Vrănceanu connections and foliations with bundle-like metrics

Transcription

1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 118, No. 1, February 2008, pp Printed in India Vrănceanu connections and foliations with bundle-like metrics AUREL BEJANCU and HANI REDA FARRAN Department of Mathematics and Computer Science, Kuwait University, P. O. Box 5969, Safat 13060, Kuwait MS received 1 August 2006; revised 30 September 2007 Abstract. We show that the Vrănceanu connection which was initially introduced on non-holonomic manifolds can be used to study the geometry of foliated manifolds. We prove that a foliation is totally geodesic with bundle-like metric if and only if this connection is a metric one. We introduce the notion of a foliated Riemannian manifold of constant transversal Vrănceanu curvature and the notion of a transversal Einstein foliated Riemannian manifold. The geometry of these two classes of manifolds is studied and the relationship between them is determined. Keywords. Foliation; Vrănceanu connection; foliation with bundle-like metric; foliated Riemannian manifold of constant transversal Vrănceanu curvature; transversal Einstein foliated Riemannian manifold. 0. Introduction Vrănceanu [11] introduced a linear connection for the study of differential geometry of non-holonomic manifolds. We show that this connection can be successfully used for a study of foliated Riemannian manifolds. In the first section we present the Vrănceanu connection on a foliated Riemannian manifold by using both the local coordinates and the coordinate-free approaches. We show here that a totally geodesic foliation with bundle-like metric is characterized by means of this connection. In the next section we introduce the transversal Vrănceanu curvature tensor field and relate it with the projection of the curvature tensor field of the Levi Civita connection on the transversal bundle. In the case when the Riemannian metric is bundlelike for the foliation [6], we develop a theory for foliated Riemannian manifolds of constant transversal Vrănceanu curvature. We show that Euclidean spaces do not admit (even locally) foliations of negative constant transversal Vrănceanu curvature. However, we find a large class of manifolds of positive constant transversal Vrănceanu curvature. Finally, we introduce transversal Einstein foliated Riemannian manifolds and show their interrelations with foliated Riemannian manifolds of constant transversal Vrănceanu curvature. 1. Vrănceanu connection on a foliated Riemannian manifold Let (M, g, F) be a foliated Riemannian manifold, where M is an (n + p)-dimensional manifold, g is a Riemannian metric and F is an n-foliation. Denote by D the distribution that is tangent to F and by D the complementary orthogonal distribution to D in TM. We call D the transversal distribution to the foliation F. 99

2 100 Aurel Bejancu and Hani Reda Farran Throughout the paper all manifolds are smooth and paracompact. We denote by F(M) the algebra of smooth functions (differentiable of class C )onm and by Ɣ(T M) the F(M)-module of smooth sections of TM. Similar notations will be used for any other manifold or vector bundle. Also, we use the Einstein convention, that is, repeated indices with one upper index and one lower index denote summation over their range. If not stated otherwise, we use indices: i,j,k,... {1,...,n} and α,β,γ,... {n + 1,...,n+ p}. The Vrănceanu connection on (M, g, F) is defined as follows: X Y = P PXPY + Q QX QY + P [QX, P Y ] + Q[PX,QY], (1.1) for any X, Y Ɣ(T M), where is the Levi Civita connection on (M, g), and P and Q are the projection morphisms of Ɣ(T M) on Ɣ(D) and Ɣ(D ) respectively. The coordinatefree formula (1.1) was given by Ianus [3] in the context of almost product manifolds endowed with a linear connection. The linear connection was defined first, using local coordinates, by Vrănceanu [11] on a non-holonomic manifold endowed with a linear connection (see also p. 235 of [12]). Here, by non-holonomic manifold we mean a manifold that is endowed with two complementary distributions, at least one of which is non-integrable. We would like to point out that many of the connections that are used in the literature to study foliated manifolds can be related to the Vrănceanu connection in one way or the other. For example, we mention some of them here. The Bott connection can be considered as the differential operator (cf. p. 19 of [9]) : Ɣ(D) Ɣ(D ) Ɣ(D ); PX QY = Q[PX,QY]. Then it is easy to see that the Bott connection is just the restriction of Vrănceanu connection on Ɣ(D) Ɣ(D ). Also, the linear connection defined by the formula (3.3) in p. 21 of [9] is the restriction of Vrănceanu connection on Ɣ(T M) Ɣ(D ). The adapted connection F defined by Reinhart (p. 147 of [7]) coincides with the Vrănceanu connection on (M, g, F). Finally, we note that the second connection defined by Vaisman [10] on (M, g, F) is also the Vrănceanu connection. From (1.1) we see that is an adapted linear connection on (M, g, F), that is, both distributions D and D are parallel with respect to. Moreover, by using Theorem 1.2 from [1] we deduce that is the only adapted linear connection on (M, g, F) satisfying ( PX g)(p Y, P Z) = 0, ( QX g)(qy, QZ) = 0, (c) P(T (X, P Y )) = 0, (d) Q(T (X, QY )) = 0, (1.2) for any X, Y, Z Ɣ(T M), where T is the torsion tensor field of. As most of the results of this paper refer to the geometry of the transversal distribution D we introduce the following geometric objects. The second fundamental form h of D is defined by h(qx, QY ) = P( QX QY ), X, Y Ɣ(D ). (1.3)

3 Vrănceanu connections and foliations 101 The symmetric second fundamental form h s of D is given by h s (QX, QY ) = 1 (h(qx, QY ) + h(qy, QX)). (1.4) 2 Also, for any PX Ɣ(D) we define the shape operator A PX of D by A PX QY = Q( QY PX). (1.5) As in the theory of submanifolds we deduce that g(h(qx, QY ), P Z) = g(a PZ QX, QY ), X, Y, Z Ɣ(T M). (1.6) The second fundamental form B of the foliation F is defined by B(PX,PY) = Q( PX PY), X, Y Ɣ(T M). (1.7) Thus B, restricted to a leaf of F, is just the second fundamental form of that leaf. Therefore B is a Ɣ(D )-valued symmetric bilinear form on Ɣ(D). We note that the Vrănceanu connection is neither torsion-free nor a metric connection. More precisely, we state the following. PROPOSITION 1.1 Let be the Vrănceanu connection on (M, g, F). Then we have the following assertions: (i) The torsion tensor field of is given by T (X, Y ) = h(qy, QX) h(qx, QY ), X, Y Ɣ(T M). (1.8) (ii) The covariant derivative of g with respect to satisfies (1.2a), (1.2b) and ( PX g)(p Y, QZ) = 0, ( PX g)(qy, QZ) = 2g(PX, hs (QY, QZ)), (c) ( QX g)(p Y, QZ) = 0, (d) ( QX g)(p Y, P Z) = 2g(QX, B(P Y, P Z)), (1.9) for any X, Y, Z Ɣ(T M). Proof. The first assertion follows by using (1.1), (1.3), and taking into account that is a torsion-free connection and D is an integrable distribution. Next, we have (1.9a) and (1.9c) since is an adapted linear connection on (M, g, F). Finally, (1.9b) and (1.9d) follow by direct calculations using (1.1), (1.3), (1.4) and (1.7), and taking into account that is a torsion-free and metric connection. Now, we recall two special classes of foliations. First if each leaf of F is totally geodesic immersed in (M, g) then F is called totally geodesic. Thus F is totally geodesic if and only if B = 0. Also, we say that g is a bundle-like metric for F if each geodesic in (M, g) which is tangent to the transversal distribution D at one point remains tangent for its

4 102 Aurel Bejancu and Hani Reda Farran entire length [6]. Then a necessary and sufficient condition for g to be bundle-like for F is that (cf. p. 46 of [9]) g(px, QY QZ + QZ QY ) = 0, X, Y, Z Ɣ(T M). (1.10) An interesting characterization of these two classes of foliations by means of the Vrănceanu connection is stated in the next theorem. Theorem 1.1. Let (M, g, F) be a foliated Riemannian manifold. Then we have: (i) F is totally geodesic if and only if ( QX g)(p Y, P Z) = 0, X, Y, Z Ɣ(T M). (1.11) (ii) g is bundle-like for F if and only if ( PX g)(qy, QZ) = 0, X, Y, Z Ɣ(T M). (1.12) Proof. The first assertion follows from (1.9d). Next, by using (1.10), (1.3) and (1.4) we deduce that g is bundle-like for F if and only if h s (QY, QZ) = 0, Y, Z Ɣ(T M). (1.13) Thus (1.12) follows from (1.9b) via (1.13). COROLLARY 1.1 Let (M, g, F) be a foliated Riemannian manifold. Then F is totally geodesic with bundlelike metric g if and only if the Vrănceanu connection is a metric connection. Proof. It follows from Theorem 1.1 taking into account (1.2a), (1.2b), (1.9a) and (1.9c). Next, we consider a foliated atlas on (M, g, F) and denote by (x i,x α ), i {1,...,n}, α {n + 1,...,n+ p}, the local coordinates on M, where (x i ) are the leaf coordinates, that is, / x i Ɣ(D). Then there exist functions A i α such that [6] x α = x α Ai α, α {n + 1,...,n+ p}, (1.14) xi is a local frame field in Ɣ(D ). We call { / x i, /x α } an adapted frame field on (M, g, F). By direct calculations using (1.14) we obtain [ ] x α, x β = Iαβ i x i, [ ] x α, x i = A j iα x (1.15) where we put I i αβ = Ai α x β Ai β x α, A j iα = Aj α x i. (1.16)

5 Vrănceanu connections and foliations 103 Also, g is locally represented by ( ) g ij = g x i, x j, ( ) g αβ = g x α, x β. (1.17) Then g is bundle-like for F if and only if [6] g αβ x i = 0, α, β {n + 1,...,n+ p}, i {1,...,n}. (1.18) We denote by g ij and g αβ the entries for the inverse matrices of [g ij ] and [g αβ ] respectively. PROPOSITION 1.2 The Vrănceanu connection on (M, g, F) is locally represented as follows: x α = F γ αβx γ, (c) (d) x β x i x α x j x α = 0, x i = A j iα x i = P k ij x j, where A j iαare given by (1.16b) and we put x k, (1.19) F γ αβ = 1 ( gμα 2 gγμ x β + g μβ x α g αβ x μ Pij k = 1 ( ghi 2 gkh x j + g hj x i g ij x h ), ). (1.20) Proof. By using (1.1) and (1.15b) we obtain (1.19b) and (1.19c). Now, from (1.1) we deduce that x β x j x α = Q x β x i = P x j x α and x i. (1.21) Then we recall that the Levi Civita connection on (M, g) is given by (cf. p. 61 of [5]) 2g( X Y, Z) = X(g(Y, Z)) + Y(g(Z, X)) Z(g(X, Y)) + g([x, Y ],Z) g([y, Z],X)+ g([z, X],Y), (1.22) for any X, Y, Z Ɣ(T M). Then (1.19a) and (1.19d) are obtained by using (1.21), (1.22) and (1.15a).

6 104 Aurel Bejancu and Hani Reda Farran Now, we consider the curvature tensor field R of and put (c) R ( x γ, R ( x i, ) x β ) x β R ( x j, x i x α = R μ αβγ x μ, x α = R μ αβix μ, ) x α = R μ αij x μ. (1.23) Taking into account that R is given by (cf. p. 133 of [4]) R (X, Y )Z = X Y Z Y X Z [X,Y ] Z, (1.24) and using (1.23), (1.15), (1.19a) and (1.19b) we obtain (c) R μ αβγ = F μ αβ x γ F αγ μ x β + F αβ ɛ F ɛγ μ F αγ ɛ F μ ɛβ, R μ αβi = Fμ αβ x i, R μ αij = 0. (1.25) In particular, we state the following. PROPOSITION 1.3 Let (M, g, F) be a foliated manifold such that g is bundle-like for F. Then we have (c) R μ αβγ = Fμ αβ x γ Fμ αγ x β + F αβ ɛ F ɛγ μ F αγ ɛ F μ ɛβ, R μ αβi = 0, R μ αij = 0. (1.26) Proof. By using (1.14) and (1.18) in (1.20a) we deduce that F γ αβ = 1 2 gγμ ( gμα x β + g μβ x α g ) αβ x μ. Hence F γ αβ are functions of (xn+1,...,x n+p ) alone since g αβ and g αβ are so. Then (1.26) follows from (1.25) via (1.14). Finally, we introduce some geometric objects related to the differential geometry of the transversal distribution. A transversal tensor field of type (r, s) on (M, g, F) is an F(M) (r + s)-multilinear mapping T : (Ɣ(D )) r (Ɣ(D )) s F(M),

7 Vrănceanu connections and foliations 105 where D is the dual vector bundle to D. Locally, T is given by p r+s smooth functions satisfying T α 1...α r β 1...β s T α 1...α r x γ 1 xγr β 1...β s x α 1 x α r = T γ 1...γ r x ɛ 1 xɛs ɛ 1...ɛ s x β 1 x β. s It is easy to see that a transversal tensor field is a particular adapted tensor field on (M, g, F) [2]. As examples of transversal tensor fields we present g αβ,g αβ and R μ αβγ. Next, by using the Vrănceanu connection we define the transversal Vrănceanu covariant derivative of T as follows: T α 1...α r β 1...β s γ = T α1...αr β 1...β s x γ + Then from (1.2b) we deduce that r x=1 T α 1...ɛ...α r β 1...β s F α x s ɛγ y=1 T α 1...α r β 1...ɛ...β s F ɛ β y γ. (1.27) g αβ γ = 0 and g αβ γ = 0. (1.28) If in particular, the local components of a transversal tensor field T are functions of (x n+1,...,x n+p ) alone then we say that T is a basic transversal tensor field. Then by using (1.18) and (1.26) we obtain the following. PROPOSITION 1.4 Let (M, g, F) be a foliated Riemannian manifold where g is bundle-like for F. Then g αβ,g αβ and R μ αβγ define basic transversal tensor fields on (M, g, F). 2. Foliated Riemannian manifolds of constant transversal Vrănceanu curvature Let R be the curvature tensor field of Vrănceanu connection on the foliated Riemannian manifold (M, g, F). Then we define R : (Ɣ(D )) 4 F(M); R (QU, QZ, QX, QY ) = g(r (QX, QY )QZ, QU), (2.1) and call it the transversal Vrănceanu curvature tensor field. Now, we want to relate the above R with the curvature tensor field R of the Levi Civita connection on (M, g). First we prove the following. Lemma 2.1. Let (M, g, F) be a foliated Riemannian manifold. Then we have QR(QX, QY )QZ = R (QX, QY )QZ + A h(qx,qz) QY for any X, Y, Z Ɣ(T M). Proof. First, by using (1.1) and (1.3) we obtain A h(qy,qz) QX + A P [QX,QY ] QZ, (2.2) QX QY = QX QY + h(qx, QY ), X, Y Ɣ(T M). (2.3)

8 106 Aurel Bejancu and Hani Reda Farran Then by direct calculation using (2.3), (1.1) and (1.5) we infer that Q QX QY QZ = QX QY QZ A h(qy,qz)qx. (2.4) Also, taking into account that is torsion-free and by using again (1.1) and (1.5) we deduce that Q [QX,QY ] QZ = Q P [QX,QY ] QZ + Q[QX,QY ] QZ = Q P [QX,QY ] QZ + [QX,QY ] QZ Q[P [QX, QY ],QZ] = Q QZ P [QX, QY ] + [QX,QY ] QZ = [QX,QY ] QZ A P [QX,QY ]QZ. (2.5) Then (2.2) follows by using (2.4), (2.5) and formulas as (1.24) for both R and R. Now, we put and state the following. R(QU, QZ, QX, QY ) = g(r(qx, QY )QZ, QU), (2.6) Theorem 2.1. Let (M, g, F) be a foliated Riemannian manifold such that g is bundle-like for F. Then we have R(QU, QZ, QX, QY ) for any X, Y, Z, U Ɣ(T M). = R (QU, QZ, QX, QY ) + g(h(qx, QZ), h(qy, QU)) g(h(qx, QU), h(qy, QZ)) + 2g(h(QX, QY ), h(qz, QU)), (2.7) Proof. First, taking into account that is torsion-free and using (1.3) and (1.13) we deduce that P [QX, QY ] = 2h(QX, QY ), X, Y Ɣ(T M). (2.8) Then (2.7) follows from (2.2) by using (2.1), (2.6), (1.6) and (2.8). Next, we write down the Bianchi identities for restricted to the transversal distribution (see p. 135 of [4]) {( QX T )(QY, QZ) + T (T (QX,QY),QZ) R (QX, QY )QZ} =0 (QX,QY,QZ) and (QX,QY,QZ) {( QX R )(QY, QZ) + R (T (QX,QY),QZ)}(QU) = 0, where (QX,QY,QZ) means cyclic sum with respect to (QX, QY, QZ). (2.9) (2.10)

9 Vrănceanu connections and foliations 107 Theorem 2.2. Let (M, g, F) be a foliated Riemannian manifold such that g is bundle-like for F. Then the transversal Vrănceanu curvature tensor field satisfies the identities: R (QU, QZ, QX, QY ) + R (QU, QZ, QY, QX) = 0, R (QU, QZ, QX, QY ) + R (QZ, QU, QX, QY ) = 0, (c) {R (QU, QZ, QX, QY )} =0, (QX,QY,QZ) (d) (e) R (QU, QZ, QX, QY ) = R (QX, QY, QU, QZ), {( QZ R )(QU, QV, QX, QY )} =0, (2.11) (QX,QY,QZ) for any X, Y, Z, U, V Ɣ(T M). Proof. First we note that (2.11a) is a well-known property of any curvature tensor field. Then (2.11b) follows from (2.7) by using (1.13) and taking into account that R satisfies an identity of that type. Now we examine (2.9). By (1.8) we deduce that T (X, Y ) Ɣ(D), X, Y, Ɣ(T M). (2.12) Then we apply again (1.8) and obtain T (T (QX,QY),QZ)= 0, X, Y, Z Ɣ(T M). (2.13) Also taking into account that D is parallel with respect to and by using (2.12) we infer that ( QX T )(QY, QZ) Ɣ(D), X, Y, Z, Ɣ(T M). (2.14) Thus (2.11c) follows from (2.9) by using (2.13) and (2.14). Next, (2.11d) is obtained by a combinatorial exercise using (2.11a), (2.11b) and (2.11c) (see p. 75 of [5] for R). Finally, we inspect (2.10). By (2.12) and (1.26b) we deduce that R (T (QX,QY),QZ)= 0, X, Y, Z Ɣ(T M). (2.15) Then (2.11e) follows from (2.10) by using (2.1), (1.2b) and (2.15). Now, we consider a 2-dimensional subspace of Dx which we call a transversal plane to F. Then take a basis {U,V } in and define the number K (U, V ) = R (U,V,U,V), (2.16) (U, V ) where (U, V ) is a positive number given by (U, V ) = g(u, U)g(V, V ) g(u, V ) 2. (2.17) By using (1.11a), (1.11b), (2.16) and (2.17) it is easy to check that K (U, V ) is independent of the basis {U,V } of. Thus we assign to any transversal plane at x the number K () = K (U, V ), (2.18)

10 108 Aurel Bejancu and Hani Reda Farran where {U,V} is a basis of. The transversal Vrănceanu sectional curvature of M is the real-valued function K on the set of transversal planes of F given by (2.18). When K is a constant on M, we say that (M, g, F) is of constant transversal Vrănceanu curvature. By similar reasons as in the case of Riemannian spaces of constant curvature (see p. 80 of [5]) we conclude that when (M, g, F) is of constant transversal Vrănceanu curvature c, then the transversal Vrănceanu curvature tensor field is expressed as follows: R (QU, QZ, QX, QY ) = c{g(qx, QU)g(QY, QZ) g(qy, QU)g(QX, QZ)}, (2.19) for any X, Y, Z, U Ɣ(T M). Theorem 2.3. Let M be an open submanifold of the Euclidean space (R n+p,g) and (M, g, F) be a foliated Riemannian manifold, where g is a bundle-like metric for the n- foliation F. Then we have: (i) At any point of M, the transversal Vrănceanu sectional curvature of M must be nonnegative. (ii) If D is non-integrable and (M, g, F) is of constant transversal Vrănceanu curvature c, then c>0. Proof. By using (1.13) in (2.7) and taking into account that R = 0 we obtain R (U,V,U,V)= 2 h(u, V ) 2, U V Ɣ(D ). (2.20) As (U, V ) > 0, from (2.18), (2.16) and (2.20) we deduce that K () 0, for any transversal plane to F. This proves the assertion (i). To prove the assertion (ii) we suppose that c = 0. Then by (2.20) and (2.19) we infer that h(u, V ) = 0 for any U,V Ɣ(D ). Hence by (1.3) we have P [U,V ] = 0, that is, D is integrable. This is a contradiction of the hypothesis, and by assertion (i) it follows that c>0. We close the section with an example of foliated Riemannian manifolds of positive constant transversal Vrănceanu curvature. Let us consider the family of 3-dimensional manifolds { M (α,k) = (x,y,z) R 3 :0<k(y+ z) + α< π }, 2 where α R and k>0. Next, for each pair (α, k) we define on M (α,k) the function f(x,y,z) = 2 tan(k(y + z) + α). Then the vector fields U = f y x and V = f z + x, define a non-integrable distribution D on M (α,k). Moreover, the complementary orthogonal distribution D to D in TM (α,k) defines a foliation with bundle-like metric g, where g is the Euclidean metric of R 3. Then by direct calculation we deduce that U V V V U V = 0,

11 Vrănceanu connections and foliations 109 and [U,V] V = 2k2 f 2 (t)(f (t)) 2 (2 + f 2 (t)) 2 (V U), where we put t = k(y + z) + α, which imply R (U,V,U,V)= 2k2 f 2 (t)(f (t)) f 2. (t) Also, by using (2.17) we obtain (U, V ) = f 2 (t)(2 + f 2 (t)). Finally, by (2.18) and (2.16) we infer that K () = k 2, for any transversal plane. 3. Transversal Ricci tensor and transversal scalar curvature Throughout this section we suppose that (M, g, F) is an (n + p)-dimensional foliated Riemannian manifold, where g is a bundle-like metric for the n-foliation F. We consider the transversal Vrănceanu curvature tensor field R (see (2.1)) and define n+p Ric (QX, QY ) = α=n+1 {R (E α,qx,qy,e α )}, (3.1) where {E α } is an orthonormal frame field for the transversal distribution D. It is easy to see that Ric is independent of the choice of the orthornormal frame field {E α}. Moreover, by (2.11a), (2.11b) and (2.11d) we deduce that Ric is a symmetric transversal tensor field of type (0,2) on (M, g, F). We call Ric the transversal Ricci tensor on (M, g, F). Next, we consider the frame field {/x α },α {n + 1,...,n+ p} in Ɣ(D ) and put E α = E γ α Then we deduce that x γ and x α = Ēγ α E γ. (3.2) g αβ = n+p Ēα γ Ēγ β γ =n+1 and g αβ = Also we put R ic n+p γ =n+1 E α γ Eβ γ. (3.3) ( ) x α, x β = Rαβ,

12 110 Aurel Bejancu and Hani Reda Farran and using (3.1), (3.3) and (1.23a) we obtain R αβ = R γ αβγ. (3.4) Finally, by (3.1) and (2.16) we infer that R ic (E γ,e γ ) = n+p α=n+1 α γ K (E γ,e α ), γ {n + 1,...,n+ p}, (3.5) since (E γ,e α ) = 1, for α γ. The transversal scalar curvature of (M, g, F) is a function on M denoted by S and defined by S = n+p α=n+1 Then by using (3.6), (3.2a) and (3.3b) we obtain R ic (E α,e α ). (3.6) S = g αβ R αβ. (3.7) Also, by using (3.5) in (3.6) we can express the transversal scalar curvature by means of the transversal Vrănceanu sectional curvature as follows: S = 2 α<β K (E α,e β ). (3.8) Theorem 3.1. Let (M, g, F) be an (n + p)-dimensional foliated connected Riemannian manifold, where F is a foliation of codimension p>2and g is a bundle-like metric for F. If the transversal Ricci tensor satisfies Ric = λg, where λ is a smooth function on M, then λ must be a constant. Proof. First we put R αβγ μ = g βɛr ɛ αγμ, and from (2.11a), (2.11b) and (2.11d) we obtain R αβγ μ = R βαγμ = R αβμγ = R γμαβ. (3.9) Then by using (3.4) and the hypothesis on the transversal Ricci tensor we deduce that R αβ = gγμ R αγβμ = λg αβ. (3.10) Locally, we express the identity (2.11e) as follows: Rαβγ μ ɛ + R αβμɛ γ + R αβɛγ μ = 0, (3.11) where we use the transversal Vrănceanu covariant derivative (see (1.27)). Contracting (3.11) by g αγ g βμ and using (1.28), (3.9) and (3.10) we infer that (p 2)λ ɛ = 0, for any ɛ {n + 1,...,n+ p}, (3.12)

13 which implies Vrănceanu connections and foliations = λ ɛ = λ x ɛ = λ x ɛ Ai ɛ λ x i, since p>2. Taking into account that R μ αβγ is a basic transversal tensor field, from (3.4) we deduce that Rαβ is so. Therefore λ from (3.10) depends on (xn+1,...,x n+p ) only since g αβ define a transversal tensor field too. Hence λ x i = 0, and from (3.12) we obtain λ x ɛ = 0, i {1,...,n}, ɛ {n + 1,...,n+ p}. As M is connected, we deduce that λ is a constant on M. Next, we say that (M, g, F) is a transversal Einstein foliated Riemannian manifold if the transversal Ricci tensor satisfies R ic = λg, (3.13) where λ is a constant on M. By using (3.13) and (3.6) we deduce that λ = S /p and therefore (3.13) becomes R ic = S p g or R αβ = S p g αβ. Theorem 3.2. Let (M, g, F) be a foliated Riemannian manifold, where g is bundle-like for F. If(M, g, F) is of constant transversal Vrănceanu curvature then it is transversal Einstein. Proof. Let {E α } be an orthonormal frame field for Ɣ(D ). Then we have g(qx, QY ) = n+p α=n+1 {g(qx, E α )g(qy, E β )}, X, Y Ɣ(T M). (3.14) Then by direct calculations using (3.1), (2.19) and (3.14) we infer that n+p Ric (QX, QY ) = c α=n+1 {g(qx, QY ) g(qx, E α )g(qy, E α )} = c(p 1)g(QX, QY ). (3.15) Thus (M, g, F) is transversal Einstein. For a particular dimension of the manifold we can state the following converse of Theorem 3.2. Theorem 3.3. Let (M, g, F) be an (n + 3)-dimensional transversal Einstein foliated Riemannian manifold, where F is an n-foliation and g is bundle-like for F. Then (M, g, F) is of constant transversal Vrănceanu curvature.

14 112 Aurel Bejancu and Hani Reda Farran Proof. Let {E 1,E 2,E 3 } be an orthonormal frame field in Ɣ(D ). Then by using (3.5) and (3.13) we calculate R ic (E t,e t ), t {1, 2, 3}, and obtain Thus we have K (E 1,E 2 ) + K (E 1,E 3 ) = K (E 2,E 1 ) + K (E 2,E 3 ) K (E 1,E 2 ) = K (E 2,E 3 ) = K (E 3,E 1 ) = λ 2, = K (E 3,E 1 ) + K (E 3,E 2 ) = λ. (3.16) which completes the proof of the theorem. Remark. If in particular we consider the trivial foliation by points of M, then the Vrănceanu connection is just the Levi Civita connection on the 3-dimensional Einstein manifold M. Thus Theorem 3.3 is a generalization of a result of Schouten and Struik [8]. 4. Final comments In the present paper we emphasized the important role that the Vrănceanu connection can play in studying foliated manifolds with bundle-like metrics. Apart from fibre bundles, there is a large class of manifolds which falls in the above class of foliated manifolds. Namely, any Riemannian manifold M endowed with a Killing vector field is a foliated manifold with bundle-like metric (cf. [7]). In this case, the transversal distribution D is known as the contact distribution on M. Moreover, D is non-integrable provided M is a Sasakian manifold, and therefore M is a non-holonomic manifold. Applying the general theory we developed in the present paper we may consider on a Sasakian manifold objects like the following: sectional curvature, Ricci tensor, and scalar curvature of the contact distribution. In this way, we expect to obtain in a forthcoming paper, relations between the geometry of M on one side, and the geometry of the contact distribution on the other side. Acknowledgements The authors would like to thank the referee for his remarks which improved both the presentation and the content of the paper. References [1] Bejancu A and Farran H R, On the geometry of semi-riemannian distributions, Anal. St. Univ. Al. I. Cuza Iasi, 51, s.i., f.1, (2005) [2] Bejancu A and Farran H R, Structural and transversal geometry of foliations, Int. J. Pure Appl. Math. 9(4) (2003) [3] Ianus S, Some almost product structures on manifolds with linear connections, Kodai Math. Sem. Rep. 23 (1971) [4] Kobayashi S and Nomizu K, Foundations of Differential Geometry (New York: Interscience) (1963) vol I [5] O Neill B, Semi-Riemannian geometry with applications to relativity (New York: Academic Press) (1983)

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