On Infinitesimal Conformal Transformations of the Tangent Bundles with the Generalized Metric

E. Peyghani, A. Tayebiii

ABSTRACT Let (,)M g be an n-dimensional Riemannian manifold, and TM be its tangent bundle with the lift metric G . Then every infinitesimal fiber-preserving conformal transformation X induces an infinitesimal homothetic transformation V on M . Furthermore, the correspondence XV→ gives a homomorphism of the Lie algebra of infinitesimal fiber-preserving conformal transformations on TM onto the Lie algebra of infinitesimal homothetic transformations on M , and the kernel of this homomorphism is naturally isomorphic onto the Lie algebra of infinitesimal isometries of M . KEYWORDS Infinitesimal conformal transformation, homothetic transformation, Lagrange metric, isometry

∂v j 1. INTRODUCTION g =⊗∇=−Γgdxij dx, v k v ij i j∂xi ij k In the present paper everything will be always and discussed in the C ∞ category, and Riemannian manifolds will be assumed to be connected and dimM > 1. i ∂ Vv= i . Let M be an n-dimensional Riemannian manifold ∂x with a metric g and φ be a transformation on M . Then Raising j and contracting with i in (2) it is easily seen φ is called a conformal transformation if it preserves the that ∇ vvnij+∇ =2 Ω angles. Let V be a vector field on M and {}ϕt be the ij local one-parameter group of local transformations on Hence M generated by V . Then V is called an infinitesimal 1 conformal transformation, if each ϕ is a local conformal Ω= div() V t n transformation of M . It is well known that V is an where nM= dim . infinitesimal conformal transformation if and only if there exists a scalar function Ω on M such that Example. Let n ≥ 3 . On the n-dimensional Euclidean £g=Ω2 g (1) n V space, which is simply the manifold R with the where £ denotes the Lie derivation with respect to the ij V Riemannian metric tensor field g =⊗gdxij dx where vector field V , especially V is called an infinitesimal gij is a constant for each 1,≤≤ij n, equation (2) homothetic one when Ω is constant [7]. reduces to In the presence of a chart i , the equation xx= ()1≤≤in (3) ∂v j ∂vi (1) reduce to +=Ω2 g ∂xxij∂ ij ∇+∇=Ωvv2 g (2) ij ji ij from which it can be deduced that Ω must be of the form where k Ω =+bxk c for some constants bck , ∈ R and

i Corresponding Author, E. Peyghan is with the Faculty of Science, Department of Mathematics, Arak University, Arak, Iran (e-mail: [email protected]). ii A. Tayebi is with the Faculty of Science, Department of Mathematics, Qom University, Qom, Iran (e-mail: [email protected]).

65 Amirkabir / MISC / Vol . 41 / No.2 / Fall 2009 1 ijk, , ,... will run from 1 to ndimM= . vAxH=+kkr + xxagagag() + − ii ki krirkiikr iik 2 The functions Nxyjjk(),:=Γ () xy are the local where Aiki, HR∈ are constants and coefficients of a nonlinear connection, that is the local vector fields HH+=2 cg. We notice that this conformal vector ki ik ki k δii= ∂−Nxy i(), , ∂ field is homothetic if and only if the vector aa= ()i k 1≤in≤ where ∂ = ∂ span distribution on TM called vanishes. k ∂yk Let TM be the tangent space of M , and let Φ be a horizontal which is supplementary to the vertical transformation of TM . Then Φ is called a fiber- distribution u→= Vuu TM ker()τ ∗ where uTM∈ . preserving transformation, if it preserves the fibers. Let Let us denote by uHTM→ u the horizontal distribution X be a vector field on TM , and let us consider the and let {}δi ,∂i be the basis adapted to the local one-parameter group {}Φt of local transformations decomposition of TM generated by X . Then X is called an TTM= H TM⊕ VTM infinitesimal fiber-preserving transformation on TM , if uuu where uTM∈ . The basis dual of it is {}dxii,δ y with each Φt is a local fiber-preserving transformation of iii k TM . Clearly an infinitesimal fiber-preserving δ ydyNxydx=+k () , . transformation on TM induces an infinitesimal transformation in the base space M [4]. Let g be a We can easily prove the following lemma:

(pseudo)-Riemannian metric of TM . An infinitesimal Lemma 1. The Lie brackets satisfy the following: fiber-preserving transformation X on TM is said to be rm an infinitesimal fiber-preserving conformal []δδi, j=∂,yK jir m transformation, if there exists a scalar function ρ on m []δijim,∂=Γ∂,j TM such that £g= 2ρ g, where £ denotes the Lie X X []0∂,∂ = , derivation with respect to X [7]. ij The purpose of the present paper is to prove the m where K jir denote the components of the curvature following theorem: tensor of M . Theorem . Let (,)M g be an n-dimensional

Riemannian manifold, and TM be its tangent space The metric II+ III on TM is as follows: with the lift metric G . Then every infinitesimal fiber- II+ III=+.2() g x dxijδδδ y g () x y ij y preserving conformal transformation X of TM induces ij ij an infinitesimal homothetic transformation on M . V If in the term of G one replaces gij ()x with the Furthermore, the correspondence XV→ gives a components hxy(), of a generalized Lagrange homomorphism of the Lie algebra of infinitesimal fiber- ij preserving conformal transformations on TM onto the metric([5]) one gets a metric Lie algebra of infinitesimal homothetic transformations ij ij Gxy()2(),= hij xydxy ,δ + h ij () xy ,δδ y y. on M , and the kernel of this homomorphism is naturally isomorphic onto the Lie algebra of infinitesimal isometries of M . In particular, hxyij (), could be a deformation of

gij ()x , a case studied by M. Anastasiei in [3]. 2. GENERALIZED METRIC G

Let ()M , g be a (pseudo)-Riemannian manifold and In this paper, we are concerning with the metric G in

∇ be its Levi-Civita connection. In a local chart the case when hxyij (), is the following special (())Ux, i we set gg=∂,∂(), where ∂ := ∂ and ij i j i ∂xi deformation of gij ()x i 2 we denote by Γ jk the Christoffel symbols of ∇ . Let hxyij()()(), =, aLgx ij ()()xii,≡,yxy be the local coordinates on the manifold TM projected on M by π . The indices

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2 ij j 2 ij ij where Lgxyyygxy=,=ij() i ij () and £GX=+ £ X(( aL ))(2 gdxy ijδδδ g ij y y ) 2 2 ij aImL:⊆⎯→() R++ R with a > 0 . + aL() £Xij (2 gdxδ y + gyyδδij)(4) 3. INFINITESIMAL CONFORMAL TRANSFORMATION ij From lemma 2 we conclude the following result: Let X be an infinitesimal fiber-preserving ij ij hh transformation on TM and (,vv ) be the components £gdxygyyXij(2δδδ+= ij ) hh bc m b m m i j of X with respect to the adapted frame {}dx,δ y . −−Γ−2{gim y v K jcb v bjδ j ()} v dx dx Then X is fiber-preserving if and only if vh depend mm +−∇+∂2{£gVijimj g v g imj ( v ) h only on variables ()x . Clearly X induces an bc m b m m i j −+Γ+gmjyvK icb g mj v bi g mjδδ i ()} v dx y infinitesimal transformation V with the components vh mij in the base space M [4]. We have the following lemma: +∂2()gvyymi j δδ (5)

Lemma 2. The Lie derivative of the adapted frame and 2 2 the dual basis are given as follows: Since δh ()0L = and ∂=h ()2L yh , we have: a (1) £vXhδδ=−∂ h a 22 £aLX ( ( ))= XaL ( ( )) bc a b a a +−Γ−{()},yvKhcb v bhδ h v δ a hh22 =+∂vaLvδh (( )) (( aL )) ba a h (2)£vX ∂=hh { Γhb −δ ( v )}δ a , h 2 = 2()vyaLh ′ (6) hhm (3) £dxXm=∂ vdx , hbchbhhm (4)£yXmcbbmmδδ=− { yvKv − Γ − ( vdx )} By taking (5) and (6) in (4), we have the proof. −Γ−{()}vvybhδδ h m . mb m Let X be an infinitesimal fiber-preserving conformal Proof. Proof of this lemma, is similar to proof of the transformation on TM with respect to metric G , that is, Proposition 2.2 of Yamauchi [7]. there exists a scalar function ρ on TM such that

£GX = 2ρ G Lemma 3. The Lie derivative is in the following £GX Then from lemma 3, we have form: gyvKv{()}bc m−Γ− b mδ v m 2 bc m b m im jcb bj j £GXimjcbbj=−2( aLg ) { yvK − v Γ gyvKv{()}0(7)bc m b m v m mij +−Γ−=,jm icb biδ i −δ j ()}vdxdx

2 ++2(aL ){2ϕ gij £ V g ij mm 2()ϕ g£ggij+−∇+∂ V ij im j v g im j v mm −∇gvgim j +∂ im j () v bc m b m m −−Γ−=,gyvKvmj{()}2(8) icb biδρ i v g ij bc m b m −+ΓgyvKmj icb gv mj bi + gvdxyδδ()}mij mm mj i 2()()2(9)ϕρggij+∂ miji v +∂ g mj v = g ij . 2 mij ++∂2(aL )(ϕ gij g mi j ( v ))δδ y y Let Ω =−ρ ϕ . From (8) and (9) we conclude ′ 2 following relations: where ϕ = vyh aL() . h aL()2 mm £gVijimj−∇ g v +∂ g imj () v

Proof. From the definition of Lie derivative we have: bc m b m m −−Γ−=Ω,gyvKvmj{ icb biδ i ( v )} 2 g ij (10)

mm gvgvmi∂+∂=Ω.ji() mj ()2 g ij (11)

67 Amirkabir / MISC / Vol . 41 / No.2 / Fall 2009 Proposition 4. The vector field V with the m which implies that gvmi∂ r ∂=j ()0. This shows h components ()v is an infinitesimal conformal m h that ∂ j ()v depends only on the variables ()x . Hence transformation on M . h hrhh h v can be written as vyAB= r + , where Ar and Proof. By replace i and j in (10) and addition new h B are certain function on M . The coordinate relation with (10), we get h h transformation rule implies Ar and B are the 2()()£g−∇ g vmmm +∂ g v −∇+ g v g ∂ v m Vijimj imji jmi jm components of a certain (1, 1) tensor field A and a −−Γ−−−ΓgyvKv{()}{bc m b mδ v m gyvKv bc m b m mi icb bi i mi jcb bj certain contravariant vector field B . m −=Ω,δ jij()}4vg Proposition 6. The vector field B = ()Bh is an By attention to (7), (11) and equation infinitesimal isometry on M . mm £gVij=∇ g imj v +∇ g jmi v, we have Proof. Substituting equation (12) into the equation £gVij=Ω2 g ij. This shows the scalar function Ω on (10) and (11), then by Proposition 4, we can get h TM depends only on the variables ()x with respect to AvBij− ∇+∇=, j i j i 0 (13) hh hhr (14) the induced coordinates ()x , y , thus we can regard Ω ∇ jiAKv+=, rij 0 as a function on M , and V is an infinitesimal conformal Aij+ Ag ji=Ω2 ij . (15) transformation on M . From equation (13), (15) and Proposition 4, we have In the following we write Ω instead of Ω .

£gBij= ∇+∇= ji B ij B 0. h Proposition 5. The vertical components ()v of X Thus the vector field B is an infinitesimal isometry on can be written as the following form: M . hrhh vyAB=+,r (12) Proposition 7. The scalar function Ω on M is a where Ah and Bh are the components of a certain r constant function. (1, 1) tensor field A and a certain contravariant vector field B on M , respectively. Proof. From equation (11), (12), we have

2(∇kijkijjiΩ=∇+gAA ) ( )

Proof. By derivation of (11) respect ∂r , we get aa = −−Kvakji Kv akij =.0 mm gvgvgmi∂∂ rji() + mj ∂∂ r ()2 = ij ∂ r () Ω. Thus the scalar function Ω on M is constant. Since the scalar function Ω on TM depends only on the h By proposition 7, the vector field V on M become variables ()x , thus we have ∂Ω=() 0. Therefore we r infinitesimal homothetic transformation. get Conversely, let Vv= ()h be an infinitesimal mm gvgvmi∂∂ rji() + mj ∂∂ r ()0 = . homothetic transformation on M that is, there exists a

Then we have constant c such that £gVij= 2 cg ij. Then we define the mm m gmi∂∂ rjii()vg =− mj ∂∂ r () v =−∂ ( gv mj ∂ r ()) vector field X on TM as follows hah m . XvXy=+∇ha vXh =−∂ij(()2) −gvmr ∂ + Ω g jr Proposition 8. The vector field X on TM defined mm =∂∂=∂∂gvmrij() j ( gv mr i ()) above is an infinitesimal conformal transformation. m Proof. From lemma 3 we have: =∂j (()2) −gvmi ∂ r + Ω g ri mm =−gvgvmi ∂jj ∂ r() =− mi ∂ r ∂ () ,

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2 bc m a b m 2 ji £GXjmicbabi=−2( aLg ) { yvK − y ∇ v Γ =+4(aL )(ϕδ cgdx )ij y ab ji 2 ji −∇δia()}y v dx dx ++2(aL )(ϕ cg )ijδδ y y 2 m ++−∇2(aL ){2ϕ gij 2 cg ij g jm i v =+2(ϕ cG ) am ji Thus we have £GX = 2Ω G. This shows the vector +∂gyvdxyjmi ()} ∇ a δ 2 bc m a b m field X on TM is an infinitesimal conformal +−∇Γ2(aL ) gmi { yv K jcb y a v bj transformation. −∇δδ()}yvdxyab ji ia 2 am ji Proof of Theorem. Summing up proposition 1 to ++∂∇2(aL ){ϕ gij g jmi ( y a v )}δδ y y proposition 5, it is clear that the correspondence 2 ac mbm =−2(aL ) gjm y { v Kicb−∇ av Γ bi XV→ gives a homomorphism of the Lie algebra of mbmji infinitesimal fiber-preserving conformal transformations −∂∇iav +Γ aia ∇ v} dx dx on TM onto the Lie algebra of infinitesimal homothetic 2 ji ++4(aL )(ϕδ cgdx )ij y transformations on M , and the kernel of this ++∇2(aL2 )(ϕδδ g g vmji ) y y homomorphism is naturally isomorphic onto the Lie ij jm i algebra of infinitesimal isometries of M .

4. REFERENCES [1] M. T. K. Abbassi, Note on the classification theorems of g-natural [6] R. Miron, and M. Anastasiei, Vector bundles and Lagrange spaces metrics on the tangent bundle of a Riemannian manifold (M, g), with application to Relativity. Geometry Balkan Press, Romania, Commnet. Math. Univ. Carolinae, 45 (4) (2004), 591-596. (1981). [2] H. Akbar-Zadeh, Transformations infinitesimals conformes des [7] K. Yamauchi , On infinitesimal conformal transformations of the varietes finsleriennes compactes, Ann. Polon. Math., 36 (1979), tangent bundles over Riemannian manifolds, Ann. Rep. 213-229. Asahikawa. Med. Coll.Vol. 15. 1994. [3] M. Anastasiei, Locally conformal Kaehler structures on tangent [8] K. Yano, The theory of Lie Derivatives and Its Applications, North manifold of a space form, Libertas Math., 19 (1999), 71-76. Holland, (1957). [4] I. Hasegawa and K. Yamauchi, Infinitesimal projective [9] K. Yano and S. Ishihara, Tangent and Cotangent Bundles, Marcel transformations on tangent bundles with lift connection, Scientiae Dekker, New York, (1973). Mathematicae Japonicae 52 (2003), 469-483. [10] K. Yano and S. Kobayashi, Prolongations of tensor fields and [5] R. Miron, and M. Anastasiei,, The Geometry of Lagrange spaces: connection to tangent bundle I, General theory, J. Math. Soc. Theory and applications, Kluwer Acad. Publ, FTPH, no.59,(1994). Japan, 18 (1996), 194 -210.

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