A Kähler Einstein structure on the tangent bundle of a space form

© Hindawi Publishing Corp.

A KÄHLER EINSTEIN STRUCTURE ON THE TANGENT

BUNDLE OF A SPACE FORM

VASILE OPROIU

(Received 18 September 1998)

Abstract.We obtain a Kähler Einstein structure on the tangent bundle of a Riemannian manifold of constant negative curvature. Moreover, the holomorphic sectional curvature of this Kähler Einstein structure is constant. Similar results are obtained for a tube around zero section in the tangent bundle, in the case of the Riemannian manifolds of constant positive curvature.

2000 Mathematics Subject Classification. Primary 53C07, 53C15, 53C55.

1. Introduction. The tangent bundleT Mof a Riemannian manifold(M,g)can be organized as an almost Kählerian manifold (see [3, 10, 18]) by using the Sasaki metric and an almost complex structure defined by the splitting of the tangent bundle to

T Minto the vertical and horizontal distributionsVT MandHT M—(the last one being determined by the Levi Civita connection onM)—see also [8, 17, 19]. However, this structure is Kähler only in the case where the base manifold is locally Euclidean. The Sasaki metric is not a “good” metric in the sense of [1] since its Ricci curvature is not constant, that is, the Sasaki metric is not, generally, Einstein.

In the present paper, we are interested in finding a Kähler Einstein structure on the tangent bundle of a space form, following an idea from [2] (see also [1]). We have changed the metricGon the tangent bundle (so that it is no longer a Sasaki metric) by using a certain tensor field onT Mobtained in the following way. Denote byτ:T M→

Mthe canonical projection of the tangent bundle on the base manifold. Letybe an element ofT M, denote by

t=1₂y2₌1

2gτ(y)(y,y)= 1

2gik(x)yiyk (1.1) the value of the energy density in y and let gy ∈T∗T M be the cotangent vector obtained fromy by the “musical” isomorphism betweenT MandT∗_{M defined byg} (the “lowering” of indices). Identify the tensor fieldgwith its pullback byτ onT M. Then, we may consider the following symmetric tensor field of type(0,2)onT M,

˜

G=u(t)g+v(t)gy⊗gy, (1.2)

whereu,v : [0,∞)→R are smooth real-valued functions depending on t only. Of course, we assume that the functionsuandvfulfil the conditions under which the bilinear form defined by ˜Gis positive definite.

an almost complex structureJonT M, related to the considered metric, such that we obtain, in fact, a Kähler Einstein structure onT Min the case where(M,g)has constant (negative) curvature. As a matter of fact, we obtain the existence of a Kähler Einstein structure even in the case where(M,g)has positive constant curvature, but only on a tube around the zero section in T M (Theorem 4.2). The surprising fact was that we have obtained onT M a structure of Kähler manifold with constant holomorphic sectional curvature (Theorem 5.1).

The manifolds, tensor fields, and geometric objects we consider in this paper, are assumed to be differentiable of classC∞_{(i.e., smooth). We use the computations in} local coordinates in a fixed local chart, but many results from this paper may be ex-pressed in an invariant form. The well-known summation convention is used through-out this paper, the range for the indices i,j,k,l,h,s,r being always {1,...,n}(see [4, 5, 6, 7, 14, 15]). We denote byΓ(T M)the module of smooth vector fields onT M.

2. Almost Kähler structures on the tangent bundle. Let (M,g) be a smooth

n-dimensional Riemannian manifold and denote its tangent bundle byτ:T M→M. Recall thatT Mhas a structure of 2n-dimensional smooth manifold induced from the smooth manifold structure ofM. A local chart(U,ϕ)=(U,x1_,...,xn_{)onMinduces a} local chart(τ−1_(U),Φ)=(τ−1_(U),x1_,...,xn_,y1_,...,yn_{)onT M, where the local} coor-dinatesxi_,yi_{, i=1,...,nare defined as follows. The firstnlocal coordinates are the} local coordinates in the local chart(U,ϕ)of the base point of a tangent vector from

τ−1_{(U), that is,x}i_=xi_{◦τ, i=1,...,n, by an abuse of notation. The lastnlocal} coor-dinatesyi_{, i=1,...,nare the vector space coordinates of the same tangent vector,} with respect to the natural local basis in the corresponding tangent space, defined by the local chart(U,ϕ).

This special structure ofT Mallows us to introduce the notion ofM-tensor field on it (see [9] for detailed explanations). AnM-tensor field of type(p,q)onT Mis defined by sets of functions

Ti1,...,ip

structures are given by the Sasaki metric onT M defined byg (see [3, 17]) and the complete lift type pseudo-Riemannian metric defined byg(see [11, 12, 18, 19]). Recall that the Levi Civita connection ˙∇ofgdefines a direct sum decomposition

T T M=VT M⊕HT M (2.2)

of the tangent bundle toT Minto the vertical distributionVT M=kerτ∗and the hor-izontal distributionHT M. The set of vector fields(∂/∂y1_{,...,∂/∂y}n_{)defines a local} frame field forV T M and forHT Mwe have the local frame field(δ/δx1_,...,δ/δxn_), where

δ δxi=

∂

∂xi−Γih0_∂y∂h, Γih0=Γikhyk (2.3) andΓh

ik(x)are the Christoffel symbols defined by the Riemannian metricg.

The distributionsV T M andHT Mare isomorphicto each other and it is possible to derive an almost complex structure onT Mwhich, together with the Sasaki metric, determines a structure of almost Kählerian manifold onT M (see [3]). Consider now the energy density (kineticenergy)

t=1 2y2=

1

2gik(x)yiyk (2.4)

defined onT Mby the Riemannian metricgofM, wheregikare the components ofgin the local chart(U,ϕ). Letu,v:[0,∞)→Rbe two real smooth functions. Consider the following symmetricM-tensor field of type (0,2) onT M, defined by the components, (see [13, 16]),

Gij=u(t)gij+v(t)g0ig0j, (2.5)

whereg0i=ghiyh. The matrix(Gij)is symmetricand its inverse (when it exists) has the entries

Hkl₌ 1

ugkl+wykyl, (2.6)

wheregkl_{are the components of the inverse of the matrix(g} ij)and

w=w(t)= −_u(u+v_2tv). (2.7)

The conditions under which the matrix(Gij)is positive definite, hence nondegener-ate, can be obtained easily by studying the property of the expressionGijzizj, z1,...,zn ∈Rto be positive. These conditions are

u >0, u+2tv >0, ∀t≥0. (2.8)

obtained from the componentsHkl_{by “lowering” the indices}

Hij=gikHklglj=_u1gij+wg0ig0j, (2.9) as well as the following M-tensor fields onT M obtained by usual algebraictensor operations

Gkl_=gki_G

ijgjl=ugkl+vykyl, Gi

k=Gihghk=Gkhghi=uδik+vyig0k, Hi

k=Hihghk=Hkhghi=_u1δik+wyig0k.

(2.10)

Remark that the matrix(Hi

k)is the inverse of the matrix(Gik). The following Riemannian metricmay be considered onT M

G=Gijdxidxj+Hij∇˙yi∇˙yj, (2.11)

where ˙∇yi_=dyi_+Γi

j0dxjis the absolute differential ofyiwith respect to the Levi Civita connection ˙∇ofg. Equivalently, we have

G _δ

δxi, δ δxj

=Gij, G _∂

∂yi, ∂ ∂yj

=Hij,

G ∂ ∂yi,

δ δxj

=G δ δxj,

∂ ∂yi

=0.

(2.12)

Remark thatHT MandV T M are orthogonal to each other with respect toGbut the Riemannian metrics induced fromG on HT M and V T M are not the same, so the considered metric GonT M is no longer a metricof Sasaki type. Remark also that the system of 1-forms(dx1_,...,dxn_,∇˙y1_,...,∇˙yn_{)defines a local frame ofT}∗_{T M,} dual to the local frame(δ/δx1_,...,δ/δxn_,∂/∂y1_{,...,∂/∂y}n_{)adapted to the direct sum} decomposition (2.2).

An almost complex structureJmay be defined onT Mby

J_δxδ_i=Gk

i_∂y∂k; J ∂

∂yi= −Hik_δxδk. (2.13) Then we obtain the following theorem.

Theorem2.1. (T M,G,J)is an almost Kählerian manifold.

Proof. First of all, we may check easily thatJ2_{= −I, by using the local expression} (2.13) ofJ, due to the property of the matrix(Hi

k)to be the inverse of the matrix(Gik). Then, we have

G

J_δxδ_i,J_δxδ_j

=GikgkaGjhghbG _∂

∂ya, ∂ ∂yb

=GikgkaGjhghbgacGcdgdb=Gij=G _δ

δxi, δ δxj . (2.14) The relations G

J_∂y∂_i,J_∂y∂_j

=G _∂

∂yi, ∂ ∂yj

, G

J_∂y∂_i,J_δxδ_j

=G_∂y∂_i,_δxδ_j

may be obtained in a similar way, thusGis almost Hermitian with respect toJ. The associated 2-formΩ, defined by

Ω(X,Y )=G(X,JY ), X,Y∈Γ(T M), (2.16)

is given by

Ω _δ

δxi, δ δxj

=Ω

_∂ ∂yi,

∂ ∂yj

=0, Ω _∂

∂yi, δ δxj

=gij. (2.17)

Hence, we have

Ω=gij∇˙yi∧dxj (2.18)

andΩis closed since it does coincide with the 2-form associated to the Sasaki metric onT M(see [3]).

3. A Kähler structure on T M. In this section, we study the integrability of the almost complex structure defined by J onT M. To do this, we need the following well-known formulae for the brackets of the vector fields∂/∂yi_,δ/δxi_{, i=1,...,n,}

_∂ ∂yi,

∂ ∂yj

=0;

_∂ ∂yi,

δ δxj

= −Γh

ij_∂y∂h;

_δ δxi,

δ δxj

= −Rh

0ij_∂y∂h, (3.1) whereRh

0ij=Rhkijykand Rkijh are the local coordinate components of the curvature tensor field of ˙∇onM.

Theorem 3.1. The almost complex structure J onT M is integrable if(M,g) has constant sectional curvaturecand the functionvis given by

v= c−uu

2tu_−u. (3.2)

Of course we have to study the conditions under whichu,v fulfil the conditions

u >0, u+2tv >0,for allt≥0.

Proof. First of all, the following formulae can be checked by straightforward com-putation:

˙

∇iGjk=_δxδ_iGjk−ΓijhGhk−ΓikhGjh=0,

˙

∇iGjk=_δxδiG j

k+ΓihjGkh−ΓikhGhj=0.

(3.3)

In a similar way there are obtained the formulae ˙∇iHjk=0,∇˙iHkj=0. Then, by using the definition of the Nijenhuis tensor fieldNJofJ, that is,

NJ(X,Y )=[JX,JY ]−J[JX,Y ]−J[X,JY ]−[X,Y ], ∀X,Y∈Γ(T M), (3.4)

we have

NJ _δ

δxi, δ δxj = Gh i ∂Gk j

∂yh−Gjh∂G k i ∂yh+Rk0ij

∂ ∂yk

− _δ

δxjGki−_δxδiGkj+GihΓhjk −GhjΓihk

Hl k_δxδl.

The coefficient ofHl

k(δ/δxl)is just−∇˙jGki+∇˙iGkj=0 so, we have to study the van-ishing of the coefficient of∂/∂yk_{. By using the expression (2.10) ofG}k

i, we get

Rh

0ij+Gik_∂y∂kGjh−Gkj_∂y∂kGhi =

uu_{+2tuv−uv g}

0iδhj−g0jδhi +Rh0ij=0. (3.6) Differentiating with respect toyk_{, takingy=0 and using Schur theorem, it follows} that the curvature tensor field of ˙∇(in the case whereMis connected and dimM >2) must have the expression

Rk hij=c

δk

ighj−δkjghi , (3.7) wherecis a constant. Then we obtain the expression (3.2) ofv.

Next, it follows by a straightforward computation that NJ(∂/∂yi,δ/δxj) = 0, NJ(∂/∂yi,∂/∂yj)=0, wheneverNJ(δ/δxi,δ/δxj)=0.

Hence,(M,g)must have constant sectional curvaturec,vis given by (3.2) and then we obtain easily the expression of the functionw,

w=w(t)=_uuu−c

2tc−u2 . (3.8)

4. A Kähler Einstein structure onT M. In this section, we study the property of

(T M,G)to be Einstein. We find the expression of the Levi Civita connection∇of the metricGonT M, then we get the expression of the curvature tensor field of∇. Then, by computing the corresponding traces, we find the components of the Ricci tensor field of∇. Asking for the Ricci tensor field to be proportional with the metricG, we find a second-order ordinary differential equation which must be fulfilled by the functionu. Fortunately, we have been able to find the general solution of this differential equation by elementary methods. For a special value of one of the integration constants we obtained the property of G to be Einstein. At the same time we have been able to study the conditions under whichu >0, u+2tv >0,for allt≥0.

Recall that the Levi Civita connection∇on a Riemannian manifold(M,g)is obtained from the formula

2g∇XY ,Z =Xg(Y ,Z) +Yg(X,Z) −Zg(X,Y ) +g[X,Y ],Z

−g[X,Z],Y −g[Y ,Z],X , ∀X,Y ,Z∈χ(M) (4.1)

and is characterized by the conditions

∇G=0, T=0, (4.2)

whereT is the torsion tensor of∇(see [4, 5, 6, 7]).

Theorem4.1(see [16]). The Levi Civita connection∇ofGhas the following expres-sion in the local adapted frame(∂/∂y1_{,...,∂/∂y}n_,δ/δx1_,...,δ/δxn_),

∇∂/∂yi ∂

∂yj=Qhij_∂y∂h, ∇δ/δxi ∂

∂yj=Γijh_∂y∂h+Pjih_δxδh, ∇∂/∂yi δ

δxj=Pijh_δxδh, ∇δ/δxi δ

δxj =Γijh_δxδh+Sijh_∂y∂h,

where theM-tensor fieldsPh

ij,Qhij,Sijh are given by

Ph ij=1₂

∂Gjk

∂yi +HilRl0jk

Hkh_,

Qh

ij=1₂Ghk

∂Hjk ∂yi +

∂Hik ∂yj −

∂Hij ∂yk

,

Sh ij=1₂

−Rh

0ij−∂G_∂yijkGkh

.

(4.4)

Taking into account the expressions (2.5), (2.9) of Gij and Hij and by using the formulae (3.2), (3.8), and (2.7) we may obtain the following expressions:

Ph ij= u

2uδhjg0i+uv_2u−₂cδhig0j−(c+₂uv)w_v gijyh +vw(uv−c)+uw

u_v−uv

2uv g0ig0jyh,

Qh ij= −u

2u

δh

jg0i+δhig0j −v

u_+2u2_w

2u3_w gijyh−

v2u_w+uw

2u2_w g0ig0jyh,

Sh

ij=c−₂uvδhjg0i−c+₂uvδhig0j+ u _v

2uwgijyh+

vv_−2uvw

2uw g0ig0jyh.

(4.5)

Remark that in the obtained formulae we have used the formulae (2.7) in order to replace the energy densityt, such that it is not involved explicitly. Of course, we can replace v,v_{,w,w as functions of u,u,u, and t but we obtain much more} complicated expressions forPh

ij,Qhij,Sijh.

The curvature tensor field K of the connection ∇ is defined by the well-known formula

K(X,Y )Z= ∇X∇YZ−∇Y∇XZ−∇[X,Y ]Z, X,Y ,Z∈Γ(T M). (4.6)

By using the local frame (δ/δxi_,∂/∂yi_{), i=1,...,n, we obtain, after a standard} straightforward computation,

K δ δxi,

δ δxj

_δ

δxk =XXXkijh _δxδj, K _δ

δxi, δ δxj

_∂

∂yk=XXYkijh _∂y∂h,

K _∂

∂yi, ∂ ∂yj

_δ

δxk =Y Y Xkijh _δxδh, K _∂

∂yi, ∂ ∂yj

_∂

∂yk =Y Y Ykijh _∂y∂h,

K _∂

∂yi, δ δxj

_δ

δxk =Y XXhkij_∂y∂h, K _∂

∂yi, δ δxj

_∂

∂yk=Y XYkijh _δxδh,

where

XXXh

kij=Rhkij+PlkhRl0ij+PlihSjkl −PljhSikl , XXYh

kij=Rhkij+QhlkR0lij+SilhPkjl −SjlhPkil, Y Y Xh

kij=_∂y∂iPjkh−_∂y∂jPikh+PilhPjkl −PjlhPikl,

Y Y Yh

kij=_∂y∂iQhjk−_∂y∂jQhik+QhilQljk−QhjlQlik,

Y XXh

kij=_∂y∂iSjkh+QilhSjkl −SjlhPikl,

Y XYh

kij=_∂y∂iPkjh+PilhPkjl −PljhQlik.

(4.8)

Remark that, as a first step, the formulae for the local expressions ofKalso con-tained some other terms involving the Christoffel symbolsΓk

ij. However, after some computations similar to that made for obtaining the formulae (3.3) we have been able to show that those other terms are zero.

Now, we have to replace the expressions (4.5) of theM-tensor fieldsPh

ij,Qhij, Sijhand the expressions (3.2), (3.8), and (2.7) of the functions v,w and of their derivatives in order to obtain the components from (4.8) as functions ofu,u_,u,u(3)_{only. The} obtained expressions are quite complicated and, at this stage, we decided to use the mathematica package RICCI in order to do the necessary tensor calculations. It has been useful to considerc,t,u,v,w,u_,v,w,u,v,w,u(3) _{as constants, the} tan-gent vectoryas a first-order tensor, the componentsGij,Hijas second-order tensors and so on, on the Riemannian manifoldM, the associated indices beingh,i,j,k,l,r ,s. It was not convenient to think ofu,v,w as functions oftsince RICCI did not make some useful factorizations after the command TensorSimplify.

The components of the Ricci tensor field of the connection∇, defined as Ric(Y ,Z)= trace(X→K(X,Y )Z), X,Y ,Z∈Γ(T M)are obtained as follows:

Ric _δ

δxj, δ δxk

=RicXXjk=XXXkhjh +Y XXkhjh ,

Ric _∂

∂yj, ∂ ∂yj

=RicY Yjk=Y Y Ykhjh −Y XYkjhh ,

Ric ∂

∂yj, δ δxk

=Ric δ

δxk, ∂ ∂yj

=0.

(4.9)

The expressions of RicXXjk, RicY Yjk are quite complicated. In order to present a summary description of these expressions, we introduce the function

a=nu−2tu _{2cu−2ctu−u}2_{u +22ct−u}2 _tuu+uu−tu2 _{. (4.10)} Then

RicXXjk= a

2u−2tu 2gjk+α

t,u,u_,u,u(3) _g 0jg0k, RicY Yjk= a

2u2_u−2tu 2gjk+β

t,u,u_,u,u(3) _g 0jg0k,

(4.11)

To study the conditions under which(T M,G) is Einstein, we consider the differ-ences

DiffXXjk=RicXXjk− a

2uu−2tu 2Gjk, DiffY Yjk=RicY Yjk− a

2uu−2tu 2Hjk.

(4.12)

Then, we obtain

DiffXXjk= 1 2u2_u−2tu 4

×nu2_{−2ct 2tu−u u}2_u−2tu3_+2uu2

−8c2_tu3_u+4cu5_u+16c2_t2_u2_u2_+4ctu4_u2_−6u6_u2_−24c2_t3_uu3 −8ct2_u3_u3_+10tu5_u3_+16c2_t4_u4_−4t2_u4_u4_−24c2_t2_u3_u

+20ctu5_u−4u7_u+16c2_t3_u2_uu−4tu6_uu−16ct3_u3_u2_u +8t2_u5_u2_u−32c2_t4_u2_u2_+32ct3_u4_u2_−8t2_u6_u2_−8c2_t3_u3_u(3) +8ct2_u5_u(3)_−2tu7_u(3)_+16c2_t4_u2_uu(3)_−16ct3_u4_uu(3)

+4t2_u6_uu(3)_g 0jg0k, DiffY Yjk= 1

2u2_u2_{−2ct u−2tu} 2

×nu2_{−2ct 2tu−u u}2_u−2tu3_+2uu2

+4cu3_u−8ctu2_u2_−6u4_u2_+12ct2_uu3_+10tu3_u3_−8ct3_u4 −4t2_u2_u4_+12ctu3_u−4u5_u−8ct2_u2_uu−4tu4_uu+8t2_u3_u2_u +16ct3_u2_u2_−8t2_u4_u2_+4ct2_u3_u(3)_−2tu5_u(3)_−8ct3_u2_uu(3) +4t2_u2_uu(3)_g

0jg0k.

(4.13)

Our task is to find a positive function u(t) such that DiffXXjk=DiffY Yjk=0. It was hopeless to try to find directly a general common solution of the system of the third-order ordinary differential equations obtained by imposing the conditions DiffXXjk=DiffY Yjk=0. First, it was quite obvious that the functionu=constant is a solution of the obtained system. The case where u=1 has been discussed by the author in [13]. The obtained Kähler Einstein structure onT M in the case where

c <0 or on a tube around the zero section inT M in the case wherec >0 is even locally symmetric. Another case which can be considered is that where u2_=2ct. Remark that in this case we have alsou−2tu_{=0. It is a singular case and it has} been studied separately by the author and N. Papaghiucin [16]. It has been obtained when we tried (inspired by an idea of Calabi) to get a Kähler structure onT Mstarting with a Lagrangian depending ont only. The main result in [16] is that the bundle

If we try to find another functionufor which DiffXXjk=DiffY Yjk=0 and which does not depend on the dimensionnofM, then we have to find the positive solutions of the second-order differential equation

u2_u−2tu3_+2uu2_{=0, (4.14)} obtained from the vanishing of the coefficients ofnin DiffXXjk,DiffY Yjk. Remark that ifuis a solution of this equation, then −uis a solution too. We have got the general solution of the (4.14) by transforming it in a second-order differential equation fulfilled by the inverse functiont=t(u)of the functionu. We have obtained a second order Euler differential equation with the general solution

t=C1u2+C2u, (4.15) whereC1andC2are constants. Then the general solution of (4.14) may be written as

u=A2_{+Bt+A, (4.16)} where A and B are constants. The solution u = −√A2_{+Bt+A has the property}

u(0)=0 and should be excluded. However, the corresponding case can be treated as a singular case. We can takeB≠0 since in the contrary case we obtain the solu-tion=constant and this situation is discussed elsewhere. IfB >0, then we have the positive solution u=A+√A2_{+Btfor everyA∈R. If B <0, thenuis real only if} 0≤t <−A2_{/Band we must haveA >0. If we replace the expression (4.16) ofuin} DiffXXjkand DiffY Yjkwe obtain quite complicate expressions but it is quite obvious that we have DiffXXjk=DiffY Yjk=0 if and only ifB= −2c. Thenu=√A2−2ct+A and, by using the formulae (3.2) and (3.8), we obtain

v(t)=₂1_t

A−4_Act−A2_−2ct

, w(t)=−A3+3Act+

A2_−2ct 3/2

4ct2_A2_−2ct . (4.17) Remark thatv(0)is defined and we must haveA≠0. From the found expressions ofuandv, we arrived at the following situations when the symmetricmatrix(Gjk) is positive.

(1)c <0, A >0 and we have

u=A+A2_{−2ct >0, v=} 1 2t

A−4ct

A −

A2_{−2ct>0, t≥0. (4.18)} (2)c >0, A >0 and we have

u=A+A2_{−2ct, v=} 1 2t

A−4_Act−A2_−2ct

, 0≤t <A_2c2. (4.19) Hence, we may state our main result.

Theorem4.2. (1)Assume that(M,g)has constant negative curvaturec. Then(T M, G,J), withu,vgiven by (4.18), whereA >0, is a Kähler Einstein manifold.

(2)Assume that(M,g)has constant positive curvaturec. Then the tube around the zero section inT M, defined by the condition

y2_=g

jkyjyk=2t <A 2

c , (4.20)

Remark4.3. Another singular case is obtained whenC1=0 in the general solution of the Euler equation. In this case we haveu=At, A >0, henceu−tu_{=0. This case} can be discussed separately.

5. The holomorphic sectional curvature of(T M,G,J). In this section, we obtain the components of the curvature tensor fieldKof∇in the case whereu,vare given by (4.18), with c <0, A >0. Similar computations may be done in the remaining cases. Replacing Ph

ij,Qhij,Sijh in formulae (4.8) with their expressions in (4.5), where u,u_,u,u(3)_{,v,v,v,w,w, ware computed by using (4.18), we obtain the} follow-ing quite simple expressions of the components (of course, we have used RICCI to do the corresponding tensor calculations)

XXXh kij=₂c_A

δh

iGjk−δhjGik , XXYkijh =₂c_A

gjkGhi−gikGhj ,

Y Y Xh kij=₂c_A

gjkHih−gikHjh , Y Y Ykijh =₂c_A

δh

iHjk−δhjHik ,

Y XXh kij=₂c_A

δh

iGjk+gikGjh+2gijGHk ,

Y XYh

kij= −₂c_A

δh

jHik+gjkHih+2gijHkh .

(5.1)

Recall that a Kähler manifold(M,g,J)has holomorphic constant sectional curva-turekif its curvature tensor fieldRis given by

R(X,Y )Z

=k₄g(Y ,Z)X−g(X,Z)Y+g(JY ,Z)JX−g(JX,Z)JY+2g(X,JY )JZ, X,Y ,Z∈χ(M).

(5.2)

Comparing expressions (5.1) of the components ofKwith those obtained from (5.2), when we takeM→T M, g→G and for the vector fieldsX,Y ,Z involved in (5.2) we take the elements of the local frame (δ/δxi_,∂/∂yi_{), i=1,...,n, we obtain a quite} interesting result.

Theorem 5.1. If A >0, c <0, and(T M,G,J) has the Kähler Einstein structure defined by the expressions (4.18) ofuandv, then(T M,G,J)is a complex space form with negative constant holomorphic curvature2c/A.

Similar results are obtained in the cases whereu,vare given by (4.19). The components of the Ricci tensor field Ric of∇are

RicXXjk=(n+_A1)cGjk, RicY Yjk=(n+1_A )cHjk, RicXYjk=RicY Xjk=0, (5.3) thus

not written down in this paper but it is indicated how they can be obtained by using RICCI. The author is grateful to Victor Fecioru from Brisbane, Australia, for helping him with some advice, software and hardware, necessary in the computations made by using RICCI.

This work was partially supported by the Grant 34/64/1998, Ministerul Educa¸tiei Na¸tionale, România.

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Vasile Oproiu: Faculty of Mathematics, University Al. I. Cuza, Ia¸si, Romania