Homogeneous Geodesics of Left Invariant Randers Metrics on a Three-Dimensional Lie Group

Transcription

1 Int. J. Contemp. Math. Sciences, Vol. 4, 009, no. 18, Homogeneous Geodesics of Left Invariant Randers Metrics on a Three-Dimensional Lie Group Dariush Latifi Department of Mathematics Universit of Mohaghegh Ardabili P.O. Box , Ardabil, Iran latifi@uma.ac.ir Asadollah Razavi Facult of Mathematics and Computer Science Amirkabir Universit of Technolog 44 Hafez Ave., Tehran, Iran arazavi@aut.ac.ir Abstract In this paper we stud homogeneous geodesics in a three-dimensional connected Lie group G equipped with a left invariant Randers metric and investigates the set of all homogeneous geodesics. We show that there is a three-dimensional unimodular Lie group with a left invariant non-berwaldian Randers metric which admits exactl one homogeneous geodesic through the identit element. Mathematics Subject Classification: 53C60; 53C30 Kewords: Randers metrics; Homogeneous geodesics; geodesic vectors 1 Introduction A connected Riemannian manifold M,g is said to be homogeneous if a connected group of isometries G acts transitivel on it. Then M can be viewed as a coset space G with a G invariant metrics, where H is the isotrop subgroup H at a fixed point o of M. A geodesic γt through the origin o = eh is called a homogeneous geodesic if it is an orbit of a one-parameter subgroup of G, i.e γt = exptzo,

2 874 D. Latifi and A. Razavi where Z is a non-zero vector in the Lie algebra g of G. Homogeneous geodesic in a Lie group were studied b V. V. Kajzer in [3] where he proved that a Lie group G with a left-invariant metric has at least one homogeneous geodesic through the identit. A generalization of Kajzer s result was obtained b O. Kowalski and J. Szenthe [4]. In [10] J. Szenthe proved that if a compact connected and semisimple Lie group has rank greater that 1, then for ever left-invariant metric there are infinitel man homogeneous geodesics through the identit element. In [7] R. A. Marinosci investigated the set of all homogeneous geodesics in a 3-dimensional Lie group. In [5] the first author studied homogeneous geodesics in homogeneous Finsler spaces. The definition of homogeneous geodesics in Finsler spaces is similar to that in the Riemannian case. In this paper, we stud homogeneous geodesics in three-dimensional unimodular and non-unimodular Lie groups endowed with a left-invariant Randers metric. Preliminaries In this section, we recall briefl some known facts about Finsler spaces. For details, see [1]. Let M be a n-dimensional C manifold and TM = x M T xm the tangent bundle. If the continuous function F : TM R + satisfies the conditions that it is C on TM\{0}; F tu =tf u for all t 0 and u TM, i.e, F is positivel homogeneous of degree one; and for an tangent vector T x M \{0}, the following bilinear smmetric form g : T x M T x M R is positive definite : g u, v = 1 s t [F x, + su + tv] s=t=0, then we sa that M,F is a Finsler manifold. Let π TM be the pull-back of the tangent bundle TM b π : TM \{0} M. Unlike the Levi-Civita connection in Riemannian geometr, there is no unique natural connection in the Finsler case. Among these connections on π TM, we choose the Chern connection whose coefficients are denoted b Γ i jk see [1]. This connection is almost g compatible and has no torsion. Here gx, =g ij x, dx i dx j = 1F i jdxi dx j is the Riemannian metric on the pulled-back bundle π TM. The Chern connection defines the covariant derivative D V U of a vector field U χm in the direction V T p M. Since, in general, the Chern connection coefficients Γ i jk in natural coordinates have a directional dependence, we must sa explicitl that D V U is defined with a fixed reference vector. In particular, let σ :[0,r] M be a smooth curve with velocit field T = T t = σt. Suppose that U and W are vector fields defined along σ. We define D T U with

3 Homogeneous geodesics of left invariant Randers metrics 875 reference vector W as [ ] du i D T U = dt + U j T k Γ i jk σ,w x i σt. A curve σ :[0,r] M, with velocit T = σ is a Finslerian geodesic if [ ] T D T = 0, with reference vector T. F T We assume that all our geodesics σt have been parameterized to have constant Finslerian speed. That is, the length F T is constant. These geodesics are characterized b the equation D T T = 0, with reference vector T. Since T = dσi dt, this equation sas that x i d σ i dt + dσj dt dσ k dt Γi jk σ,t =0. If U, V and W are vector fields along a curve σ, which has velocit T = σ, we have the derivative rule d dt g W U, V =g W D T U, V +g W U, D T V whenever D T U and D T V are with reference vector W and one of the following conditions holds: i U or V is proportional to W, or ii W=T and σ is a geodesic. 3 Homogeneous geodesics of left invariant Finsler metrics Let G be a connected Lie group with Lie algebra g = T e G. We ma identif the tangent bundle TG with G g b means of the diffeomorphism that sends g, X tol g X T g G. Definition 3.1 A Finsler function F : TG R + will be called G-invariant if F is constant on all G-orbits in TG = G g; that is, F g, X =F e, X for all g G and X g.

4 876 D. Latifi and A. Razavi The G-invariant Finsler functions on TGma be identified with the Minkowski norms on g. IfF : TG R + is an G-invariant Finsler function, then we ma define F : g R + b F X =F e, X, where e denotes the identit in G. Conversel, if we are given a Minkowski norm F : g R +, then F arises from an G-invariant Finsler function F : TG R + given b F g, X = F X for all g, X G g. Definition 3. Let G be a connected Lie group, g = T e G its Lie algebra identified with the tangent space at the identit element, F : g R+ a Minkowski norm and F the left-invariant Finsler metric induced b F on G. A geodesic γ : R + G is said to be homogeneous if there is a Z g such that γt =exptzγ0, t R + holds. A tangent vector X T e G {0} is said to be a geodesic vector if the 1-parameter subgroup t exptx, t R +,is a geodesic of F. The basic formula characterizing geodesic vector in the Finslerian case was derived in [5], Theorem 3.1. In the following theorem we present a new elementar proof of this theorem for left invariant Finsler metrics on Lie groups. Theorem 3.1 Let G be a connected Lie group with Lie algebra g, and let F be a left-invariant Finsler metric on G. Then X g {0} is a geodesic vector if and onl if g X X, [X, Z] = 0 holds for ever Z g. Proof: Following the conventions of [] a left-invariant vector field associated to an element X in T e G is denoted b X : G TG; that is X x = L x X.For an left invariant vector fields X,Ỹ, Z on G, we have Similarl, Ỹg e X Z, X =g e X D ey Z, X+g e X Z,D e Y X with reference X 1 Zg e X Ỹ, X =g e X D ezỹ, X+g e X Ỹ,D ez, X Xg e X Z, X =g e X D e X Z, X+g e X Z,D e X, X 3 All covariant derivatives have X as reference vector. Subtracting from the summation of 1 and 3 we get gx e Z,D X+g X+ e ey X e X Ỹ,D X ez = Ỹg ex Z, X ZgX e Ỹ, X+ XgX e Z, X gx e [Ỹ, Z], X gx e [ X, Z], X,

5 Homogeneous geodesics of left invariant Randers metrics 877 where we have used the smmetr of the connection, i.e., D e Z X D e X Z = [ Z, X]. Set Ỹ = X Z in the above equation, we obtain g e X Z,D e X X = Xg e X Z, X Zg e X X, X g e X [ X, Z], X. 4 Since F is left-invariant, dl x is a linear isometr between the spaces T e G = g and T x G, x G. Therefore for an left-invariant vector field X, Z on G, we have g e X Z, X =g X Z, X i.e., the functions g e X Z, X,g e X X, X are constant. Therefore from 4 the following is obtained g e X Z,D e X X e = g e X [ X, Z], X e = g X [X, Z],X. Consequentl the assertion of the theorem follows. 4 Homogeneous geodesics of left invariant Randers metrics on three-dimensional Lie groups Let M be a smooth n-dimensional manifold, a Randers metric on M consists of a Riemannian metric ã = ã ij dx i dx j on M and a 1-form b = b i dx i, [1], [9]. Here ã and b define a function F on TM b F x, =αx, +βx, x M, T x M where αx, = ã ij i j, βx, =b i x i. F is Finsler structure if b = bi b i < 1 where b i = ã ij b j, and ã ij is the inverse of ã ij. The Riemannian metric ã = ã ij dx i dx j induces the musical bijections between 1-forms and vector fields on M, namel : T x M T x M given b ã x, and its inverse : T x M T x M. In the local coordinates we have b i = ã ij j T x M θ i = ã ij θ j θ T x M Now the corresponding vector field to the 1-form b will be denoted b b, obviousl we have b = b and βx, =b =ã x b,. Thus a Randers metric F with Riemannian metric ã = ã ij dx i dx j and 1-form b can be showed b F x, = ã x, +ã x b, x M, T x M

6 878 D. Latifi and A. Razavi where ã x b,b < 1 x M. Let F x, = ã x, +ã x X, be a left invariant Randers metric. It is eas to check that the underling Riemannian metric ã and the vector field X are also left invariant. Let G be a three-dimensional connected Lie group endowed with a leftinvariant Riemannian metric ã. Case I Let G be an unimodular Lie group. According to a result due to J. Milnor see [8],Theorem 4.3, p.305], [7] there exist an orthonormal basis {e 1,e,e 3 } of the Lie algebra g such that [e 1,e ]=λ 3 e 3, [e,e 3 ]=λ 1 e 1, [e 3,e 1 ]=λ e. Let F be a left invariant Randers metric on G defined b the Riemannian metric ã and the vector field X = x 1 e 1 + x e + x 3 e 3, where 0 < ãx, X < 1, i.e. F p, = ã p, +ã p X,. We note, b using Theorem 0.1 of [6], that G, F is not of the Berwald tpe. We want to describe all geodesic vectors of G, F. For s, t R F + su + tv = ã + su + tv, + su + tv+ã X, + su + tv + ã + su + tv, + su + tvãx, + su + tv B definition g u, v = 1 r s F + ru + sv r=s=0. So b a direct computation we get g u, v = ãu, v+ãx, uãx, v vãx, ãv, ãu, ãx, +ãu, ã, ã, ã, vãu, ãx, uãv, +ãx, ã, + ã,.

7 Homogeneous geodesics of left invariant Randers metrics 879 So for all z g we have g, [, z] = ã X + ã,, [, z] F 5 B using Theorem 3.1 and 5 a vector = 1 e 1 + e + 3 e 3 of g is a geodesic vector if and onl if ã x 1 e 1 + x e + x 3 e 3 + 1e 1 + e + 3 e , [ 1 e 1 + e + 3 e 3,e j ] =0 for each j =1,, 3. So we get : λ λ 3 λ x x 3 λ =0, λ 3 λ 1 x 3 1 λ 3 x 1 3 λ =0, 3 λ 1 λ x 1 λ 1 x 1 λ + 1 = As a special case, if X = x 1 e 1,0<x 1 < 1 and λ 1 = λ = λ 3 0we conclude that all geodesic vectors are those from the set Span{e 1 }. Consequentl, there is onl one homogeneous geodesic. Case II Let G be a non-unimodular Lie group. According to a result due to J. Milnor see [8], Lemma 4.10, p.309], [7] there exists an orthogonal basis {e 1,e,e 3 } of the Lie algebra g such that [e 1,e ]=αe + βe 3, [e,e 3 ]=0, [e 1,e 3 ]=γe + δe 3, where α, β, γ, δ are real numbers such that the matrix α β γ δ has trace α + δ = and αγ + βδ =0. Let F be a left invariant Randers metric on G defined b the Riemannian metric ã and the vector field X = x 1 e 1 + x e + x 3 e 3, ãx, X < 1.

8 880 D. Latifi and A. Razavi B using Theorem 3.1 and 5, a vector = 1 e 1 + e + 3 e 3 of g is a geodesic vector if and onl if ã x 1 e 1 + x e + x 3 e 3 + 1e 1 + e + 3 e , [ 1 e 1 + e + 3 e 3,e j ] =0 for each j =1,, 3. This condition leads to the sstem of equations x α + x 3 γ + x 3 β + x 3 3 δ + α + 3 γ + 3 β + 3δ =0, x 1 α + x 3 1 β + 1 α β =0, x 1 γ + x 3 1 δ + 1 γ δ = Putting X = x 1 e 1 and α =,δ =0,γ = 0 the above equations take the form + β 3 =0, 1 + β 3 =0. So a vector of g is a geodesic vector if and onl if : - Spane 1,e 3 for β =0. - Spane 1 Spane 3 Span β e e 3 for β 0 References [1] D.Bao, S.S. Chern and Shen, An Introduction to Riemann-Finsler geometr, Springer-Verlag, New-York, 000. [] Helgason, Differential Geometr, Lie groups and Smmetric Spacea, Academic Press, New York, [3] V.V. Kajzer, Conjugate points of left-invariant metrics on Lie groups, Soviet Math , [4] O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata, , Erratum: Geom. Dedicata, ,

9 Homogeneous geodesics of left invariant Randers metrics 881 [5] D. Latifi, Homogeneous geodesics in homogeneous Finsler spaces, J. Geom. Phs [6] D. Latifi and A. Razavi, On homogeneous Finsler spaces, Rep. Math. Phs, , Erratum: Rep. Math. Phs, [7] R. A. Marinosci, Homogeneous geodesics in a three-dimensional Lie group, Comm. Math. Univ. Carolinae, 43 00, [8] J. Milnor, curvature of left-invariant metrics on Lie groups, Advances in Math , [9] G. Randers, On an asmmetrical metric in the four-space of general relativit, Phs. Rev , [10] J. Szenthe, homogeneous geodesics of left-invariant metrics, Univ. Iagel. Acta Math , Received: November, 008