ⊗ R) = 0. (b). Substituting (14) and X˜ h = 0 in (12), we obtain h k h h k (17) ∂iCj − ΓjiCk + ΓkiCj = 0. Substituting (14) and X˜ h = 0 in (11), we obtain h k h h k (18) ∂j Ci − ΓjiCk + ΓjkCi = 0, which means that C is parallel in Mn. (c). Interchanging i and j in (18), we have h k h h k ∂iCj − Γij Ck + ΓikCj = 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 ∈ =0(Mn). From (19), we obtain T (Y, CZ)) = −T (CZ, X) = C(T (Z, Y )) = C(T (Y, Z)) and hence C(T (Y, Z)) = T (CY, Z) = T (Y, CZ), which is the formula (c). h (d). Using (17) and (18), we eliminate all partial derivatives of Cj from (15). Then h k h k we obtain Ck ∇j Tli = ∇kTli Cj , i.e. T is a φ-tensor with respect to C [3]. (e). If we assume that the conditions (a), (b), (c) and (d) are established, then we see that X˜, given in (e), is an inﬁnitesimal aﬃne transformation. Consequently, the proof is complete. ¤

2.2. Theorem. Let C be as in Theorem 2.1. If X is an infinitesimal affine transformation of Mn with affine connection ∇, and R(X, Y, Z; ξ) is pure with respect to X and ξ, then so is CX.

3. Fibre-preserving inﬁnitesimal aﬃne transformation with H ∇

A transformation of T (Mn) is said to be fibre-preserving if it sends each fibre of T (Mn) into a fibre. An infinitesimal transformation of T (Mn) is said to be fibre-preserving if it generates a local 1-parameter group of fibre-preserving transformations. An in- X˜ h finitesimal transformation X˜ with components ¯ is fibre-preserving if and only if µX˜ h¶ X˜ h (h = 1, 2, . . . , n) depend only on the variables x1, . . . , xn with respect to the induced h h coordinates (x , y ) in T (Mn). From

0 xh = xh + X˜ h(x1, . . . , xn)∆t (21)  ¯0 ¯ ¯ xh = xh + X˜ h(x1, . . . , xn, xn+1, . . . , x2n)∆t

 X˜ h we see that a fibre-preserving infinitesimal transformation X˜ with components ¯ µX˜ h¶ h induces an infinitesimal transformation X with components X˜ in the base space Mn. k ˜ h s k ˜ h Since ∂Γji∂k¯X = 0 and y Rsji∂k¯X = 0, then from (6) we have: Infinitesimal Affine Transformations 159

3.1. Theorem. If X˜ is a fibre-preserving infinitesimal transformation of T (Mn) with H horizontal lift ∇ of a affine connection ∇ in Mn to T (Mn), then the infinitesimal transformation X induced on Mn by X˜ is also affine with respect to ∇.

3.2. Theorem. Let ∇ be an aﬃne connection in Mn. Then, H C C C C C (LC X ∇)( Y, Z) = (LX ∇)( Y, Z) + γ(LX R)( , Y, Z) 1 for any X ∈ =0(Mn). Proof. Our proposition follows from the following computations: H C C H C H C H C (LC X ∇)( Y, Z) = LC X ( ∇C Y Z) − ∇C Y (LC X Z) − ∇[C X,C Y ]Z C H C H C = LC X [ (∇Y Z) − γ(R( , Y )Z)] − ∇C Y [X, Z] − ∇C [X,Y ]Z C C C C = [ X, ∇X Y ] − [ X, γ(R( , Y )Z)] − (∇Y [X, Z]) C + γ(R( , Y ) [X, Z]) − (∇[X,Y ]Z) + γR([ X, Y ]Z)] C C C = (LX ∇X Y ) − (∇Y (LX Z)) − (∇[X,Y ]Z)

− γ(LX R( , Y )Z) + γ(R( , Y ) [X, Z]) + γ(R( , [X, Y ])Z) C C C = (LX ∇)( Y, Z) + γ(−LX R( , Y )Z + R( , Y ) [X, Z] + R( , [X, Y ])Z) C C C = (LX ∇)( Y, Z) + γ(LX R)( , Y, Z), where R( , X)Y denotes a tensor field W of type (1, 1) in Mn such that W (Z) = 1 R(Z, X)Y for any Z ∈ =0(Mn). ¤ Let X˜ and X be as in Theorem 3.1. From Theorem 3.2 we see that, if X is an infinitesimal automorphism with respect to W [3], then cX is an infinitesimal affine h H c X transformation of T (Mn) with ∇. Since X has the components , it follows µ∂Xh¶ c H that X˜ − X is a vertical infinitesimal affine transformation in T (Mn) with ∇. Thus we have

3.3. Theorem. If X˜ is a fibre-preserving infinitesimal affine transformation of T (Mn) with lift H ∇, and X is an infinitesimal automorphism with respect to W , then X˜ =c X +v D + γC, where D and C are tensor fields of type (1, 0) and (1, 1), respectively, satisfying conditions (a), (b) and (c) of Theorem 2.1

Acknowledgement. The authors are grateful to Professor A. A. Salimov for his valuable suggestions. References [1] Yano K. and Ishihara, S. Tangent and cotangent bundles (Marcel Dekker, New York, 1973). [2] Magden A. and Salimov, A. A. Horizontal lifts of tensor ﬁelds to sections of the tangent bundle. (in Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 3, 77–80, 2001. [3] Kobayashi S. and Nomizu K. Foundations of Diﬀerential Geometry (Wiley-Inter Science Publications, New York, 1963).