G-NATURAL METRICS on the COTANGENT BUNDLE

International Electronic Journal of Geometry Volume 6 No.1 pp. 129-146 (2013) °c IEJG

g-NATURAL METRICS ON THE COTANGENT BUNDLE

FIL˙ IZ˙ AGCA˘

(Communicated by Arif SALIMOV)

Abstract. The main aim of this paper is to investigate curvature properties and geodesics of the g-natural metric on the cotangent bundle of Riemannian manifold.

1. Introduction Let (M n, g) be an n-dimensional Riemannian manifold, T ∗M n its cotangent bundle and π the natural projection T ∗M n → M n. A system of local coordinates (U, xi), i = 1, ..., n on M n induces on T ∗M n a system of local coordinates −1 i ¯i ¯i (π (U), x , x = pi), ¯i := n + i (i = 1, ..., n), where x = pi are the components ∗ n of covectors p in each cotangent space Tx M , x ∈ U with respect to the natural coframe {dxi}, i = 1, ..., n. r n r ∗ n n ∗ n ∞ We denote by =s(M )(=s(T M )) the module over F (M )(F (T M )) of C tensor ﬁelds of type (r, s), where F (M n)(F (T ∗M n)) is the ring of real-valued C∞ functions on M n(T ∗M n). i ∂ i n Let X = X ∂xi and ω = ωidx be the local expressions in U ⊂ M of a vector 1 n 0 n and a covector (1-form) ﬁeld X ∈ =0(M ) and ω ∈ =1(M ), respectively. Then C H 1 ∗ n 1 n the complete and horizontal lifts X, X ∈ =0(T M ) of X ∈ =0(M ) and the V 1 ∗ n 0 n vertical lift ω ∈ =0(T M ) of ω ∈ =1(M ) are given, respectively, by

X C i ∂ h ∂ (1.1) X = X − ph∂iX , ∂xi ∂x¯i i X H i ∂ h j ∂ (1.2) X = X + phΓ X , ∂xi ij ∂x¯i i X V ∂ (1.3) ω = ωi , ∂x¯i i

Date: December 11, 2012 and Accepted: March 3, 2013. 2000 Mathematics Subject Classification. 53C07, 53C22, 53C25. Key words and phrases. g-natural metric, cotangent bundle, vertical and horizontal lift, curvature tensor, geodesics. 129 130 FIL˙ IZ˙ AGCA˘ with respect to the natural frame { ∂ , ∂ }, where Γh are the components of the ∂xi ∂x¯i ij n Levi-Civita connection ∇g on M (see [19] for more details). Theorem 1.1. Let M n be a Riemannian manifold with metric g, ∇ be the Levi- Civita connection and R be the Riemannian curvature tensor. Then the Lie bracket of the cotangent bundle T ∗M n of M n satisfies the following i)[V ω, V θ] = 0, H V V (1.4) ii)[ X, ω] = (∇X ω), iii)[H X, H Y ] = H [X,Y ] + γR(X,Y ) = H [X,Y ] + V (pR(X,Y )) 1 n 0 n for all X,Y ∈ =0(M ) and ω, θ ∈ =1(M ). (See [19, p.238, p.277] for more details) Definition 1.1. Let M n be a Riemannian manifold with metric g. A Riemannian metricg ¯ on cotangent bundle T ∗M n is said to be natural with respect to g on M n if i)g ¯(H X, H Y ) = g(X,Y ), ii)g ¯(H X, V ω) = 0(1.5) 1 n 0 n for all X,Y ∈ =0(M ) and ω, θ ∈ =1(M ). Theorem 1.2. Let M n be a Riemannian manifold with metric g and T ∗M n be the cotangent bundle of M n. If the Riemannian metric g¯ on T ∗M n is natural with respect to g on M n then the corresponding Levi-Civita connection ∇¯ satisfies ¯ H H i)g ¯(∇H X Y, Z) = g(∇X Y,Z), H V 1 V V ii)g ¯(∇¯ H Y, ω) = g¯( ω, (pR(X,Y ))), X 2 V H 1 V V iii)g ¯(∇¯ H ω, Z) = g¯( (pR(Z,X)), ω, ), X 2 V V 1¡H V V V V iv)g ¯(∇¯ H ω, θ) = X(¯g( ω, θ)) − g¯( ω, (∇ θ)) X 2 X V V ¢ +¯g( θ, (∇X ω)) ,

H H 1 V V (1.6) v)g ¯(∇¯ V Y, Z) = − g¯( ω, (pR(Y,Z))), ω 2 H V 1¡H V V V V vi)g ¯(∇¯ V Y, θ) = Y (¯g( ω, θ)) − g¯( ω, (∇ θ)) ω 2 Y V V ¢ −g¯( θ, (∇Y ω)) ,

V H 1¡ H V V V V vii)g ¯(∇¯ V θ, Z) = − Z(¯g( ω, θ)) +g ¯( ω, (∇ θ)) ω 2 Z V V +¯g( θ, (∇Z ω))),

V V 1 V V V V V V V V V viii)g ¯(∇¯ V θ, ξ) = ( ω(¯g( θ, ξ) + θ(¯g( ξ, ω) − ξ(¯g( ω, θ)) ω 2 1 n 0 n for all X,Y ∈ =0(M ) and ω, θ ∈ =1(M ) [4]. Corollary 1.1. Let M n be a Riemannian manifold with metric g and g¯ be a natural metric on the cotangent bundle T ∗M n of M n. Then the Levi-Civita connection ∇¯ satisﬁes

H H 1 V (1.7) ∇¯ H Y = (∇ Y ) + (pR(X,Y )) X X 2 g-NATURAL METRICS ON THE COTANGENT BUNDLE 131

1 n for all X,Y ∈ =0(M ) [4]. For each x ∈ M n the scalar product g−1 = (gij) is defined on the cotangent −1 ∗ n space π (x) = Tx M by −1 ij g (ω, θ) = g ωiθj 0 n for all ω, θ ∈ =1(M ). Definition 1.2. A g-natural metricg ˜ is defined on T ∗M n by the following three equations ¡ ¢ ¡ ¢ (1.8) g˜ H X, H Y = V g(X,Y ) = g(X,Y ) ◦ π, ¡ ¢ (1.9) g˜ V ω, H Y = 0, ¡ ¢ g˜ V ω, V θ = ϕ(z)g−1(ω, θ) + ψ(z)g−1(ω, p)g−1(θ, p)(1.10) 1 n 0 n for any X,Y ∈ =0(M ) and ω, θ ∈ =1(M ). Here ϕ and ψ are some functions of 1 1 −1 argument z = 2 |p| = 2 g (p, p) such that ϕ > 0 and ϕ + 2zψ > 0. Since any tensor field of type (0,2) on T ∗M n is completely determined by its action on vector fields of type H X and V ω, it follows thatg ˜ is completely determined by its equations (1.8), (1.9) and (1.10). The Sasaki metric is obtained for ϕ(z) = 1 and ψ(z) = 0, while the Cheeger- 1 2 −1 Gromoll metric for ϕ(z) = ψ(z) = 1+r2 , r = g (p, p). Sasaki, Cheeger-Gromoll and g-natural metrics are in the class of natural metric. C 1 n We now see, from (1.1) and (1.2), that the complete lift X of X ∈ =0(M ) is expressed by

¡ ¢ (1.11) C X = H X − V p(∇X) ,

i h where p(∇X) = pi(∇hX )dx . Using (1.8), (1.9), (1.10) and (1.11), we have ¡ ¢ ¡ ¢ ¡ ¢ g˜ C X, C Y = V g(X,Y ) + ϕ(z) g−1(p(∇X), p(∇Y )) (1.12) + ψ(z)g−1(p(∇X), p)g−1(p(∇Y ), p), −1 ij l k −1 ij where g (p(∇X), p(∇Y )) = g (pl∇iX )(pk∇jY ), g (p(∇X), p) = g pi(p(∇X))j. 0 ∗ n Since the tensor fieldg ˜ ∈ =2(T M ) is completely determined also by its action on vector fields type C X and C Y (see[19, p.237]), we have an alternative charac- terization ofg ˜ on T ∗M n:g ˜ is completely determined by the condition (1.12). The main purpose of this paper is to introduce Levi-Civita connection of g- natural type metric on the cotangent bundle T ∗M n of Riemannian manifold M n and investigate curvature properties and geodesics on T ∗M n with respect to the Levi-Civita connection ofg ˜. Since the construction of lifts to the cotangent bundle is not similar to the definition of lifts to the tangent bundle, we have some differences for g-natural metrics on cotangent bundles. g-natural metric includes the Sasaki metric ([7], [12], [13]) and the Cheeger-Gromoll metric (see also [2], [4], [5], [6], [9], [11], [14], [15], [16], [18]) as a special cases. In [1]-[3] Abbasi and Sarih characterized the g-natural metric on the tangent bundle. In [17] Sukhova studied a class of Riemannian almost product metrics on the tangent bundle of a smooth manifold and investigated the scalar curvature of the tangent bundle. In [10] Munteanu computed the Levi-Civita connection, the curvature tensor, the 132 FIL˙ IZ˙ AGCA˘ sectional curvature and the scalar curvature of the g-natural metric on the tangent bundle.

2. Levi-Civita connection of g˜ We put ∂ X = , θ(i) = dxi, i = 1, ..., n. (i) ∂xi

H V (i) Then from (1.2) and (1.3) we see that X(i) and θ have respectively local expressions of the form X H ∂ a ∂ (2.1) e˜ = X = + paΓ , (i) (i) ∂xi hi ∂xh¯ h

V (i) ∂ (2.2) e˜ ¯ = θ = . (i) ∂x¯i

H V (i) We call the set {e˜(α)} = {e˜(i), e˜(¯i)} = { X(i), θ } the frame adapted to Levi- Civita connection ∇g. The indices α, β, ... = 1, ..., 2n indicate the indices with respect to the adapted frame. We now, from the equations (1.2), (1.3), (2.1) and (2.2) see that H X and V ω have respectively components µ ¶ Xi (2.3) H X = Xie˜ , H X = (H Xα) = , (i) 0

X µ ¶ V V V α 0 (2.4) ω = ωie˜(¯i), ω = ( ω ) = ωi i

i with respect to the adapted frame {e˜(α)}, where X and ωi being local components 1 n 0 n of X ∈ =0(M ) and ω ∈ =1(M ), respectively. From (1.8), (1.9) and (1.10) we see that

¡ ¢ V ¡ ¢ g˜ij =g ˜ e˜(i), e˜(j) = g(∂i, ∂j) = gij, ¡ ¢ g˜¯ij =g ˜ e˜(¯i), e˜(j) = 0, ¡ ¢ −1 i j −1 i −1 j g˜¯i¯j =g ˜ e˜(¯i), e˜(¯j) = ϕ(z)g (dx , dx ) + ψ(z)g (dx , pk)g (dx , pl) ij ik lj = ϕ(z)g + ψ(z)g g pkpl, i.e.g ˜ has components µ ¶ gij 0 (2.5) g˜ = ij ik lj 0 ϕ(z)g + ψ(z)g g pkpl with respect to the adapted frame {e˜(α)}. For the Levi-Civita connection of the g-natural metric we have the following.

Theorem 2.1. Let M n be a Riemannian manifold with metric g and ∇˜ be the Levi-Civita connection of the cotangent bundle T ∗M n equipped with the g-natural g-NATURAL METRICS ON THE COTANGENT BUNDLE 133 metric g˜. Then ∇˜ satisﬁes

H H 1 V i) ∇˜ H Y = (∇ Y ) + (pR(X,Y ), X X 2 V V ϕ(z) H −1 ii) ∇˜ H ω = (∇ ω) + (p(g ◦ R( ,X)˜ω), X X 2 H ϕ(z) H −1 (2.6) iii) ∇˜ V Y = (p(g ◦ R( ,Y )˜ω), ω 2 0 V ϕ (z) ¡ V V V V ¢ iv) ∇˜ V θ = − g˜( ω, γδ) θ +g ˜( θ, γδ) ω ω 2ϕ(z)(ϕ(z) + 2zψ(z)) 2ψ(z) − ϕ0(z) + g˜(V ω, V θ)γδ 2ϕ(z)(ϕ(z) + 2zψ(z)) ψ0(z)ϕ(z) − ϕ0(z)ψ(z) − 2ψ2(z) + g˜(V ω, γδ)˜g(V θ, γδ)γδ 2ϕ(z)(ϕ(z) + 2zψ(z))3 1 n 0 n −1 1 n for all X,Y ∈ =0(M ), ω, θ ∈ =1(M ), where ω˜ = g ◦ ω ∈ =0(M ), R( ,X)˜ω ∈ 1 n −1 2 n 1 1 −1 =1(M ), g ◦ R( ,X)˜ω ∈ =0(M ), z = 2 |p| = 2 g (p, p), ϕ > 0, ϕ + 2zψ > 0, R and γδ denotes respectively the curvature tensor of ∇ and the canonical vertical ∗ n vector field on T M with expression γδ = pie(¯i). Proof. i) The first statement is just Corollary 1.1. ii) Following Definition 1.1 and Theorem 1.2 we see that ¡ ¢ ˜ V H V V 2˜g(∇H X ω, Y ) =g ˜ (pR(Y,X)), ω = ϕ(z)g−1(pR(Y,X), ω) + ψ(z)g−1(pR(Y,X), p)g−1(ω, p) ¡ ¢ = ϕ(z)˜g H (p(g−1 ◦ R( ,X)˜ω)), H Y and ¡ ¢ ˜ V V H V V V V V V 2˜g(∇H X ω, θ) = X(˜g( ω, θ))−g˜( ω, (∇X θ))+˜g( θ, (∇X ω)) V V V V V V V V =g ˜( ω, (∇X θ))+˜g( θ, (∇X ω))−g˜( ω, (∇X θ))+˜g( θ, (∇X ω)) V V V V = 2˜g( θ, (∇X ω)) = 2˜g( (∇X ω), θ) Using −1 kl g (pR(Y,X), ω) = (g (pR(Y,X))kωl) kl s i j s i j kl = (g psRijk Y X ωl) = (psRijk Y X g ωl) s i j k a s i j k = (psRijk Y X ω˜ ) = (gaipsR.jk Y X ω˜ ) ¡ ¢ = g p(g−1◦R( ,X)˜ω),Y ¡ ¢ =g ˜ H (p(g−1 ◦ R( ,X)˜ω)), H Y ,

−1 ij s a b g (pR(Y,X), p) = (g psRabi Y X pj) ts a b i a b i t = (psg RabitY X p˜ ) = (RabitY X p˜ p˜ ) = (R Y aXbp˜ip˜t) = (g R f Y aXbp˜ip˜t) ¡ itab ¢ fb ita. = g R(˜p, p˜)Y,X = 0, H X(ϕ(z)) = 0, H X(ψ(z)) = 0 and H V V V V V V X(˜g( ω, θ) =g ˜( ω, (∇X θ))+˜g( θ, (∇X ω)) we have 134 FIL˙ IZ˙ AGCA˘

˜ V V ϕ(z) H −1 ∇H X ω = (∇X ω) + 2 (p(g ◦ R( ,X)˜ω) iii) Calculations similar to those in ii) give

¡ ¢ ˜ H V H V V V V V V 2˜g(∇V ω Y, θ) = Y (˜g( ω, θ))−g˜( ω, (∇Y θ))−g˜( θ, (∇Y ω)) V V V V V V V V =g ˜( ω, (∇Y θ))+˜g( θ, (∇Y ω))−g˜( ω, (∇Y θ))−g˜( θ, (∇Y ω)) = 0 and

¡ ¢ ˜ H H V V 2˜g(∇V ω Y, Z) = −g˜ ω, (pR(Y,Z)) = −ϕ(z)g−1(ω, pR(Y,Z)) − ψ(z)g−1(pR(Y,Z), p)g−1(ω, p) ¡ ¢ = ϕ(z)˜g H (p(g−1 ◦ R( ,Y )˜ω)), H Z .

Thus we have

H ϕ(z) H −1 ∇˜ V Y = (p(g ◦ R( ,Y )˜ω). ω 2 iv) Appliying Theorem 1.2 we yield

¡ ¢ ˜ V H H V V V V V V 2˜g(∇V ω θ, Z) = − Z(˜g( ω, θ))+˜g( ω, (∇Z θ))+˜g( θ, (∇Z ω)) V V V V V V V V = −g˜( ω, (∇Z θ))−g˜( θ, (∇Z ω))+˜g( ω, (∇Z θ))+˜g( θ, (∇Z ω)) = 0

Using V ω(ϕ(z)) = ϕ0(z)g−1(ω, p), V ω(ψ(z)) = ψ0(z)g−1(ω, p)

V ω(˜g(V θ, V ξ)) = ϕ0(z)g−1(ω, p)g−1(θ, ξ) +ψ0(z)g−1(ω, p)g−1(θ, p)g−1(ξ, p) ¡ ¢ +ψ(z) g−1(ω, θ)g−1(ξ, p) + g−1(θ, p)g−1(ω, ξ) and

g˜(V ω, γδ) = ϕ(z)g−1(ω, p) + ψ(z)g−1(ω, p)g−1(p, p) ¡ ¢ = g−1(ω, p) ϕ(z) + 2zψ(z) , g-NATURAL METRICS ON THE COTANGENT BUNDLE 135 we have ¢ ˜ V V V V V V V V V V V g˜(∇V ω θ, ξ) = ω(˜g( θ, ξ)) + θ(˜g( ξ, ω)) − ξ(˜g( ω, θ)) = ϕ0(z)g−1(ω, p)g−1(θ, ξ) + ϕ0(z)g−1(θ, p)g−1(ξ, ω) −ϕ0(z)g−1(ω, θ)g−1(ξ, p) + ψ0(z)g−1(θ, p)g−1(ω, p)g−1(ξ, p) +ψ0(z)g−1(θ, p)g−1(ξ, p)g−1(ω, p) − ψ0(z)g−1(θ, p)g−1(ξ, p)g−1(ω, p) +2ψ(z)g−1(ξ, p)g−1(θ, ω) ϕ0(z) ϕ0(z) = g−1(ω, p)˜g(V θ, V ξ) − g−1(ξ, p)˜g(V ω, V θ) ϕ(z) ϕ(z) ϕ0(z) + g−1(θ, p)˜g(V ξ, V ω) − ψ0(z)g−1(θ, p)g−1(ω, p)g−1(ξ, p) ϕ(z) +2ψ(z)g−1(ω, θ)g−1(ξ, p) ¡ ϕ0(z) =g ˜ (˜g(V ω, γδ)V θ +g ˜(V θ, γδ)V ω) ϕ(z)(ϕ(z) + 2zψ(z)) 2ψ(z) − ϕ0(z) + g˜(V ω, V θ)γδ ϕ(z)(ϕ(z) + 2zψ(z)) ψ0(z)ϕ(z) − ϕ0(z)ψ(z) − 2ψ2(z) ¢ + g˜(V ω, γδ)˜g(V θ, γδ)γδ, V ξ . ϕ(z)(ϕ(z) + 2zψ(z))3 Thus 0 V ϕ (z) V V V V ∇˜ V θ = (˜g( ω, γδ) θ +g ˜( θ, γδ) ω) ω ϕ(z)(ϕ(z) + 2zψ(z)) 2ψ(z) − ϕ0(z) + g˜(V ω, V θ)γδ ϕ(z)(ϕ(z) + 2zψ(z)) ψ0(z)ϕ(z) − ϕ0(z)ψ(z) − 2ψ2(z) + g˜(V ω, γδ)˜g(V θ, γδ)γδ. ¤ ϕ(z)(ϕ(z) + 2zψ(z))3 ˜ ˜δ ∗ n We write ∇eα eβ = Γαβeδ with respect to the adapted frame {eα} of T M , ˜δ where Γαβ denote the Christoﬀel symbols constructed byg ˜. From Theorem 2.1, we immediately have Corollary 2.1. Let M n be a Riemannian manifold with metric g and ∇˜ be the Levi-Civita connection of the cotangent bundle T ∗M n equipped with the g-natural ˜δ metric g˜. The particular values of Γαβ for diﬀerent indices, on taking account of (2.6) are then found to be ˜k k ˜k ˜k¯ Γij = Γij, Γ¯i¯j = Γ¯ij = 0, ¯ ¯ 1 Γ˜k = −Γj , Γ˜k = p R a, i¯j ik ij 2 a ijk ϕ(z) ϕ(z) (2.7) Γ˜k = p Rk ia, Γ˜k = p Rk ja, ¯ij 2 a .j. i¯j 2 a .i. 0 0 ¯ ϕ (z) 2ψ(z) − ϕ (z) Γ˜k = (piδj + pjδi ) + gijp ¯i¯j 2ϕ(z) k k 2ϕ(z)(ϕ(z) + 2zψ(z)) k ψ0(z)ϕ(z) − 2ϕ0(z)ψ(z) + pipjp 2ϕ(z)(ϕ(z) + 2zψ(z)) k i j j i ij i j = L(p δk + p δk) + Mg pk + Np p pk. 136 FIL˙ IZ˙ AGCA˘

i it k ia kt is a with respect to the adapted frame, where p = g pt,R.j. = g g Rtjs .

3. Curvature properties of g˜ We now consider local 1-formsω ˜α in π−1(U) deﬁned by

α ¯α B ω˜ = A Bdx , where Ã ! µ ¶ A¯i A¯i i −1 ¯α j ¯j δj 0 (3.1) A = (A B) = ¯¯i ¯¯i = a j A j A ¯j −paΓij δi

The matrix (3.1) is the inverse of the matrix

µ i i ¶ µ i ¶ A Aj A¯j δj 0 (3.2) A = (Aβ ) = ¯i ¯i = a j Aj A¯j paΓij δi

A of the transformatione ˜β = Aβ ∂A (see (2.1) and (2.2)). We easily see that the set α α ¯α B {ω˜ } is the coframe dual to the adapted frame {e˜(β)}, i.e.ω ˜ (˜e(β)) = A BAβ = α δβ . Since the adapted frame {e˜(β)} is non-holonomic, we put

α [˜eγ , e˜β] = Ωγβ e˜α from which we have

α A A ¯α Ωγβ = (˜eγ Aβ − e˜βAγ )A A.

According to (2.1), (2.2), (3.1) and (3.2), the components of non-holonomic object α Ωγβ are given by ( ¯i ¯i j Ωl¯j = −Ω¯jl = Γli, (3.3) ¯i a Ωlj = paRlji ,

a all the others being zero, where Rlji being local components of the curvature tensor R of ∇g. Let R˜ be a curvature tensor of ∇˜ . Then we obtain ˜ ˜ ˜ ˜ ˜ ε ˜ R(˜e(α), e˜(β))˜e(γ) = ∇α∇βe˜(γ) − ∇β∇αe˜(γ) − Ωαβ ∇εe˜(γ),

˜ ˜ ˜ where ∇α = ∇e˜(α) . The curvature tensor R has components

˜ σ ˜σ ˜σ ˜σ ˜ε ˜σ ˜ε ε ˜σ Rαβγ =e ˜αΓβγ − e˜βΓαγ + ΓαεΓβγ − ΓβεΓαγ − Ωαβ Γεγ with respect to the adapted frame {e˜(α)}. g-NATURAL METRICS ON THE COTANGENT BUNDLE 137

Taking account of (2.7) and (3.3), we ﬁnd

ϕ(z) R˜ l = R l − p p R aRl tm kij kij 2 m a kit .j. ϕ(z) + p p (Rl tmR a − Rl tmR a), 4 m a .k. ijt .i. kjt l ϕ(z) l im R˜ ¯ = p ∇ R , kij 2 m k .j. l ϕ(z) l jm l jm R˜ ¯ = p (∇ R − ∇ R ), kij 2 m k .i. i .k. ¯ 1 R˜ l = p (∇ R m − ∇ R m), kij 2 m k ijl i kjl ¯l j ϕ(z) a t ja m t ja R˜ ¯ = R + p p (R R − R R ) kij ikl 4 m a ktl .i. itl .k. ϕ0(z) 2ψ(z) − ϕ0(z) − p pjR a − p p R ja, 2ϕ(z) a kil 2(ψ(z) + 2zϕ(z)) l a ki.

¯l 1 k ϕ(z) m t ka R˜¯ = R − p p R R kij 2 ijl 4 m a itl .j. ϕ0(z) 2ψ(z) − ϕ0(z) + p pkR a + p p R ka, 4ϕ(z) a ijl 4(ψ(z) + 2zϕ(z)) l a ij. 0 l ϕ (z) k l ia i l ka ϕ(z) l ik l ki R˜¯¯ = p (p R − p R ) + (R − R ) kij 2 a .j. .j. 2 .j. .j. ϕ2(z) + p p (Rl kmRt ia − Rl imRt ka), 4 m a .t. .j. .t. .j. 0 l ϕ(z) l jk ϕ (z) k l ja j l ka R˜¯ ¯ = R + p (p R − p R ) kij 2 .i. 4 a .i. .i. ϕ2(z) + p p Rl kmRt ja, 4 m a .t. .i. ˜ ¯l jk i ij k Rk¯¯i¯j = [L − M(1 + 2zL)](g δl − g δl ) 0 2 kj i ij k +[N − (2M + M + 2zMN)](g p pl − g p pl) 0 2 i k j k i j +[2L − L − N(1 + 2zL)](δl p p − δl p p ), ˜ ¯l ˜ ¯l ˜ l ˜ ¯l (3.4) Rk¯i¯j = Rk¯¯ij = Rk¯¯i¯j = Rki¯ ¯j = 0,

ϕ0(z) 2ψ(z)−ϕ0(z) ψ0(z)ϕ(z)−2ϕ0(z)ψ(z) where L = 2ϕ(z) , M = 2ϕ(z)(ϕ(z)+2zψ(z)) , N = 2ϕ(z)(ϕ(z)+2zψ(z)) .

It is known (see [8, p.200]) that the sectional curvature on (T ∗M n, CGg) for P (U, V ) is given by

˜ k m i j ˜ Rkmij U V U V (3.5) K(P ) = − k m i j , (˜gkig˜mj − g˜kjg˜mi)U V U V

i where P (U, V ) = (U, V ) denotes the plane spanned by (U, V ). Let {Xi} and {ω }, i = 1, ..., n be a local ortonormal frame and coframe on M n, respectively. Then H H V 1 V n from (8)-(10) we see that { X1, ..., Xn, ω , ..., ω } is a local ortonormal frame on T ∗M n. Let K˜ (H X, H Y ), K˜ (H X, V θ) and K˜ (V ω, V θ) denote the sectional curvature of the plane spanned by (H X, H Y ), (H X, V θ) and (V ω, V θ) on (T ∗M n, g˜), 138 FIL˙ IZ˙ AGCA˘ respectively. Then, using (2.3), (2.4), (2.5) and (3.4), we have from (3.5)

R˜ H X˜ kH Y˜ iH X˜ jH Y˜ s i) K˜ (H X, H Y ) = − kijs H kH iH jH s (˜gkjg˜is − g˜ksg˜ij) X˜ Y˜ X˜ Y˜ l H kH iH jH s ¯l H kH iH jH s R˜ g˜ X˜ Y˜ X˜ Y˜ + R˜ g˜ ¯ X˜ Y˜ X˜ Y˜ = − kij sl kij sl H kH iH jH s (˜gkj g˜is − g˜ksg˜ij) X˜ Y˜ X˜ Y˜ ¡ ¢ ¡ ¢ ϕ(z) gtf pR(X,Y ) pR(X,Y ) = K(X,Y ) + 2 t f g(X,X)g(Y,Y ) − g(X,Y )g(X,Y ) ¡ ¢ ¡ ¢ ¡ ¢ ¡ ¢ ϕ(z) gtf pR(X,Y ) pR(X,Y ) ϕ(z) gtf pR(Y,Y ) pR(X,X) − 4 t f + 4 t f g(X,X)g(Y,Y ) − g(X,Y )g(X,Y ) g(X,X)g(Y,Y ) − g(X,Y )g(X,Y )

3ϕ(z)¯ ¯2 = K(X,Y ) − ¯(pR(X,Y )¯ . 4

H kV ¯iH jV s¯ R˜ ¯ X˜ ω˜ X˜ ω˜ ii) K˜ (H X, V θ) = − kijs¯ H ˜ kH ¯iH ˜ jV s¯ (˜gkjg˜¯is¯ − g˜k¯s¯g˜¯ij) X ω˜ X ω˜ l k j ¯l k j R˜ ¯ g˜sl¯ X ωiX ωs + R˜ ¯ g˜ ¯X ωiX ωs = −¡ k¡ij kij s¯l¢¢ is ia sb k j gkj ϕ(z)g + ψ(z)g g papb) X ωiX ωs 2 ¡ ¢ ¡ ¢ ϕ (z) gtf pR( ,X)˜ω pR( ,X)˜ω = 4 t f (ϕ(z)g(X,X)g−1(ω, ω) + ψ(z)g(X,X)(g−1(ω, p))2) 2 ϕ (z) ¯ ¯2 = ¡ ¢¯(pR( ,X)˜ω)¯ 4 ϕ(z) + ψ(z)(g−1(ω, p))2

V kV¯ ¯iV ¯jV s¯ R˜¯¯¯ ω˜ θ˜ ω˜ θ˜ iii) K˜ (V ω, V θ) = − kijs¯ V kV¯ ˜¯iV ¯jV ˜s¯ (˜gk¯¯jg˜¯is¯ − g˜k¯s¯g˜¯i¯j) ω˜ θ ω˜ θ l ¯l R˜¯¯¯ g˜sl¯ ωkθiωjθs + R˜¯¯¯ g˜ ¯ωkθiωjθs = − kij kij s¯l (˜gk¯¯jg˜¯is¯ − g˜k¯s¯g˜¯i¯j)ωkθiωjθs · A(δipkpj − δkpipj) + B(gkj pip − gij pkp ) = − l l l l P ¸ C(gjkδi − gijδk) ¡ ¢ + l l ϕ(z)(gsl + ψ(z)gsaglbp p ) ω θ ω θ P a b k i j s A(g−1(ω, p))2 = − ϕ(z) + ψ(z)[(g−1(θ, p))2 + (g−1(ω, p))2] ¡ ¢ B(ϕ(z) + 2zψ(z))(g−1(θ, p))2 + C ϕ(z) + ψ(z)(g−1(θ, p))2 − , ϕ2(z) + ϕ(z)ψ(z)[(g−1(θ, p))2 + (g−1(ω, p))2] g-NATURAL METRICS ON THE COTANGENT BUNDLE 139 where

P = (˜g¯¯g˜¯is¯ − g˜¯ g˜¯i¯j)ωkθiωjθs · kj ks¯ kj ka jb is it sf = (ϕ(z)g + ψ(z)g g papb)(ϕ(z)g + ψ(z)g g ptpf ) ¸ ks kc sd ij iu jv −(ϕ(z)g + ψ(z)g g pcpd)(ϕ(z)g + ψ(z)g g pupv) ωkθiωjθs

¡ ¢2 = ϕ2(z)g−1(ω, ω)g−1(θ, θ) + ϕ(z)ψ(z)g−1(ω, ω) g−1(θ, p) ¡ ¢2 ¡ ¢2¡ ¢2 +ϕ(z)ψ(z)g−1(θ, θ) g−1(ω, p) + ψ2(z) g−1(ω, p) g−1(θ, p) ¡ ¢2 −ϕ2(z) g−1(ω, θ) − ϕ(z)ψ(z)g−1(ω, θ)g−1(ω, p)g−1(θ, p) ¡ ¢2¡ ¢2 −ϕ(z)ψ(z)g−1(ω, θ)g−1(ω, p)g−1(θ, p) − ψ2(z) g−1(ω, p) g−1(θ, p) £¡ ¢2 ¡ ¢2¤ = ϕ2(z) + ϕ(z)ψ(z) g−1(ω, p) + g−1(θ, p) .

0 0 0 0 ϕ (z) 2ψ(z)−ϕ (z) ψ (z)ϕ(z)−2ϕ (z)ψ(z) 0 2 and L = 2ϕ(z) , M = 2ϕ(z)(ϕ(z)+2zψ(z)) , N = 2ϕ(z)(ϕ(z)+2zψ(z)) , A = 2L − L − N(1 + 2zL), B = N − (2M 0 + M 2 + 2zMN), C = L − M(1 + 2zL) Thus we have

Theorem 3.1. Let (M n, g) be a Riemannian manifold and T ∗M n be its cotangent bundle equipped with the g-natural metric g˜. Then the sectional curvature K˜ of (T ∗M n, g˜) satisfy the following:

3ϕ(z)¯ ¯2 i)K(H X,˜ H Y ) = K(X,Y ) − ¯(pR(X,Y )¯ , 4 2 ϕ (z) ¯ ¯2 ii)K˜ (H X, V ω) = ¡ ¢¯(pR( ,X)˜ω)¯ , 4 ϕ(z) + ψ(z)(g−1(ω, p)) A(g−1(ω, p))2 iii)K˜ (V ω, V θ) = − ϕ(z) + ψ(z)[(g−1(θ, p))2 + (g−1(ω, p))2] ¡ ¢ B(ϕ(z) + 2zψ(z))(g−1(θ, p))2 + C ϕ(z) + ψ(z)(g−1(θ, p))2 − , ϕ2(z) + ϕ(z)ψ(z)[(g−1(θ, p))2 + (g−1(ω, p))2]

n −1 ij 1 n where K is a sectional curvature of (M , g) and ω˜ = g ◦ ω = (g ωj) ∈ =0(M ), 0 0 1 n ϕ (z) 2ψ(z)−ϕ (z) R( ,X)˜ω ∈ =1(M ), L = 2ϕ(z) , M = 2ϕ(z)(ϕ(z)+2zψ(z)) , 0 0 ψ (z)ϕ(z)−2ϕ (z)ψ(z) 0 2 0 2 N = 2ϕ(z)(ϕ(z)+2zψ(z)) , A = 2L −L −N(1+2zL), B = N −(2M +M +2zMN) and C = L − M(1 + 2zL).

Theorem 3.2. Let (M n, g) be a Riemannian manifold of constant sectional curvature K. Let T ∗M n be its cotangent bundle equipped with the g-natural metric g˜. 140 FIL˙ IZ˙ AGCA˘

Then the sectional curvature K˜ of (T ∗M n, g˜) satisfy the following:

3ϕ2(z) ¡ ¢ i)K˜ (H X, H Y ) = K − K2 (g−1(p, X˜))2 + (g−1(p, Y˜ ))2 , 4  ¡ ¡ ¢2¢  ϕ2(z)K2 2z−2g−1(X,p˜ )g−1(ω,p)+ g−1(X,p˜ )  ¡ ¢ , g(X, ω˜) = 1, H V 4 ϕ(z)+ψ(z)(g−1(ω,p))2 ii)K˜ ( X, ω) = ¡¡ ¢2¢  ϕ2(z)K2 g−1(X,p˜ )  ¡ ¢, g(X, ω˜) = 0, 4 ϕ(z)+ψ(z)(g−1(ω,p))2 ¡ ¢2 A g−1(ω, p) iii)K˜ (V ω, V θ) = − £ ¤ ϕ(z) + ψ(z) (g−1(θ, p))2 + (g−1(ω, p))2 ¡ ¢ B(ϕ(z) + 2zψ(z))(g−1(θ, p))2 + C ϕ(z) + ψ(z)(g−1(θ, p))2 − £ ¤ , ϕ2(z) + ϕ(z)ψ(z) (g−1(θ, p))2 + (g−1(ω, p))2

−1 ij 1 n i ij −1 ˜ 1 n where ω˜ = g ◦ ω = (g ωj) ∈ =0(M ), X = g Xj = g ◦ X ∈ =0(M ), L = 0 0 0 0 ϕ (z) 2ψ(z)−ϕ (z) ψ (z)ϕ(z)−2ϕ (z)ψ(z) 0 2 2ϕ(z) , M = 2ϕ(z)(ϕ(z)+2zψ(z)) , N = 2ϕ(z)(ϕ(z)+2zψ(z)) , A = 2L −L −N(1+2zL), B = N − (2M 0 + M 2 + 2zMN) and C = L − M(1 + 2zL).

s s s Proof. Let Rkmj = K(δkgmj − δmgkj ).

3ϕ(z)¯ ¯2 i)K˜ (H X, H Y ) = K(X,Y ) − ¯(pR(X,Y )¯ 4 3ϕ(z) = K − gij (pR(X,Y )) (pR(X,Y )) 4 i j 3ϕ(z) = K − gij p K(δag − δag )p K(δb g − δb g )XkY lXf Y m 4 a k li l ki b f mj m fj 3ϕ(z) £ = K − K2 g−1(Y,˜ Y˜ )g−1(X,˜ p)g−1(X,˜ p) 4 −g−1(X,Y )g−1(X, p)g−1(Y, p) − g−1(X,Y )g−1(X,˜ p)g−1(Y˜ , p) ¤ +g−1(X,X)g−1(Y˜ , p)g−1(Y˜ , p) 3ϕ(z) ¡ ¢ = K − K2 (g−1(p, X˜))2 + (g−1(p, Y˜ ))2 4

ii) Using Theorem 3.1, we have

2 ¯ ¯ ˜ H V ϕ (z) ¯ ¯2 K( X, ω) = ¡ ¡ ¢2¢ (pR( ,X)˜ω) 4 ϕ(z) + ψ(z) g−1(ω, p) ¡ ¢ ¡ ¢ 2 tf ϕ (z)g pR( ,X)˜ω t pR( ,X)˜ω f = ¡ ¡ ¢2¢ 4 ϕ(z) + ψ(z) g−1(ω, p) 2 tf a i j b k m ϕ (z)g paRtij X ω˜ pbRfkm X ω˜ = ¡ ¡ ¢2¢ 4 ϕ(z) + ψ(z) g−1(ω, p) g-NATURAL METRICS ON THE COTANGENT BUNDLE 141 ¡ ¢ ¡ ¢ 2 tf a a i j b b k m ϕ (z)g pa K(δt gij − δi gtj) X ω˜ pb K(δf gkm − δkgfm) X ω˜ = ¡ ¡ ¢2¢ 4 ϕ(z) + ψ(z) g−1(ω, p) ¡ ¡ ¢2¢ ϕ2(z)K2 2z(g(X, ω˜))2 − 2g(X, ω˜)g−1(X,˜ p)g−1(ω, p) + g(˜ω, ω˜) g−1(X,˜ p) = ¡ ¡ ¢2¢ 4 ϕ(z) + ψ(z) g−1(ω, p)  ¡ ¡ ¢2¢  ϕ2(z)K2 2z−2g−1(X,p˜ )g−1(ω,p)+ g−1(X,p˜ )  ¡ ¡ ¢2¢ , g(X, ω˜) = 1, 4 ϕ(z)+ψ(z) g−1(ω,p) = ¡¡ ¢2¢  ϕ2(z)K2 g−1(X,p˜ )  ¡ ¢, g(X, ω˜) = 0, 4 ϕ(z)+ψ(z)(g−1(ω,p))2 where g(X , ω˜b) = g Xi (˜ωb)j = g Xi gjkωb = δkXi ωb a ij a ij a k½ i a k 1, a = b, = Xkωb = ωb(X ) = δb = a k a a 0, a 6= b,

i j is jk s jk g(˜ω, ω˜) = gij ω˜ ω˜ = gij g ωsg ωk = δj ωsg ωk sk −1 = g ωsωk = g (ω, ω) = 1. iii) The statement is obtained by iii) of Theorem 3.1. ¤

∗ n Let (x, p) be a point on T M with p 6= 0 and {e1, ..., en} be an orthonormal basis n n 1 n for the tangent space TxM of M at x. Also, let {ω , ..., ω } be a dual orthonormal ∗ n n 1 p basis for the cotangent spaces Tx M of M at x such that ω = |p| , where |p| is the norm of p with respect to the metric g on M n. Then for i ∈ {1, ..., n} and k ∈ H V ω1 {2, ..., n} deﬁne the horizontal and vertical lifts by fi = ei, fn+1 = √ ϕ(z)+2zψ(z) 1 V k 1 −1 and fn+k = √ ( ω ), z = |p| = g (p, p). Then {f1, ..., f2n} is an orthonormal ϕ(z) 2 ∗ n basis for the cotangent space T(x,p)M with respect to the g-natural metricg ˜. Using Theorem 3.1, we have

3ϕ(z)¯ ¯2 i)K˜ (f , f ) = K˜ (H e , H e ) = K(e , e ) − ¯pR(e , e )¯ , i j i j i j 4 i j V 1 H ω ii)K˜ (fi, fn+1) = K˜ ( ei, p ) ϕ(z) + 2zψ(z) 2 V 1 ϕ (z) ¯ ω ¯2 = ¯(pR( , e )p )¯ ¡ ¡ V 1 ¢2¢ i 4 ϕ(z) + ψ(z) g−1(√ ω , p) ϕ(z) + 2zψ(z) ϕ(z)+2zψ(z) = 0 by virtue of ¡ ¢ ¡ ¢ 1 m k p s k p s l 1 k s l pR( , ei)˜ω = pmR ,ks ei ( |p| ) = R ,kslei ( |p| ) p = |p| (R ,kslei p p ) = 0. ¯ ¯ 2 ¯ ω˜ k ¯2 V k ϕ (z) (pR( , ei)√ ) ω ϕ(z) iii)K˜ (f , f ) = K˜ (H e , p ) = i n+k i ¡ ¡ V k ¢2¢ ϕ(z) 4 ϕ(z) + ψ(z) g−1(√ ω , p) ϕ(z)

1¯ ¯2 = ¯(pR( , e )˜ωk)¯ , 4 i 142 FIL˙ IZ˙ AGCA˘

V ω1 V ωk iv)K˜ (fn+1, fn+k) = K˜ (p , p ) ϕ(z) + 2zψ(z) ϕ(z) ¡ ¡ V 1 ¢¢2 A g−1 √ ω , p ϕ(z)+2zψ(z) = − £¡ V k ¢2 ¡ V 1 ¢2¤ ϕ(z) + ψ(z) g−1(√ ω , p) + g−1(√ ω , p) ϕ(z) ϕ(z)+2zψ(z) ¡ V k ¢2 ¡ ¡ V k ¢2¢ B(ϕ(z) + 2zψ(z)) g−1(√ ω , p) + C ϕ(z) + ψ(z) g−1(√ ω , p) ϕ(z) ϕ(z) − £¡ V k ¢2 ¡ V 1 ¢2¤ ϕ2(z) + ϕ(z)ψ(z) g−1(√ ω , p) + g−1(√ ω , p) ϕ(z) ϕ(z)+2zψ(z) 2zA + (ϕ(z) + 2zψ(z))C = − , (ϕ(z) + 2zψ(z))ϕ(z) + 2zψ(z) ¡ V ωk V ωl ¢ v)K˜ (fn+k, fn+l) = K˜ p , p ϕ(z) ϕ(z) ¡ ¡ V k ¢¢2 A g−1 √ ω , p ϕ(z) = − £¡ l ¢2 ¡ k ¢2¤ ϕ(z) + ψ(z) g−1(√ω , p) + g−1(√ω , p) ϕ(z) ϕ(z) ¡ l ¢2 ¡ ¡ l ¢2¢ B(ϕ(z) + 2zψ(z)) g−1(√ω , p) + C ϕ(z) + ψ(z) g−1(√ω , p) ϕ(z) ϕ(z) − £¡ l ¢2 ¡ k ¢2¤ ϕ2(z) + ϕ(z)ψ(z) g−1(√ω , p) + g−1(√ω , p) ϕ(z) ϕ(z) C 2ϕ(z)ψ(z) − 2ϕ(z)ϕ0(z) − zϕ02(z) = − = , ϕ(z) 2ϕ2(z)(ϕ(z) + 2zψ(z)) where A = 2L0 − L2 − N(1 + 2zL), B = N − (2M 0 + M + 2zMN) and C = L − M(1 + 2zL). Thus we have

∗ n Theorem 3.3. Let (x, p) be a point on T M and {f1, ..., f2n} be an orthonormal ∗ n ˜ basis for the cotangent spaces Tx M as above. Then the sectional curvature K satisfy the following equation

3ϕ(z)¯ ¯2 i) K˜ (f , f ) = K(e , e ) − ¯pR(e , e )¯ , i j i j 4 i j

ii) K˜ (fi, fn+1) = 0, 1¯ ¯2 iii) K˜ (f , f ) = ¯(pR( , e )˜ωk)¯ , i n+k 4 i 2zA + (ϕ(z) + 2zψ(z))C iv) K˜ (f , f ) = − , n+1 n+k (ϕ(z) + 2zψ(z))ϕ(z) + 2zψ(z) C 2ϕ(z)ψ(z) − 2ϕ(z)ϕ0(z) − zϕ02(z) v) K˜ (f , f ) == − = n+k n+l ϕ(z) 2ϕ2(z)(ϕ(z) + 2zψ(z)) where K is a sectional curvature of (M n, g) and ω˜k = g−1 ◦ ωk, for i ∈ {1, ..., n} and k, l ∈ {2, ..., n}.

Corollary 3.1. Let (M n, p) be a Riemannian manifold and the cotangent bundle T ∗M n be equipped with the g-natural metric g˜. If we have constant sectional cur- ∗ n 2L0−L2 vature, then T M ﬂat and A = C = 0 for any z. It follows N = 1+2zL and g-NATURAL METRICS ON THE COTANGENT BUNDLE 143

2 L 2ϕ(z)ϕ0(z)+zϕ0 (z)−2ϕ(z)ψ(z) M = 1+2zL . We use C = 2ϕ(z)(ϕ(z)+2zψ(z)) = 0, then

2ϕ(z)ϕ0(z) + zϕ02(z) − 2ϕ(z)ψ(z) = 0.

So,

¡ zϕ0(z)¢ ψ(z) = ϕ0(z) 1 + . 2ϕ(z) i) ψ(z) = kϕ0(z) where k is a real constant. If ϕ0(z) = 0 then and ϕ(z) is constant. If ϕ0(z) 6= 0 then ϕ(z) = az2(k−1) (k > 1 or k ≤ 0, a > 0). √ ϕ(z) −1± 1+2z ii) ψ(z) = ϕ(z), then we obtain ϕ0(z) = z which gives √ e2 1+2z ϕ(z) = a¡ √ ¢2 , a > 0 1 + 1 + 2z or √ e−2 1+2z ϕ(z) = a¡√ ¢2 . 1 + 2z − 1

So we have to deal with non zero vector.

∗ n Let now {f1, ..., f2n} be an orthonormal basis for the cotangent space Tx M as P ˜ above, then the scalar curvaturer ˜ = i6=j K(fi, fj) is given by

X r˜ = K˜ (fi, fj) i6=j Xn Xn Xn CG CG = 2 K˜ (fi, fj) + 2 K(fi, fn+j) + 2 K(fn+i, fn+j) i,j=1 i,j=1 i,j=1 i

Theorem 3.4. Let (M n, g) be a Riemannian manifold and T ∗M n be its cotangent bundle equipped with the g-natural metric g˜. Let r be the scalar curvature of g and 144 FIL˙ IZ˙ AGCA˘ r˜ be the scalar curvature of g˜. Then the following equation holds n n 3ϕ(z) X ¯ ¯2 1 X ¯ ¯2 r˜ = r − ¯pR(e , e )¯ + ¯(pR( , e )˜ωj)¯ 4 i j 2 i i,j=1 i,j=1 2zA + (ϕ(z) + 2zψ(z))C C − 2(n − 1) − (n − 1)(n − 2) . (ϕ(z) + 2zψ(z))ϕ(z) + 2zψ(z) ϕ(z) Theorem 3.5. Let (M n, g), n > 2 be a Riemannian manifold of constant scalar curvatura κ. Then the scalar curvature r˜ of (T ∗M n, g) is £ r˜ = (n − 1) nκ + z(2 − 3ϕ(z))κ2 ¡ 4zA + 2(ϕ(z) + 2zψ(z))C (n − 2)C ¢¤ − − , (ϕ(z) + 2zψ(z))ϕ(z) + 2zψ(z) ϕ(z) 0 0 0 0 ϕ (z) 2ψ(z)−ϕ (z) ψ (z)ϕ(z)−2ϕ (z)ψ(z) 0 2 where L = 2ϕ(z) , M = 2ϕ(z)(ϕ(z)+2zψ(z)) , N = 2ϕ(z)(ϕ(z)+2zψ(z)) , A = 2L − L − N(1 + 2zL) and C = L − M(1 + 2zL). s s s Proof. Using the formulas Rkmj = κ(δkgmj − δmgkj), r = n(n − 1)κ and Theorem 8, we get the conclusion. ¤ 2 ∗ n Example. Using the Theorem 3.5. If ϕ(z) = 3 and ψ(z) = 0, then (T M , g) has constant scalar curvaturer ˜ = n(n − 1)κ.

4. Geodesics of g˜ n h h Let C be a curve in M expressed locally by x = x (t) and ωh(t) be a covector ﬁeld along C. Then, in the cotangent bundle T ∗M n, we deﬁned a curve C˜ by

h h h¯ def x = x (t), x = ph = ωh(t)(4.1) If the curve C satisfies at all the points the relation δω dω dxj h = h − Γi ω = 0, dt dt jh dt i then the curve C˜ is said to be a horizontal lift of the curve C in M n. Thus, if the 0 initial condition ωh = ωh for t = t0 is given, there exists a unique horizontal lift expressed by (4.1). We now consider differential equations of the geodesic in the cotangent bundle T ∗M n with the metricg ˜. If t is the arc length of a curve xA = xA(t),A = (i,¯i) in T ∗M n, then equations of geodesic in T ∗M n have the usual form δ2xA d2xA dxC dxB = + ΓÃ = 0(4.2) dt2 dt2 CB dt dt i ¯i i ∗ n Ã with respect to the induced coordinates (x , x ) = (x , pi) in T M , where ΓCB are components of ∇˜ defined by (2.7). We find it more convenient to refer equations (4.2) to the adapted frame {eα}. H From (2.1) and (2.2) we see that the matrix of change of frames eβ = Aβ ∂H has components of the form (3.2). Using (3.1), now we write α ¯α A θ = A Adx , i.e. h ¯h A h i h θ = A Adx = δi dx = dx g-NATURAL METRICS ON THE COTANGENT BUNDLE 145 for α = h and h¯ ¯h¯ A a j h j θ = A Adx = −paΓhjdx + δj dx = δph for α = h¯. Also we put θh dxA dxh = A¯h = , dt A dt dt h¯ A θ ¯ dx δp = A¯h = h dt A dt dt along a curve xA = xA(t) in T ∗M n. If we therefore write down the form equivalent to (4.2), namely, d ¡θα ¢ θγ θβ + Γ˜α = 0 dt dt γβ dt dt with respect to adapted frame and taking account of (2.7), then we have  2 h i  δ x k ja dx δpj  (a) dt2 + ϕ(z)paR.i. dt dt = 0, (4.3)  2 £ ¤  δ ph i j j i ij i j δpi δpj (b) dt2 + L(p δh + p δh) + Mg ph + Np p ph dt dt = 0. ϕ0(z) 2ψ(z)−ϕ0(z) ψ0(z)ϕ(z)−2ϕ0(z)ψ(z) where L = 2ϕ(z) , M = 2ϕ(z)(ϕ(z)+2zψ(z)) and N = 2ϕ(z)(ϕ(z)+2zψ(z)) . Thus the equations (4.3) are the equations of the geodesic in T ∗M n with the h h h¯ metricg ˜. Let now C˜ : x = x (t), x = ph(t) = ωh(t) be a horizontal lift ¡ ¢ 2 h δph δωh h h δ x n dt = dt = 0 of the geodesic C : x = x (t)( dt2 = 0) in M of ∇g. Then by virtue of (4.3), we have

Theorem 4.1. The horizontal lift of a geodesic in (M n, g) is always geodesic in T ∗M n with the g-natural metric g˜.

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Department of Mathematics, Faculty of Science, Karadeniz Technical University, Trabzon-TURKEY E-mail address: filiz [email protected]