ON THE GEOMETRY OF TANGENT BUNDLE OF A (PSEUDO-) RIEMANNIAN MANIFOLD

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1 ANALELE ŞTIINŢIFICE ALE UNIVERSITĂŢII AL.I.CUZA IAŞI Tomul XLIV, s.i.a, Matematică, 1998, f1 ON THE GEOMETRY OF TANGENT BUNDLE OF A (PSEUDO-) RIEMANNIAN MANIFOLD BY V. OPROIU and N. PAPAGHIUC 0. Introduction. In [5], [6] the authors have studied the properties of a pseudo Riemannian metric G on the cotangent bundle T M of a manifold M by using an arbitrary symmetric nonlinear connection on this bundle. Remark that this pseudo Riemannian metric G on T M is very much similar to the Riemann extension considered in [7], [8], [9]. In [3] the first author has studied some aspects of the differential geometry of the tangent bundle of a Langange manifold when this bundle is endowed with a pseudo Riemannian metric obtained from the fundamental tensor field by a method similar to the obtaining of the complete lift of a (pseudo )Riemannian metric on a differentiable manifold. So, the Levi Civita connection associated with the considered pseudo Riemannian metric has been obtained, next the local coordinate expressions of its curvature tensor field and the corresponding Bianchi identities have been written down. In the present paper we study the properties of a pseudo Riemannian metric G of type complete lift on the tangent bundle T M of a (pseudo ) Riemannian manifold (M, g). The considered pseudo Riemannian metric G on T M is defined by using the Levi Civita cannection of the (pseudo ) Riemannian metric g on M and an arbitrary M tensor field of type (0,2) on the tangent bundle T M. First we show that the geometric properties of (T M, G) to be either flat, or projectively flat, or conformally flat or locally symmetric are expressed only in the terms of the Levi Civita connection of g on M and of the symmetric part of the considered M tensor field on T M. Next, in the particular case where the considered M tensor field is independent of the tangential coordinates we get the necessary ans suffi-

2 68 V. OPROIU and N. PAPAGHIUC 2 cient conditions under which the pseudo Riemannian manifold (T M, G) is either flat, or projectively flat, or conformally flat, or lacally symmetric. In section 2 we define an almost complex structure J on T M and we get the conditions under which the almost complex manifold with Norden metric (T M, J, G) is a Kaehlerian manifold with Norden metric. In section 3, we consider a naturally defined almost product structure P on T M and study the conditions under which the almost parahermitian manifold (T M, G, P ) is a parakaehlerian manifold. Finally, some classes of manifolds whose tangent bundles carry parakaehlerian structures are also presented (Theorems 7, 8, 9 and 10). The manifolds, tensor fields and geometric objects we consider in this paper, are assumed to be differentiable of class C. We use the well known summation convention, the range for the indices i, j, k, h, l, s, t being always {1, 2,..., n}. We shall denote by Γ(T M) the module of smooth vector fields on T M. 1. The pseudo Riemannian manifold (T M, G). Let M be an n dimensional manifold and denote by τ: T M M its tangent bundle. Then T M is a 2n dimensional manifold and some special local charts on T M induced from local charts on M may be used. Namely, if (U, x i ); i = 1,..., n is a local chart on M, then the local chart (τ 1 (U), x i τ, y i ); i = 1,..., n is defined on T M where y 1,..., y n are the vector space coordinates of a element from τ 1 (U) with respect to the natural frame ( x,..., 1 x ) n defined by the local chart (U, x i ); i = 1,..., n. We shall denote, by an abuse of notation, x i = x i τ, so x i are considered simultaneously as local coordinates on M and on T M. The M tensor fields and the linear M connections may be considered on T M and the usual tensor fields and linear connections on the base manifold M may be thought of naturally as M tensor fields and linear M connections on T M (sec [5], [8]). The tangent bundle T T M of T M has an integrable vector subbundle V T M = Ker τ, called the vertical distribution on T M. A nonlinear connection on T M is defined by a distribution HT M, complementary to V T M in T T M (called the horizontal distribution) (1) T T M = V T M HT M. In the following we assume that M is a (pseudo )Riemannian manifold with the (pseudo )Riemannian metric g = g ij (x)dx i dx j

3 3 (PSEUDO-) RIEMANNIAN MANIFOLD 69 and denote by g ij the components of the inverse of the matrix (g ij ); i, j, = 1,..., n, i.e. g ik g jk = δ j i. Denote by Γ k ij the connection coefficients of the Levi Civita connection of g (i.e. the Christoffel symbols) and let a ij ; i, j = 1,..., n be the components of an arbitrary M tensor field of type (0,2) on T M. We think of g ij as the components of an M tensor field of type (0,2) on T M and consider the functions (2) N i j = Γ i jky k + g ih a hj ; i, j = 1,..., n. It follows that these functions are the connection coefficients of a nonlinear connection which defines a horizontal distribution denoted by HT M on T M. The system of the local vector fields ( i = y ); i = 1,..., n is a i local frame in V T M and the system of local vector fields (δ i = δ δx ); i = i 1,..., n, where (3) δ i = δ δx i = x i N j i y j is a local frame in HT M. Then ( i, δ i ); i = 1,..., n is a local frame on T M adapted to the direct sum decomposition (1). The system of local 1 froms (δy i, dx i ); i = 1,..., n, where (4) δy i = dy i + N i jdx j is the dual local frame of the local frame ( i, δ i ); i = 1,..., n. Then we have, as usual: (5) where [ ] y i, δ δx j = Φ k ji y k ; [ ] δ δx i, δ δx j = Rij k y k, (6) Φ k ji = N k j y i ; Rk ij = δn k j δx i δn k i δx j and the integrability of the system defined by HT M on T M is equivalent to the vanishing of the components R k ij on τ 1 (U).

4 70 V. OPROIU and N. PAPAGHIUC 4 Remark that the components Φ k ji define a linear M connection and the components Rij k define an M tensor field of type (1,2) on T M. Consider the following pseudo Riemannian metric G of complete lift type on T M (7) G = 2g ij δy i dx j = 2g ij dy i dx j + 2g ij Γ i hk yk dx h dx j + +(a ij + a ji )dx i dx j = 2g ij dy i dx j + + g ij x k y k dx i dx j + (a ij + a ji )dx i dx j. Remark that the pseudo Riemannian metric G depends only on (the pseudo )Riemannian metric g on M and on the symmetric part of the M tensor field on T M defined by the components a ij. Denote by the Levi Civita connection of the considered pseudo Riemannian metric G on T M. Then the following result is proved by a straightforward computation. Proposition 1. The local coordinate expression of in the local frame ( i, δ i ) adapted to the direct sum decomposition (1) is: where i j = 0; δi δ j = i δ j = g kh b jh y i ( y k ; δi j = Γ k ij + g kh c ) hi y j y k ; ( Γ k ij g kh c ) ij δ y h δx k gkh (R ijh R jhi R hij ) y k, R hij = g hk R k ij and c ij (respectively b ij ) denotes the simmetric part (respectively the skew symmetric part) of a ij, i.e. c ij = 1 2 (a ij + a ji ) and b ij = 1 2 (a ij a ji ). Remark. From Proposition 1 it follows that the esential coefficients of the local coordinate expression of in the local adapted frame ( i, δ i ); i = 1,..., n are expressed by using only the components of the (pseudo )Riemannian metric g on M, the coefficients of the Levi Civita connection of g, the simmetric part and the skew symmetric part of the M tensor field defined by the components a ij on T M and the M tensor field on T M defined by 1 2 (R ijh R jhi R hij ). On the other hand, taking into account that the pseudo Riemannian metric G depends only on the components of the

5 5 (PSEUDO-) RIEMANNIAN MANIFOLD 71 (pseudo )Riemannian metric g on M and on the simmetric part of the M tensor field defined by the components a ij on T M we can consider the nonlinear connection HT M defined by the connection coefficients N i j: N i j = i jky k + g ih c hj ; i, j = 1,..., n, where c ij denotes the simmetric part of a ij, i.e. the vector fields (δ i = δ δx i ; i = 1,..., n, where c ij = 1 2 (a ij + a ji ). Then δ i = δ δx i = x i N j i y j define a local frame in the horizontal distribution HT M and the vector fields ( i, δ i ) : i = 1,..., n define a local frame in T M adapted to the direct sum decomposition (8) T T M = V T M HT M. Denote by (δy i, dx i ); i = 1..., n the dual local frame of the local frame ( i, δ i ); i = 1,..., n. Then we have δy i = dy i + N i jdx j and the pseudo Riemannian metric G defined by (7) on T M becomes G = 2g ij δy i dx j = 2g ij dy i dx j + g ij x k yk dx i dx j + 2c ij dx i dx j. Since we have δ i = δ i g jh b hi j, where b ij denotes the skew symmetric part of a ij, from Proposition 1 we obtain by a straightforward computation. Proposition 2. The local coordinate expression of the Levi Civita connection of G in the local frame ( i, δ i ); i = 1,..., n adapted to the direct sum decomposition (8) is: i j = 0; i δ j = 0; δi j = ( Γ k ij + g kh c ) hi y i y k ;

6 72 V. OPROIU and N. PAPAGHIUC 6 where R ijk are defined by ( δi δj = Γ k ij g kh c ) ij δ y h δx k + gkh R ijh y k, R hij = g hk R k ij; R k ij = δ i N k j δ j N k i. Remark. From Proposition 2 it follows that the esential coefficients of the local coordinate expression of in the local adapted frame ( i, δ i ); i = 1,..., n are expresed by using only the (pseudo )Riemannian metric g on M, the coefficients of the Levi Civita connection of g, the components of the symmetric M tensor field defined by c ij and the components R kij defined above. Since the geometric properties of the pseudo Riemannian manifold (T M, G) to be either flat, or projectively flat, or conformally flat or locally symmetric are independent of the choice of the horizontal distribution HT M or HT M on T M, from Proposition 2 we have Theorem 3. The geometric properties of the pseudo Riemannian manifold (T M, G) to be either flat, or projectively flat, or conformally flat or locally symmetric which depend only on the metric G and its Levi Civita connection are expressed only in the terms of the nonlinear connection defined by N i j on T M. In the following we give the conditions under which the pseudo Riemannian manifold (T M, G) is respectively flat, projectively flat, conformally flat and locally symmetric, assuming that the M tensor field defined by the components a ij is independent of the tangential coordinates y i, i.e. the components a ij define a tensor field of type (0,2) on the base manifold M thought of as an M tensor field on T M. We obtain by a straightforward computation the following result. Theorem 4. Let (M, g) be a (pseudo )Riemannian manifold and denote by Γ k ij the coefficients of the Levi Civita connection of g. Consider on T M the nonlinear connection defined by (2), where the components a ij define a tensor field of type (0,2) on M thought of as an M tensor field on T M. Denote by (T M, G) the pseudo Riemannian manifold T M endowed with the pseudo Riemannian metric G, where G is given by (7). Then we have: (i) (T M, G) is flat if and only if (M, g) is flat and the symmetric part c ij of a ij satisfies the condition (9) i ( k c jh h c jk ) j ( k c ih h c ik ) = 0,

7 7 (PSEUDO-) RIEMANNIAN MANIFOLD 73 where i c jk are the local components of the covariant derivative of the tensor field on M defined by c jk with respect to. (ii) (T M, G) is projectively flat if and only if (T M, G) is flat. (iii) (T M, G) is conformally flat if and only if (M, g) has constant sectional curvature and the simmetric part c ij of a ij satisfies the condition (10) i ( k c jl l c jk ) j ( k c il l c ik ) = = r n(n 1) (c jkg il c jl g ik c ik g jl + c il g jk ), where r denotes the scalar curvature of (M, g) and n > 2 is the dimension of M. (iv) (T M, G) is locally symmetric if and only if (M, g) is locally symmetric and the simmetric part c ij of a ij satisfies the condition (11) h i ( k c jl l c jk ) h j ( k c il l c ik )+ + h c is R s jkl h c js R s ikl+( s c hl l c hs )R s kij ( i c hs s c hi )R s jlk+ + ( j c hs s c hj )R s ilk ( k c hs s c hk )R s lij = 0, where Rkij h are the local coordinate componenets of the curvature field of the Levi Civita connection on (M, g). Remarks. (i) If c ij = 0 then the conditions (9), (10), (11) are identically fulfilled. (ii) If c ij is parallel with respect to, then the conditions (9) and (11) are identically fulfilled too. (iii) If c ij satisfies the relation i c jk j c ik = k ω ij, where the components ω ij define a 2 form on M and if (M, g) is flat then the condition (9) is identically verified. In particular, if the 2 form defined by the components ω ij is parallel then the components c ij define a Codazzi tensor field and (9) is identically verified. 2. An almost complex structure on (T M, G). Let (M, g) be a (pseudo )Riemannian manifold and let (T M, G) be the pseudo Riemannian manifold with G defined by (7) where the considered nonlinear connection on T M is given by (2). We can define the following almost complex structure J on T M: (12) J ( ) δ δx i = y i ; J ( ) y i = δ δx i.

8 74 V. OPROIU and N. PAPAGHIUC 8 Then, according with the terminology from [2], we may verify that G and J define an almost complex structure with Norden metric on T M (or equivalently, (T M, J, G) is a hyperbolic almost Hermitian manifold), i.e. (13) G(JX, JY ) = G(X, Y ); X, Y Γ(T M). On T M we consider the following tensor field F of type (0,3) defined by (see [2]) (14) F (X, Y, Z) = G(( X J)Y, Z); X, Y, Z Γ(T M), where denotes the Levi Civita connection on (T M, G). Then the almost complex manifold with Norden metric (T M, J, G) is a Kaehlerian manifold with Norden metric (or an hyperbolic Kaehlerian manifold) if F (X, Y, Z) = 0; X, Y, Z Γ(T M), or, equivalently, if J = 0. By using (12), (14) and Proposition 1 we get by a straightforward computation ( ) ( ) δ δ F y, i y, = F j y k y, i δx, = 0; j δx ( ) ( ) k δ δ F y, (15) i y, = F j δx k y,, i δx k y = b kj j y ; ( ) i δ F δx, i y, = F ( ) δ δ δ c j y k δx, i δx, j δx = ij + c k y k ik y ; ( ) ( ) j δ δ δ δ F δx, i y, = F j δx k δx,, i δx k y = 1 j 2 (R kij + R jki R ijk ). Examining the above relations it follows that the condition F = 0 which must be fulfilled for (T M, G) to be a Kaehlerian manifold with Norden metric is reduced to (16) (i) b ij y k = 0; (ii) c ij y k + c ik y j = 0; (iii)r kij + R jki R ijk = 0. From (16)(i) and (16)(ii) it follows that a ij must be independent of y k, i.e. the components a ij define a tensor field of type (0,2) on the base manifold M thought of as an M tensor field on T M. Next, using (2), (3)

9 9 (PSEUDO-) RIEMANNIAN MANIFOLD 75 and the second relation (6) we obtain by a straightforward computation that the condition (16)(iii) becomes (17) y h R hijk + k c ij j c ik i b jk = 0, where R hijk denotes the local coordinate components of the Riemann Chris toffel tensor of on M and c ij (respectively b ij ) is the symmetric part (respectively the skew symmetric part) of a ij. Hence we may state Theorem 5. The almost complex manifold with Norden metric (T M, J, G) is a Kaehlerian manifold with Norden metric if and only if the components a ij are independent of y k, the Levi Civita connection of g is flat and the symmetric part c ij and the skew symmetric part b ij of a ij satisfy the condition (18) k c ij j c ik = i b jk. Remark. The condition (18) implies that the 2 form defined by the components b ij is closed. Then the condition (18) is equivalent to i a jk = j a ik and Theorem 5 may be formulated as follows: The almost complex manifold with Norden metric (T M, J, G) is Kaehlerian with Norden metric if and only if the components a ij define a tensor field on the base manifold M, the 2 form defined by the components b ij is closed and the horizontal distribution HT M defined by the nonlinear connection coefficients Nj i on T M is involutive. Remark also that if c ij define a Codazzi tensor field with respect to and b ij is parallel with respect to, then the condition (18) is identically satisfied. 3. An almost product structure on (T M, G). In this section we consider a nonlinear connection on T M defined by its connection coefficients of the form (2) and define and almost product structure P on T M determined by the distributions V T M and HT M. Next we obtain the conditions under which the almost parahermitian manifold (T M, G, P ) is a parakaehlerian manifold (see [1]). Consider on T M the almost product structure P natuarlly defined by the direct sum decomposition (1), i.e. (19) P ( ) y i = y i ; P ( ) δ δx i = δ δx i.

10 76 V. OPROIU and N. PAPAGHIUC 10 It follows easily that G(P X, P Y ) = G(X, Y ); X, Y Γ(T M), therefore (T M, G, P ) is an almost parahermitian manifold (see [1]). Define the 2 form Ω associated with the almost parahermitian structure (G, P ) on T M by (20) Ω(X, Y ) = G(P X, Y ); X, Y Γ(T M). According to the terminology from [1] we have that (T M, G, P ) is a parakaehlerian manifold if Ω vanishes identically on T M. Using (19), (20) and Proposition 1 we obtain by a straightforward computation (21) ( i Ω)( j, k )=( i ω)( j, δ k )=( δi Ω)( j, k )=( δi Ω)( j, δ k )=0 ( i Ω)(δ j, δ k ) = 2 b kj y i ; ( δi Ω)(δ j, δ k ) = R kij + R jki R ijk. Then the condition Ω = 0 which must be fulfilled for (T M, G, P ) to be a parakaehlerian manifold is reduced to (22) (i) b kj y i = 0; (ii) R kij + R jki = R ijk. From (22)(i) it follows that the skew symmetric part b ij of a ij does not depend on the tangential components y k, i.e. the components a ij are of the form (23) a ij (x, y) = c ij (x, y) + b ij (x), where c ij (x, y) are the components of a symmetric M tensor field of type (0,2) on T M and b ij (x) are the components of a skew symmetric tensor field of type (0,2) on M thouhgt of as a skew symmetric M tensor field on T M. Hence we have Theorem 6. The almost parahermitian manifold (T M, G, P ) where G is defined by (7) and the nonlinear connection is defined by (2) with a ij of the form (23) is a parakaehlerian manifold if and only if the components R kij satisfy the condition (22)(ii). In order to obtain some classes of manifolds whose tangent bundles carry parakaehlerian structures, we consider on T M some particular M tensor fields of the form (23) and we study the conditions under which (T M, G, P ) is a parakaehlerian manifold.

11 11 (PSEUDO-) RIEMANNIAN MANIFOLD 77 Firstly, we consider on T M the M tensor field defined by the components a ij : (24) a ij = c ij + b ij, where c ij (respectively b ij ) are the components of a symmetric (respectively skew symmetric) tensor field on M thouhgt of as an M tensor field on T M. Then, using (2), (6) and (24) we get by a straightforward computation R kij = R khij y h + i a kj j a ki, where R khij are the local coordinate components of the Riemann Christoffel tensor field defined by on (M, g). Next, the condition (22)(ii) becomes (25) R khij y h + i c kj j c ki + k b ij = 0, hence we have Theorem 7. Let the nonlinear connection Nj i on T M be defined by (2) where a ij are given by (24). Then (T M, G, P ) is a parakaehlerian manifold if and only if the (peudo )Riemannian manifold (M, g) is flat and the components c ij and b ij satisfy the condition (26) i c kj j c ki = k b ji. where Remark. The condition (25) is equivalent to R kij + i b jk = 0, (i,j,k) (i,j,k) denotes the sum consisting of three terms obtained by cyclic permutations of i, j, k. Thus, Theorem 7 becomes: Under the hypothesis of Theorem 7 we have that (T M, G, P ) is a parakaehlerian manifold if and only if the horizontal distribution HT M on T M is involutive and the 2 form defined by the components b ij is closed. Now, we consider on T M the M tensor field defined by the components a ij of the form (23), with

12 78 V. OPROIU and N. PAPAGHIUC 12 (27) a ij = kg ih g jl y h y l + t ij, where k is a nonzero constant and the components t ij define and arbitrary tensor field of type (0,2) on M thought of as an M tensor field on T M. Using (2), (6) and (27) we obtain by a strainghtforward computation (28) R ijk = [R iljk + k(t ik g jl t ij g kl + t jk g il t kj g il )]y l + j t ik k t ij. By using (28) it follows that the condition (22)(ii) becomes (29) [R iljk + k(t ik g jl t ij g kl + t jk g il t kj g il )]y l + + j h ik k h ij + i b jk = 0, where h ij (respectively b ij ) denotes the symmetric part (respectively the skew symmetric part) of t ij. The condition (29) is equivalent to (30) (i) R h ijk = k(t ikδ h j t ijδ h k + t jkδ h i t kjδ h i ); (ii) j h ik k h ij = i b kj, where Rijk h are the local coordinate components of the curvature tensor field of the Levi Civita connection an M. By taking into account that is the Levi Civita connection of g, we have from (30)(i) (31) t ij = 1 k(n 1) R ij, where R ij are the local coordinate components of the Ricci tensor defined by the curvature tensor of, i.e. we have t ij = h ij and b ij = 0. Thus, the condition (30)(i) implies that the (pseudo )Riemannian manifold (M, g) has constant sectional curvature, next replacing the expression of t ij from (31) in (30)(i) and using the second Bianchi identity we get that the condition (30)(ii) is identically verified. Hence we may state

13 13 (PSEUDO-) RIEMANNIAN MANIFOLD 79 Theorem 8. Let (M, g) be a (pseudo )Riemannian manifold with dimm = n > 2. Consider on T M the nonlinear connection defined by N k i = Γ k ijy j + kg ih y h y k + 1 k(n 1) gkh R hi, where Γ k ij are the connection coefficients of the Levi Civita connection of g, R ij are the local coordinate components of the Ricci tensor defined by the curvature tensor of and k is a nonzero constant. The the almost parahermitian manifold (T M, G, P ) where G is given by (7) and P is defined by (19) is a parakaehlerian manifold if and only if the (pseudo )Riemannian manifold (M, g) has constant sectional curvature. More parakaehlerian structures on tangent bundles can be obtained in the cases of complex and quaternion manifolds. Let (M, g, F ) be a Kaehler manifold with the almost complex structure defined by the tensor field F of type (1,1) such that F 2 = I and denote by the Levi Civita connection of g such that F = 0 (see [10]). Moreover, we have g(f X, F Y ) = g(x, Y ); X, Y Γ(M). Consider on T M the M tensor field defined by the components a ij of the form (23): (32) a ij = k(g ih g jk g is F s hg jt F t k)y h y k + t ij, where Fi h are the components of F, k is a nonzero constant and t ij are the components of an arbitrary tensor field of type (0,2) on M thought of as an M tensor field on T M. In this case, using (2), (6) and (32) we obtain by a straightforward computation R ijk = {R iljk + k [ t ik g jl t ij g kl + t jk g il t kj g il + t hj F h k F s i g sl t hk F h j F s i g sl + t hj F h i F s k g sl t hk F h i F s j g sl] }y l + j t ik k t ij, where R iljk are the local coordinate components of the Riemann Christoffel tensor on (M, g). Using the above expression of R ijk we obtain by a straightforward computation that the condition (22)(ii) which must be fulfilled for (T M, G, P ) to be a parakaehlerian manifold is equivalent to the following two conditions (33) (i) Rkij h = k(t kjδi h t kiδj h t jiδk h + t ijδk h + t lifj lf k h t lj Fi lf k h + t lifk lf j h t ljfk lf i h); (ii) i h jk j h ik + k b ij = 0,

14 80 V. OPROIU and N. PAPAGHIUC 14 where Rkij h are the local coordinate components of the curvature tensor field of on M and h ij (respectively b ij ) denotes the symmetric part (respectively the skew symmetric part) of t ij. From (33)(i) it follows (34) t ij = 1 k(n 2 4) [nr ij 2F h i F k j R hk ]. Replacing this expression of t ij in (33)(i), next using the second Bianchi identity we get that the condition (33)(ii) is identically verified. Hence we may state Theorem 9. Let (M, g, F ) be a Kaehler manifold with real dimension n > 2 and consider on T M the nonlinear connection defined by N k i = Γ k ijy j + k(g ih y k y h Fl k g is Fhy s l y h 1 ) + k(n 2 4) gkh (nr hi 2FhF l i s R ls ), where Γ k ij are the connection coefficients of the Levi Civita connection of g, R ij are the components of the Ricci tensor defined by the curvature tensor of, F j i are the components of F and k is a nonzero constant. Then the almost parahermitian manifold (T M, G, P ) where G is defined by (7) and P is defined by (19) is a parakaehlerian manifold if and only if (M, g, F ) has constant holomorphic sectional curvature. Consider now (M, g, S) a quaternion Kaehler manifold. Then M is a 4m dimensional manifold, S is a subbundle with fibre dimension 3 of the vector bundle of tensors of type (1,1) on M and, locally, S has a canonical base (F 1, F 2, F 3 ) such that F 2 α = I; F α F β = F β F α = F γ, where α = 1, 2, 3 and (α, β, γ) is any cyclic permutation of (1,2,3). Moreover, we have g(f α X, F α Y ) = g(x, Y ); X, Y Γ(T M), α = 1, 2, 3. Denote by the Levi Civita connection of g which preserves S, i.e., locally F α = η αβ F β, β=1,2,3

15 15 (PSEUDO-) RIEMANNIAN MANIFOLD 81 where η αβ are locally defined 1 forms adapted with the Levi Civita connection of g (see [4]) and η αβ = η βα. Consider on T M the M tensor field defined by the components a ij of the form (23): (35) a ij = k{g ih g jk y h y k α=1,2,3 g is (F α ) s hg jt (F α ) t ky h y k } + t ij, where (F α ) k i are the local coordinate components of F α, k is a nonzero constant and t ij are the components of an arbitrary tensor field of type (0,2) on M thought of as an M tensor field on T M. Then using (2), (6) and (35) we obtain by a straightforward computation (36) R ijk = {R iljk + k[t ik g jl t ij g kl + t jk g il t kj g il + + (t hj (F α ) h k (F α) s i g sl t hk (F α ) h j (F α) s i g sl+ α=1,2,3 +t hj (F α ) h i (F α) s k g sl t hk (F α ) h i (F α) s j g sl)]}y l + j t ik k t ij. By using (36) it follows by a straightforward computation that the condition (22)(ii) is equivalent to the following two relations (37) (i) Rkij h =k{t kjδi h t kiδj h t jiδk h+t ijδk h+ [ (F α ) l i (F α) h k t lj+ α=1,2,3 +(F α ) l j (F α) h k t li + (F α ) l k (F α) h j t li (F α ) l k (F α) h i t lj]}, (ii) i h jk j h ik = k b ji, where h ij (respectively b ij ), denotes the symmetric part (respectively the skew symmetric part) of t ij. From the condition (37)(i) we obtain by a strainghtforward computation (38) t ij = 1 8km(m + 2) [(2m + 3)R ij α=1,2,3 (F α ) h i (F α ) k j R hk ]. Replacing this expression of t ij in (37)(i), next using the second Bianchi identity we get that the condition (37)(ii) is identically verified. Hence we may state

16 82 V. OPROIU and N. PAPAGHIUC 16 Theorem 10. Let (M, g, S) be a quaterninon Kaehler manifold with dimm > 4. Consider on T M the nonlinear conection defined by Ni k, where N k i = Γ k ij yj + k[g ih y h y k + 1 α=1,2,3 8km(m+2) gkh [(2m + 3)R hi (F α ) k l g is(f α ) s h yl y h ]+ α=1,2,3 (F α ) s h (F α) t i R st], where Γ k ij are the coefficients of the Levi Civita connection on (M, g, S), k is a nonzero constant and R ij are the components of the Ricci tensor defined by the curvature tensor of. Then the almost parahermitian manifold (T M, G, P ) where G is given by (7) and P is defined by (19) is a parakaehlerian manifold if and only if the quaternion Kaehler manifold (M, g, S) has constant Q sectional curvature (see [4]). REFERENCES 1. Ga d e a, P.M., Masqué, J.M. Clasification of almost parahermitian manifolds, Rend. Mat., serie VII, 11(1991), G a n c h e v, G.T. and B o r i s o v, D.V. Note on the almost complex manifolds with Norden metric, Compt. Rend. Acad. Bulg. Sci. 39(1986), O p r o i u, V. A pseudo Riemannian structure in Lagrange geometry, An. St. Univ. Al.I.Cuza Iaşi, 33(1987), O p r o i u, V. Integrability of almost quaternal structure, An. St. Univ. Al.I.Cuza Iaşi, 30(5)(1984), O p r o i u, V. and P a p a g h i u c, N. A pseudo Riemannian structure on the cotangent bundle, An. St. Univ. Al.I.Cuza Iaşi, 36(1990), O p r o i u, V. and P a p a g h i u c, N. Locally symmetric cotangent bundles, Matematicki Vesnik, 42(1990), P a t e r s o n, E.M., and W a l k e r, A.G. Riemann extensions, Quart. J. Math., (2) 3(1952), W i l m o r e, T.J. Riemann extensions of manifolds with torsion free affine connection, Tagungsberight 48/1986, Affine Differential geometrie, Mathematisches Forschungsinstitut, Oberwolfach,

17 17 (PSEUDO-) RIEMANNIAN MANIFOLD W i l m o r e, T.J. Riemann extensions and affine differential geometry, Results in Mathematics, Vol. 13(1988), Y a n o, K. Differential Geometry on Complex and Almost Complex Spaces, Pergamon Press, New York, Received: 31.V.1996 Faculty of Mathematics University Al.I.Cuza, Iaşi ROMANIA e mail: voproiu@uaic.ro Departament of Mathematics Technical University, Iaşi ROMANIA npapag@math.tuiasi.ro