structures on the nonzero cotangent bundle of a space form

structures on the nonzero cotangent bundle of a space form

Dumitru Daniel Poro¸sniuc

Abstract

We obtain a class of locally symetric K¨ahler Einstein structures on the nonzero cotangent bundle of a Riemannian manifold of positive constant sec- tional curvature. The obtained class of K¨ahler Einstein structures depends on one essential parameter and cannot have constant holomorphic sectional curva- ture.

Mathematics Subject Classification: 53C07, 53C15, 53C55.

Key words: cotangent bundle, K¨ahler manifolds.

1 Introduction

In the study of the differential geometry of the cotangent bundleT^∗Mof a Riemannian manifold (M, g) one uses several Riemannian and semi-Riemannian metrics, induced from the Riemannian metricg onM. Next, one can get from gsome natural almost complex structures on T^∗M. The study of the almost Hermitian structures induced fromgonT^∗M is an interesting problem in the differential geometry of the cotangent bundle.

In [9] the authors have obtained a class of natural K¨ahler Einstein structures (G, J) of diagonal type induced onT^∗M from the Riemannian metricg. The obtained K¨ahler structures onT^∗M depend on two essential parametersa1 and λ, which are smooth functions depending on the energy densitytonT^∗M. In the case where the considered K¨ahler structures are Einstein they get several situations in which the parametersa1, λ are related by some algebraic relations. In the general case, (T^∗M, G, J) has constant holomorphic curvature.

In this paper we study the singular case where the parameter a1 = Atλ, A ∈ R. The class of the natural almost complex structuresJ on the nonzero cotangent bundleT₀^∗M that interchange the vertical and horizontal distributions depends on two essential parametersλandb1. These parameters are smooth real functions depending on the energy densitytonT₀^∗M. From the integrability condition forJ it follows that

Balkan Journal of Geometry and Its Applications, Vol.9, No.2, 2004, pp. 68-81.∗

°c Balkan Society of Geometers, Geometry Balkan Press 2004.

the base manifold M must have constant curvature c and the second parameter b1

must be expressed as a rational function depending on the first parameter λand its derivative. Of course, in the obtained formula there are involved the constant c and the energy densityt.

A class of natural Riemannian metrics Gof diagonal type on T₀^∗M is defined by four parametersc1, c2, d1, d2 which are smooth functions oft. From the condition for G to be Hermitian with respect to J we get two sets of proportionality relations, from which we can get the parameters c1, c2, d1, d2 as functions depending on one new parameterµand the parameterλinvolved in the expression ofJ.

In the case where the fundamental 2-form φ, associated to the class of complex structures (G, J) is closed, one finds thatµ=λ⁰.

Thus, we get a class of K¨ahler structures (G, J) on T₀^∗M, depending on one es- sential parameter λ.

Finally, we prove that the obtained class of K¨ahler structures on T₀^∗M is locally symmetric, Einstein and cannot have constant holomorphic sectional curvature.

The manifolds, tensor fields and geometric objects we consider in this paper, are assumed to be differentiable of class C^∞ (i.e. smooth). We use the computations in local coordinates but many results from this paper may be expressed in an invariant form. The well known summation convention is used throughout this paper, the range for the indices h, i, j, k, l, r, sbeing always{1, ..., n} (see [4], [6], [7]). We shall denote by Γ(T₀^∗M) the module of smooth vector fields onT₀^∗M.

2 Some geometric properties of T

M

Let (M, g) be a smoothn-dimensional Riemannian manifold and denote its cotangent bundle byπ:T^∗M −→M. Recall that there is a structure of a 2n-dimensional smooth manifold on T^∗M, induced from the structure of smoothn-dimensional manifold of M. From every local chart (U, ϕ) = (U, x¹, . . . , xⁿ) on M, it is induced a local chart (π⁻¹(U),Φ) = (π⁻¹(U), q¹, . . . , qⁿ, p1, . . . , pn), on T^∗M, as follows. For a cotangent vector p ∈ π⁻¹(U) ⊂ T^∗M, the first n local coordinates q¹, . . . , qⁿ are the local coordinates x¹, . . . , xⁿ of its base point x = π(p) in the local chart (U, ϕ) (in fact we have qⁱ =π^∗xⁱ =xⁱ◦π, i= 1, . . . n). The last n local coordinates p1, . . . , pn of p∈ π⁻¹(U) are the vector space coordinates ofp with respect to the natural basis (dx¹_π(p), . . . , dxⁿ_π(p)), defined by the local chart (U, ϕ), i.e.p=pidxⁱ_π(p).

An M-tensor field of type (r, s) on T^∗M is defined by sets of n^r+s components (functions depending onqⁱandpi), withrupper indices andslower indices, assigned to induced local charts (π⁻¹(U),Φ) on T^∗M, such that the local coordinate change rule is that of the local coordinate components of a tensor field of type (r, s) on the base manifold M (see [2] for further details in the case of the tangent bundle). An usual tensor field of type (r, s) on M may be thought of as anM-tensor field of type (r, s) onT^∗M. If the considered tensor field onM is covariant only, the corresponding M-tensor field on T^∗M may be identified with the induced (pullback by π) tensor field onT^∗M.

Some useful M-tensor fields on T^∗M may be obtained as follows. Let u, v : [0,∞) −→ R be a smooth functions and let kpk² = g⁻¹_π(p)(p, p) be the square of the norm of the cotangent vector p ∈ π⁻¹(U) (g⁻¹ is the tensor field of type (2,0)

having the components (g^kl(x)) which are the entries of the inverse of the matrix (gij(x)) defined by the components of g in the local chart (U, ϕ)). The compo- nents u(kpk²)gij(π(p)), pi, v(kpk²)pipj define M-tensor fields of types (0,2), (0,1), (0,2) onT^∗M, respectively. Similarly, the componentsu(kpk²)g^kl(π(p)),g⁰ⁱ=p_hg^hi, v(kpk²)g^0kg^0l define M-tensor fields of type (2,0), (1,0), (2,0) on T^∗M, respec- tively. Of course, all the components considered above are in the induced local chart (π⁻¹(U),Φ).

The Levi Civita connection ˙∇ ofgdefines a direct sum decomposition

(2.1) T T^∗M =V T^∗M⊕HT^∗M.

of the tangent bundle to T^∗M into vertical distributionsV T^∗M = Ker π∗ and the horizontal distributionHT^∗M.

If (π⁻¹(U),Φ) = (π⁻¹(U), q¹, . . . , qⁿ, p1, . . . , pn) is a local chart onT^∗M, induced from the local chart (U, ϕ) = (U, x¹, . . . , xⁿ), the local vector fields _∂p^∂

1, . . . ,_∂p^∂ on π⁻¹(U) define a local frame for V T^∗M over π⁻¹(U) and the local vector fieldsn

δ

δq¹, . . . ,_δq^δn define a local frame forHT^∗M overπ⁻¹(U), where δ

δqⁱ = ∂

∂qⁱ + Γ⁰_ih ∂

∂ph, Γ⁰_ih=pkΓ^k_ih and Γ^k_ih(π(p)) are the Christoffel symbols ofg.

The set of vector fields (_∂p^∂

1, . . . ,_∂p^∂

n,_δq^δ1, . . . ,_δq^δn) defines a local frame onT^∗M, adapted to the direct sum decomposition (1).

We consider

(2.2) t=1

2kpk²= 1

2g⁻¹_π(p)(p, p) = 1

2g^ik(x)pipk, p∈π⁻¹(U)

the energy density defined byg in the cotangent vectorp. We havet∈[0,∞) for all p∈T^∗M.

From now on we shall work in a fixed local chart (U, ϕ) onM and in the induced local chart (π⁻¹(U),Φ) onT^∗M.

Now we shall present the following auxiliary result.

Lemma 1.If n >1 andu, v are smooth functions on T^∗M such that ugij+vpipj= 0, p∈π⁻¹(U)

on the domain of any induced local chart on T^∗M, thenu= 0, v= 0.

The proof is obtained easily by transvecting the given relation with the components g^ij of the tensor fieldg⁻¹ andg^0j.

Remark. From the relations of the type

ug^ij+vg⁰ⁱg^0j= 0, p∈π⁻¹(U), uδⁱ_j+vg⁰ⁱpj = 0, p∈π⁻¹(U), it is obtained, in a similar way,u=v= 0.

3 A class of natural complex structures of diagonal type on T

M

The nonzero cotangent bundle T₀^∗M of Riemannian manifold (M, g) is defined by the formula: T^∗M minus zero section. Consider the real valued smooth functions λ, a1, a2, b1, b2 defined on (0,∞). We define a class of natural almost complex struc- turesJ of diagonal type onT₀^∗M , expressed in the adapted local frame by

(3.3) J δ

δqⁱ =J_ij⁽¹⁾(p) ∂

∂pj, J ∂

∂pi =−J₍₂₎^ij (p) δ δq^j. where,

(3.4) J_ij⁽¹⁾(p) =a₁(t)g_ij+b₁(t)p_ip_j, J₍₂₎^ij (p) =a₂(t)g^ij+b₂(t)g⁰ⁱg^0j, A∈R^∗. In this paper we study the singular case where

(3.5) a1(t) =Atλ(t).

The components J_ij⁽¹⁾, J₍₂₎^ij define symmetricM-tensor fields of types (0,2),(2,0) onT^∗M, respectively.

Proposition 2.The operator J defines an almost complex structure on T^∗M if and only if

(3.6) a1a2= 1, (a1+ 2tb1)(a2+ 2tb2) = 1.

Proof. The relations are obtained easily from the property J² = −I of J and Lemma 1.

From the relations (5), (6) we can obtain the explicit expression of the parameter a2, b2

(3.7) a₂= 1

Atλ, b₂= −b1

At²λ(Aλ+ 2b1).

The obtained class of almost complex structures defined by the tensor field J on T₀^∗M is called class of natural almost complex structures of diagonal type,obtained from the Riemannian metricg, by using the parametersλ, b1. We use the word diago- nal for these almost complex structures, since the 2n×2n-matrix associated toJ, with respect to the adapted local frame (_δq^δ1, . . . ,_δq^δn,_∂p^∂

1, . . . ,_∂p^∂

n) has twon×n-blocks on the second diagonal

J=

à 0 −J₍₂₎^ij J_ij⁽¹⁾ 0

! .

Remark. From the conditions (6) it follows thata1=Atλ anda2=_Atλ¹ cannot vanish and have the same sign. We assume that

(3.8) λ(t)>0 ∀t >0, A >0.

Similarly, from the conditions (6) it follows that a1+ 2tb1 and a2+ 2tb2 cannot vanish and have the same sign. We assume thata1+ 2tb1>0, a2+ 2tb2>0∀t >0, i.e.

(3.9) Aλ+ 2b1>0 ∀t >0.

Now we shall study the integrability of the class of natural almost complex struc- tures defined byJ onT₀^∗M. To do this we need the following well known formulas for the brackets of the vector fields _∂p^∂

i,_δq^δi, i= 1, ..., n

(3.10) [ ∂

∂pi, ∂

∂pj] = 0, [ ∂

∂pi, δ

δq^j] = Γⁱ_jk ∂

∂pk, [ δ δqⁱ, δ

δq^j] =R⁰_kij ∂

∂pk,

whereR^h_kij(π(p)) are the local coordinate components of the curvature tensor field of

∇˙ onM andR⁰_kij(p) =phR^h_kij. Of course, the componentsR^h_kij,R⁰_kijdefine M-tensor fields of types (1,3), (0,3) onT₀^∗M, respectively.

Recall that the Nijenhuis tensor fieldN defined byJ is given by

N(X, Y) = [JX, JY]−J[JX, Y]−J[X, JY]−[X, Y], ∀ X, Y ∈Γ(T₀^∗M).

Then, we have _δq^δkt = 0, _∂p^∂

kt =g^0k. The expressions for the components ofN can be obtained by a quite long, straightforward computation, as follows

Theorem 3. The Nijenhuis tensor field of the almost complex structure J on T₀^∗M is given by















N(_δq^δi,_δq^δj) ={At(λ+ 2tλ⁰)(b1+Aλ)(δ_i^hgjk−δ_j^hgik)−R^h_kij}ph ∂

∂pk, N(_δq^δi,_∂p^∂

j) =J₍₂₎^klJ₍₂₎^jr{At(λ+ 2tλ⁰)(b1+Aλ)(δ_i^hgrl−δ_r^hgil)−R^h_lir}ph δ δq^k, N(_∂p^∂

i,_∂p^∂

j) =J₍₂₎^irJ₍₂₎^jl {At(λ+ 2tλ⁰)(b1+Aλ)(δ^h_lgrk−δ_r^hglk)−R_klr^h }ph ∂

∂pk.

Theorem 4. Assume that exist

t→0limAt(λ+2tλ⁰)(b1+Aλ)∈R, lim

t→0

∂

∂pl

[At(λ+2tλ⁰)(b1+Aλ)] = 0, ∀l∈ {1,2, . . . , n}.

The almost complex structure J on T₀^∗M is integrable if and only if (M, g) has constant sectional curvature cand the function b1 is given by

(3.11) b1= c−A²tλ(λ+tλ⁰) At(λ+ 2tλ⁰) .

The parameter λmust fulfill the conditons (3.12) λ >0, 2c−A²tλ²

λ+ 2tλ⁰ >0 ∀t >0, A >0.

Proof. From the conditionN = 0, one obtains

{At(λ+ 2tλ⁰)(b1+Aλ)(δ^h_igjk−δ_j^hgik)−R_kij^h }ph= 0.

Differentiating with respect to pl and taking t → 0, it follows that the curvature tensor field of ˙∇ has the expression

R^l_kij= (lim

t→0At(λ+ 2tλ⁰)(b1+Aλ))(δ^l_igjk−δ^l_jgik).

Thus the sectional curvature c= limt→0At(λ+ 2tλ⁰)(b1+Aλ) of (M, g) depends only on qⁱ. Using by the Schur theorem (in the case where M is connected and dimM≥3) it follows that (M, g) has the constant sectional curvaturec= lim

t→0At(λ+

2tλ⁰)(b1+Aλ). Then we obtain the expression (3.11) ofb1.

Conversely, if (M, g) has constant curvaturecandb1is given by (3.11), it follows in a straightforward way thatN = 0.

Using by the relations (3.8), (3.9), (3.11) we obtain the conditions (3.12).

The class of natural complex structures J of diagonal type on T₀^∗M depends on one essential parameterλ. The components ofJ are given by

(3.13)







J_ij⁽¹⁾=Atλgij+^c−A_At(λ+2tλ^2tλ(λ+tλ0)⁰⁾pipj, J₍₂₎^ij =_Atλ¹ g^ij−_At^c−A2λ(2c−A^2tλ(λ+tλ2tλ⁰²⁾)g⁰ⁱg^0j.

4 A class of natural Hermitian structures on T

M

Consider the following symmetric M−tensor fields onT₀^∗M, defined by the compo- nents

(4.14) G⁽¹⁾_ij =c1gij+d1pipj, G^ij₍₂₎ =c2g^ij+d2g⁰ⁱg^0j,

wherec1, c2, d1, d2are smooth functions depending on the energy densityt∈(0,∞).

Obviously, G⁽¹⁾ is of type (0,2) andG(2) is of type (2,0). We shall assume that the matrices defined byG⁽¹⁾ andG(2) are positive definite. This happens if and only if

(4.15) c1>0, c2>0, c1+ 2td1>0, c2+ 2td2>0 ∀t >0.

Then the following class of Riemannian metrics may be considered onT₀^∗M (4.16) G=G⁽¹⁾_ij dqⁱdq^j+G^ij₍₂₎DpiDpj,

whereDpi=dpi−Γ⁰_ijdq^j is the absolute (covariant) differential ofpi with respect to the Levi Civita connection ˙∇ofg. Equivalently, we have

G( δ δqⁱ, δ

δq^j) =G⁽¹⁾_ij , G( ∂

∂pi

, ∂

∂pj

) =G^ij₍₂₎, G( ∂

∂pi

, δ

δq^j) =G( δ δq^j, ∂

∂pi

) = 0.

Remark that HT₀^∗M, V T₀^∗M are orthogonal to each other with respect to G, but the Riemannian metrics induced from G on HT₀^∗M, V T₀^∗M are not the same, so the considered metricGonT₀^∗M is not a metric of Sasaki type. The 2n×2n-matrix associated toG, with respect to the adapted local frame (_δq^δ1, . . . ,_δq^δn,_∂p^∂

1, . . . ,_∂p^∂

n) has twon×n-blocks on the first diagonal

G= Ã

G⁽¹⁾_ij 0 0 G^ij₍₂₎

! .

The class of Riemannian metrics G is called a class of natural lifts of diagonal typeofg. Remark also that the system of 1-forms (Dp1, ..., Dpn, dq¹, ..., dqⁿ) defines a local frame onT^∗T₀^∗M, dual to the local frame (_∂p^∂

1, . . . ,_∂p^∂

n,_δq^δ1, . . . ,_δq^δn) onT T₀^∗M, overπ⁻¹(U) adapted to the direct sum decomposition (1).

We shall consider another twoM-tensor fieldsH₍₁₎, H⁽²⁾onT₀^∗M, defined by the components

H₍₁₎^jk = 1

c1g^jk− d1

c1(c1+ 2td1)g^0jg^0k, H_jk⁽²⁾= 1

c2gjk− d2

c2(c2+ 2td2)pjpk.

The components H₍₁₎^jk define anM-tensor field of type (2,0) and the components H_jk⁽²⁾ define an M-tensor field of type (0,2). Moreover, the matrices associated to H(1), H⁽²⁾ are the inverses of the matrices associated toG⁽¹⁾ andG(2), respectively.

Hence we have

G⁽¹⁾_ij H₍₁₎^jk =δ^k_i, G^ij₍₂₎H_jk⁽²⁾=δ_kⁱ.

Now, we shall be interested in the conditions under which the class of the metrics Gis Hermitian with respect to the class of the complex structures J, considered in the previous section, i.e.

G(JX, JY) =G(X, Y), for all vector fieldsX, Y onT₀^∗M.

Considering the coefficients ofgij, g^ij in the conditions

(4.17)





G(J_δq^δi, J_δq^δj) =G(_δq^δi,_δq^δj), G(J_∂p^∂

i, J_∂p^∂

j) =G(_∂p^∂

i,_∂p^∂

j),

we can express the parameters c1, c2 with the help of the parameters a1, a2 and a proportionality factor which must beλ=λ(t) (see [9]). Then

(4.18) c1=λa1=Atλ², c2=λa2= 1 At, where the coefficientsa1, a2are given by (5) and (7).

Next, considering the coefficients of pipj, g⁰ⁱg^0j in the relations (17), we can express the parametersc1+ 2td1, c2+ 2td2 with help of the parametersa1+ 2tb1, a2+ 2tb2 and a proportionality factorλ+ 2tµ

(4.19)





c1+ 2td1= (λ+ 2tµ)(a1+ 2tb1), c2+ 2td2= (λ+ 2tµ)(a2+ 2tb2).

Remark that λ(t) + 2tµ(t) > 0 ∀t > 0. It is much more convenient to consider the proportionality factor in such a form in the expression of the parameters c1+ 2td1, c2+ 2td2. Using by the relations (5), (7), (11),(18) we can obtain easily from (19) the explicit expressions of the coefficientsd₁, d₂

(4.20)







d1=^{λ[c−A2tλ(λ+tλ}_At(λ+2tλ^{0)]+µt(2c−A}0) ^2tλ2), d2=^{−c+A2tλ(λ+tλ}_At2(2c−A^0)+µA2tλ^22t)^2(λ+2tλ0). Hence we may state:

Theorem 5. Let J be the class of natural, complex structure of diagonal type on T₀^∗M, given by (3) and (13). LetGbe the class of the natural Riemannian metrics of diagonal type onT₀^∗M, given by (14), (18), (20).

Then we obtain a class of Hermitian structures(G, J)onT₀^∗M, depending on two essential parametersλandµ, which must fulfill the conditions

(4.21) λ >0, 2c−A²tλ²

λ+ 2tλ⁰ >0, λ+ 2tµ >0 ∀t >0, A >0.

5 A class of K¨ ahler structures on T

M

Consider now the two-formφdefined by the class of Hermitian structures (G, J) on T₀^∗M

φ(X, Y) =G(X, JY), for all vector fieldsX, Y onT₀^∗M.

Using by the expression ofφand computing the valuesφ(_∂p^∂

i,_∂p^∂

j), φ(_δq^δi,_δq^δj), φ(_∂p^∂

i,_δq^δj), we obtain.

Proposition 6. The expression of the 2-form φ in a local adapted frame (_∂p^∂

1, . . . ,_∂p^∂

n,_δq^δ1, . . . ,_δq^δn)on T₀^∗M, is given by φ( ∂

∂pi, ∂

∂pj) = 0, φ( δ δqⁱ, δ

δq^j) = 0, φ( ∂

∂pi, δ

δq^j) =λδⁱ_j+µg⁰ⁱpj, or, equivalently

(5.22) φ= (λδⁱ_j+µg⁰ⁱpj)Dpi∧dq^j.

Theorem 7. The class of Hermitian structures (G, J)on T₀^∗M is K¨ahler if and only if

µ=λ⁰.

Proof. The expressions of dλ, dµ, dg⁰ⁱ and dDpi are obtained in a straightforward way, by using the property ˙∇kgij = 0 (hence ˙∇kg^ij= 0)

dλ=λ⁰g⁰ⁱDpi, dµ=µ⁰g⁰ⁱDpi, dg⁰ⁱ =g^ikDpk−g^0hΓⁱ_hkdq^k, dDpi =−1

2R⁰_ikldq^k∧dq^l+ Γ^l_ikdq^k∧Dpl. Then we have

dφ= (dλδ_jⁱ+dµg⁰ⁱpj+µdg⁰ⁱpj+µg⁰ⁱdpj)∧Dpi∧dq^j+ +(λδ_jⁱ+µg⁰ⁱpj)dDpi∧dq^j.

By replacing the expressions ofdλ, dµ, dg⁰ⁱandd∇y˙ ^h, then using, again, the property

∇˙kgij = 0, doing some algebraic computations with the exterior products, then using the well known symmetry properties ofgij,Γ^h_ij,and of the Riemann-Christoffel tensor field, as well as the Bianchi identities, it follows that

dφ= 1

2(λ⁰−µ)g^0hDph∧Dpi∧dqⁱ. Therefore we havedφ= 0 if and only if µ=λ⁰.

Remark.The class of natural K¨ahler structures of diagonal type defined by (G, J) onT₀^∗M depends on one essential parameterλ.

The paramaterλmust fulfill the conditions

(5.23) λ >0, 2c−A²tλ²>0, λ+ 2tλ⁰>0 ∀t >0, A >0.

It follows thatc >0.

The components of the class of K¨ahler metrics GonT₀^∗M are given by

(5.24)







G⁽¹⁾_ij =Atλ²g_ij+^c−A_At^2tλ2p_ip_j,

G^ij₍₂₎= _At¹g^ij−^c−A_At^2t[λ2(2c−A^{2+2tλ20(λ+tλ}tλ²) ^0)]g⁰ⁱg^0j. We obtain, too

(5.25)







H₍₁₎^jk = _Atλ¹2g^jk−_At2λ^c−A2(2c−A^2tλ22tλ²)g^0jg^0k, H_jk⁽²⁾=Atgjk+^c−A2t[λ_At(λ+2tλ^{2+2tλ0(λ+tλ}0)^{2 0)])}pjpk.

6 A class of locally symmetric K¨ ahler Einstein struc- tures on T

M

The Levi Civita connection ∇ of the Riemannian manifold (T₀^∗M, G) is determined by the conditions

∇G= 0, T = 0,

whereTis its torsion tensor field. The explicit expression of this connection is obtained from the formula

2G(∇XY, Z) =X(G(Y, Z)) +Y(G(X, Z))−Z(G(X, Y))+

+G([X, Y], Z)−G([X, Z], Y)−G([Y, Z], X); ∀X, Y, Z ∈Γ(T₀^∗M).

The final result can be stated as follows.

Theorem 8. The Levi Civita connection∇ of G has the following expression in the local adapted frame(_δq^δ1, . . . ,_δq^δn,_∂p^∂

1, . . . ,_∂p^∂

n) : (6.26)







∇ ∂

∂pi

∂

∂pj =Q^ij_h∂p^∂

h, ∇ δ

δqi

∂

∂pj =−Γ^j_ih∂p^∂

h +P_i^hj_δq^δh,

∇ ∂

∂pi

δ

δq^j =P_j^hi_δq^δh, ∇ δ δqi

δ

δq^j = Γ^h_ijδq^δh +Shij ∂

∂ph, where Q^ij_h, P_j^hi, Shij areM-tensor fields on T₀^∗M, defined by

(6.27)















Q^ij_h = ¹₂H_hk⁽²⁾(_∂p^∂

iG^jk₍₂₎+_∂p^∂

jG^ik₍₂₎−_∂p^∂

kG^ij₍₂₎), P_j^hi=¹₂H₍₁₎^hk(_∂p^∂

iG⁽¹⁾_jk −G^il₍₂₎R⁰_ljk), Shij =−¹₂H_hk⁽²⁾_∂p^∂

kG⁽¹⁾_ij +¹₂R⁰_hij.

Assuming that the base manifold (M, g) has positive constant sectional curvature cand replacing the expressions of the involvedM-tensor fields, one obtains

(6.28)

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Q^ij_h =_2t¹g^ijph−_2t¹(δⁱ_hg^0j+δ^j_hg⁰ⁱ)+

cλ+8ctλ⁰−2A²t²λλ⁰(λ−tλ⁰)+2t²λ⁰⁰(2c−A²tλ²)

2t²(2c−A²tλ²)(λ+2tλ⁰) g⁰ⁱg^0jph,

Pj^hi=−_2t¹g^hipj+_2t¹δjⁱg^0h+^λ+2tλ_2tλ⁰δj^hg⁰ⁱ−_2t2λ(2c−A^c(λ+2tλ20)tλ²)g^0hg⁰ⁱpj, Shij=−^λ(2c−A_2(λ+2tλ^2tλ0)²⁾gijph−^(2c−A₂^2tλ2)ghipj+^A2₂^tλ2ghjpi+

3cλ+2ctλ⁰−2A²tλ²(λ+tλ⁰) 2t(λ+2tλ⁰) phpipj.

The curvature tensor fieldK of the connection ∇ is obtained from the well known formula

K(X, Y)Z =∇X∇YZ− ∇Y∇XZ− ∇_[X,Y]Z, ∀X, Y, Z∈Γ(T₀^∗M).

The components of curvature tensor field K with respect to the adapted local frame (_δq^δ1, . . . ,_δq^δn,_∂p^∂

1, . . . ,_∂p^∂

n) are obtained easily:

(6.29)

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K(_δq^δi,_δq^δj)_δq^δk =QQQ^h_ijkδq^δh, K(_δq^δi,_δq^δj)_∂p^∂

k =QQP_ijh^k _∂p^∂

h, K(_∂p^∂

i,_∂p^∂

j)_δq^δk =P P Q^ijh_{k δq}^δh, K(_∂p^∂

i,_∂p^∂

j)_∂p^∂

k =P P P_h^ijk_∂p^∂

h, K(_∂p^∂

i,_δq^δj)_δq^δk =P QQⁱ_jkh∂p^∂

h, K(_∂p^∂

i,_δq^δj)_∂p^∂

k =P QP_j^ikh_δq^δh, where

(6.30)

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QQQ^h_ijk=λ²[^A₂^2t(δ_i^hgjk−δ^h_jgik) +^A₄²(gikpj−gjkpi)g^0h−

A²

4 (δ^h_ipj−δ^h_jpi)pk], QQP_ijh^k =−QQQ^k_ijh,

P P Q^ijh_k =−_2t¹(δⁱ_kg^jh−δ_k^jg^ih)−_4t¹2(g^ihg^0j−g^jhg⁰ⁱ)pk+

1

4t²(δ_kⁱg^0j−δ_k^jg⁰ⁱ)g^0h, P P P_h^ijk=−P P Q^ijk_h ,

P QQⁱ_jkh= ^A2₂^tλ2δⁱ_jg_hk+^λ(2c−A_4t(λ+2tλ^2tλ0²)⁾δ_kⁱp_hp_j+^λ[c−A_2t(λ+2tλ^2λt(λ+tλ0) ^0)]δ_jⁱp_hp_k+

(2c−A²tλ²)

4t δ_hⁱpjpk+^A2₄^λ2g⁰ⁱgjkph+^A2tλλ₂ ⁰g⁰ⁱghkpj+

A²λ(λ+2tλ⁰)

4 g⁰ⁱghjpk−^λ[c+2A_2t2²(λ+2tλ^{t2λ0(λ+tλ0}) ^0)]g⁰ⁱphpjpk,

P QP_j^ikh=−_2t¹δ_jⁱg^hk−_4t¹2g^ikg^0hpj−_2tλ^λ0 g^hkg⁰ⁱpj−^λ+2tλ_4t2λ⁰g^hig^0kpj−

A²λ(λ+2tλ⁰)

4t(2c−A²tλ²)δ^k_jg^0hg⁰ⁱ+^c−A_2t2(2c−A^2tλ(λ+tλ2tλ²⁰)⁾δ_jⁱg^0hg^0k−

A²(λ+2tλ⁰)²

4t(2c−A²tλ²)δ^h_jg⁰ⁱg^0k+_2t3^c(λ+2tλλ(2c−A²⁰tλ^{) 2})g^0hg⁰ⁱg^0kpj. are M-tensor fields onT₀^∗M.

Remark. From the local coordinates expression of the curvature tensor field K, we obtain that the class of K¨ahler structures (G, J) on T₀^∗M cannot have constant holomorphic sectional curvature.

The Ricci tensor field Ric of ∇is defined by the formula:

Ric(Y, Z) =trace(X −→K(X, Y)Z), ∀ X, Y, Z∈Γ(T₀^∗M).

It follows