Metric on the Cotangent Bundle

Metric on the Cotangent Bundle

Liviu Popescu

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

In this paper is studied the cotangent bundle_Tg^∗_{M =T}^∗_{M\{0}with a 0-} homogeneous liftG^∗ .The connection compatible with the homogeneous metric is determined.

Mathematics Subject classification : 53C15, 53C55, 53C60

Key words: nonlinear connection, adapted basis, homogeneous lift, metrical d- connections.

1 Introduction

Let (T^∗M, π^∗, M) be the cotangent bundle, whereM is aC^∞-differentiable, real n- dimensional manifold. If (U, ϕ) is a local chart on M and (xⁱ) are the coordinates of a point p∈ M, p ∈ ϕ⁻¹(x) ∈ U, then a point u ∈ π^∗−1(U), π^∗(u) = p has the coordinates (xⁱ, pi), (i= 1, n).The natural basis of the module X(T^∗M) is given by (∂i = ∂

∂xⁱ, ∂^r = ∂

∂pr). Given a nonlinear connection N onT^∗M ([1]) there exist a single system of functionsNia(x, p) such thatδk =∂k+Nka(x, p)∂^a, (a= 1, n) and (δk, ∂^a) is a local basis ofX(T^∗M), which is called the adapted basis toN. We have the dual basis (dxⁱ, δpa =dpa−Nka(x, p)dx^k). ForX∈ X(T^∗M) is obtained a unique decompostionX =hX+vX,hX ∈H, vX∈V, (V is the vertical distribution) and forω∈ X^∗(T^∗M) we haveω=hω+vω, where (hω)(X) =ω(hX),(vω)(X) =ω(vX).

In the adapted basis (δ_k, ∂^a) we haveX =Xⁱδ_i+X_a∂^a andω=ω_idxⁱ+ω^aδp_a.The homogeneous lift of the Riemannian and Finslerian metrics on the tangent bundle have been studied by Acad. Radu Miron ([3], [4]), while the properties of homogeneous structures on cotangent bundle were studied by P. Stavre and the author ([5], [6], [7]).

More specific, details on the homogeneous lift of a Cartan metric on cotangent bundle and on integrability conditions of homogeneous almost complex structures are given in [6], the properties of the homogeneous lift of a Riemann metric on cotangent bundle are studied in [7], and the homogeneous almost product structure case is developed in [5].

Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 43-47.∗

c

°Balkan Society of Geometers, Geometry Balkan Press 2003.

2 Existence of metrical d-connections

Let (M, gij(x)) be a Riemannian space and (T^∗M, π^∗, M) its cotangent bundle. We introduceg^rs(x) withgik(x)g^ks(x) =δ^s_i.

We consider _c

Nkr(x, p)^def= psγ_rk^s (x), (1)

whereγ_rk^s (x) are the Christoffel symbols ofg.Evidently {N^ckr (x, p)} are the coeffi- cients of a nonlinear connection on Tg^∗M =T^∗M \ {0} which is 1- homogeneous on the fibres. UsingN^ckr we consider δk =∂k+N^ckr(x, p)∂^r; δpk =dpk−N^cik(x, p)dxⁱ. We have

G=∗ hG^∗ +vG,^∗ G=^∗ gij(x)dxⁱ⊗dx^j+g^rs(x)δpr⊗δps. (2)

If we define the homothetyh^∗t: (x, p)→(x, tp),∀t∈R, then

³_∗ G◦h^∗t

´

(x, p) =gij(x)dxⁱ⊗dx^j+t²g^rs(x)δpr⊗δps6=G^∗ (x, p).

(3)

Proposition 1 G^∗ is globally defined Riemannian metric onTg^∗M and is not homo- geneous on the fibres ofT^∗M.

We consider the function

H(x, p) =g^rs(x)prps. (4)

ObviouslyH is 2-homogeneous on the fibres of cotangent bundleTg^∗M . IfG^∗ is defined by

G=∗ gij(x)dxⁱ⊗dx^j+a²

Hg^rs(x)δpr⊗δps, (5)

wherea >0 is a constant, then we get:

Proposition 2 The following properties hold:

1^◦ The pair (Tg^∗M ,G) is a Riemannian space depending only on the metric^∗ g.

2^◦G^∗ is0-homogeneous on the fibres ofTg^∗M .

3^◦ The distributionN andV are ortogonal with respect toG^∗ G∗ (hX, vY) = 0, ∀X, Y ∈ X(T^∗M).

G=∗ gij(x)dxⁱ⊗dx^j+h^rs(x, p)δpr⊗δps, (6)

where

h^rs(x, p) =a² Hg^rs(x) (7)

From [1] we have:

Definition 1 A linear connection Don T^∗M is called metrical d−linear connection with respect toG^∗ ifDG^∗ = 0andDpreserves by parallelism the horizontal distribution N.

We will prove the existence of metricald−linear connections. In the adapted frame we have:

Dδkδj =

h

F_jkⁱ δi+Fej(r)k∂^r, Dδk∂^r=−gg F_k^i(r)δi−

v

F_(j)k^(r) ∂^j, D∂^kδj=

h

C_j^i(k)δi+Ce_j(r)^(k)∂^r, D∂^k∂^r=−C^(r)i(k)gg δi−

v

C_(j)^(r)(k)∂^j, (8)

whereF^h_jkⁱ ,Fe_j(r)k, gg F_k^i(r),

v

F_(j)k^(r),

h

C_j^i(k),Ce_j(r)^(k),C^(r)i(k)gg ,

v

C_(j)^(r))k)) are the coefficients ofD.

Theorem 1 There exists a metricald−linear connectionD onTg^∗M with respect to G, which depends only on the metric tensor∗ g; its components are















Fe_j(r)k= gg

F_k^i(r)=Ce_j(r)^(k) =C^(r)i(k)gg =

h

C_j^i(k)= 0,

h

F_jkⁱ =

v

F_(j)k⁽ⁱ⁾ =γ_jkⁱ (x),

v

C_(j)^(r)(k)= 1

H(δ_j^kp^r+δ_j^rp^k−g^rkpj), (9)

whereg^rmpm=p^r.

Proof. In the general case of a vector bundle we have a canonical metrical connection given by [2],























h

F_jkⁱ = 1

2g^is(δjgsi+δkgjs−δsgjk),

v

F_(j)k^(r)=∂^rNjk+1

2h^rshjs||k,

h

C_j^i(k)=1

2gjsg^is||^k =1

2gjs∂^kg^is,

v

C_(j)^(r)(k)=−1

2hjs(∂^rh^ks+∂^kh^rs−∂^sh^rk),

where_,,|´|”and ,,||” are theh−,andv−covariant derivative with respect to the Berwald connection (B^r_jk=∂^rNjk,0).

Butg=g(x),soδjgsi=∂jgsi and∂^kg^is= 0⇒F^h_jkⁱ =γ_jkⁱ (x) and

h

C_j^i(k)= 0.

Fromh^rs(x, p) = a²

Hg^rs(x) it followshrs(x, p) = H

a²grs(x).But

∂^rh^ks(x, p) =∂^r

µ a²

g^rmp_mp_rg^ks(x)

¶

=−2a²

H²g^ks(x)g^rmpm,

v

C_(j)^(r)(k)=−1

2hjs(∂^rh^ks+∂^kh^rs−∂^sh^rk) =

=−1 2

H a²gjs(x)

µ

−2a²

H²g^ksg^rmpm−2a²

H²g^rsg^kmpm+2a²

H²g^rkg^smpm

¶

v

C_(j)^(r)(k)= 1

H(δ^k_jp^r+δ^r_jp^k−g^rkpj), g^rmpm=p^r.

v

F_(j)k^(r)=∂^rN^cjk+1

2h^rshjs||k =∂^rN^cjk+1

2h^rs[δkhjs−∂^m(N^csk)hjm−∂^m(N^cjk)hsm].

SinceNjk=γ_jk^r prand

v

F_(j)k^(r)=γ_jk^r +1 2

a²

Hg^rs(x)[H

a²∂_kg_js+ 1

a²g_js∂_kg^mlp_mp_l+ 2

a²γ_kl^mp_mg_jsg^lmp_m−

−H

a²γ^m_skgjm−H

a²γ_jk^mgsm], we obtain

v

F_(j)k^(r)=1 2γ_jk^r +1

2g^rs∂kgjs+ 1

2H∂kg^mlpmplδ^r_j+ 1

2Hg^msg^lipmpi∂kglsδ^r_j+ + 1

2Hg^smg^lipmpi∂lgksδ^r_j− 1

2Hg^smg^lipmpi∂sgklδ^r_j−

−1

4g^rs∂sgkj−1

4g^rs∂kgsj+1

4g^rs∂jgsk. Butg^ms∂k(gls) =−∂k(g^ms)gls,and consequently

v

F_(j)k^(r)=1 2γ_jk^r +1

4g^rs(∂kgjs+∂jgsk−∂sgkj) + 1

2H∂kg^mlpmplδ_j^r− 1

2H∂kg^mspmpsδ^r_j−

− 1

2Hg^ligkspmpi∂lg^smδ_j^r+ 1

2Hg^lmgkspmpi∂lg^siδ_j^r=1 2γ_jk^r +1

2γ_jk^r =γ_jk^r , which ends the proof.

2

References

[1] R. Miron, Hamilton geometry, Seminarul de mecanic˘a, No. 3, Univ. Timi¸soara, 1987.

[2] R. Miron, M. Anastasiei,The Geometry of Lagrange Spaces. Theory and Applica- tions, Kluwer Academic Publishers, no. 59 (1994).

[3] R. Miron, The homogeneous lift of a Riemannian metric, An. S¸t. Univ. “A. I.

Cuza” Iasi, (to appear).

[4] R. Miron, D. Hrimiuc, H. Shimada, S. Sab˘au, The geometry of Hamilton and Lagrange Spaces, Kluwer Academic Publishers, 2000.

[5] L. Popescu,On the homogeneous liftG in the cotangent bundle (II), Differential Geometry - Dynamical Systems 2 (2000), 1, 32-35.

[6] P. Stavre, L. Popescu, The homogeneous lift to cotangent bundle of a Cartan metric, Proceedings of National Conference of Finsler and Lagrange Geometry and Applications, Univ. of Bacau, Studies and Scientific Research, ser. Mathematics, no. 10, 2000, 273-285.

[7] P. Stavre, L. Popescu,The homogeneous liftGon the cotangent bundle, Novi-Sad Journal of Mathematics 32 (2002), 2, 1-7.

[8] P. Stavre,On the integrability of the structures (T^∗M,(G,F^∗)), Algebras, Groups and Geometries, Hadronic Press, SUA, 16 (1999), 107-114.

University of Craiova

Faculty of Economic Sciences Department of Applied Mathematics

13 ”Al.I. Cuza” Street, 1100, Craiova, Romania email: [email protected]