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 66 Amirkabir / MISC / Vol . 41 / No.2 / Fall 2009 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) (3) £dxhhm=∂ vdx , Xm 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.