# A-manifolds on a principal torus bundle over an almost Hodge A-manifold base

A-manifolds on a principal torus bundle over an almost Hodge A-manifold base An A-manifold is a manifold whose Ricci tensor is cyclic-parallel, equivalently it satisfies X Ric(X, X) = 0. This condition generalizes the Einstein condition. We construct new examples of A-manifolds on r-torus bundles over a base which is a product of almost Hodge A-manifolds. 1. Introduction. One of the most extensively studied objects in mathematics and physics are Einstein manifolds (see for example ), i.e. manifolds whose Ricci tensor is a constant multiple of the metric tensor. In his work  A. Gray defined a condition which generalizes the concept of an Einsten manifold. This condition states that the Ricci tensor Ric of the Riemannian manifold (M, g) is cyclic parallel, i.e. X Ric(Y, Z) + Y Ric(Z, X) + Z Ric(X, Y ) = 0, where denotes the Levi-Civita connection of the metric g and X, Y, Z are arbitrary vector fields on M . A Riemannian manifold satisfying this condition is called an A-manifold. It is obvious that if the Ricci tensor of (M, g) is parallel, then it satisfies the above condition. On the other hand, if Ric is cyclic-parallel, but not parallel, then we call (M, g) a strict A-manifold. A. Gray gave in  the first example of such strict A-manifold, which was the sphere S 3 with appropriately defined homogeneous metric. 2010 Mathematics Subject Classification. Primary 53C25. Key words and phrases. A-manifold, cyclic parallel Ricci, torus bundle, Einstein-like manifold, Killing tensor. The author would like to thank Prof. W. Jelonek for helping with the present paper. The first example of a non-homogeneous A-manifold was given in . This a example is a S 1 -bundle over some K¨hler­Einstein manifold. Recently Z. Tang and W. Yan in  obtained some new examples of A-manifolds as focal sets of isoparametric hypersurfaces in spheres. The result from  was generalized in  to K-contact manifolds. Namely, over every almost Hodge A-manifold with J-invariant Ricci tensor we can construct a Riemannian metric such that the total space of the bundle is an A-manifold. In the present paper we take a next step in the generalization process and we prove that there exists an A-manifold structure on every r-torus bundle over product of almost Hodge A-manifolds. Our result and that of Jelonek are based on the existence of almost Hodge A-manifolds, which was proven in . 2. Conformal Killing tensors. Let (M, g) be any Riemannian manifold. We call a symmetric tensor field of type (0, 2) on M a conformal Killing tensor field iff there exists a 1-form P such that for any X (T M ) (1) X K(X, X) = P (X)g(X, X), where is the Levi-Civita connection of g. The above condition is clearly equivalent to the following (2) CX,Y,Z X K(Y, Z) = CX,Y,Z P (X)g(Y, Z) for all X, Y, Z (T M ) where CX,Y,Z denotes the cyclic sum over X, Y, Z. It is easy to prove that the 1-form P is given by P (X) = 1 (2divK(X) + d tr K(X)) , n+2 where X (T M ) and divS and tr S are the divergence and trace of the tensor field S with respect to g. If the 1-form P vanishes, then we call K a Killing tensor. Of particular interest in this work is a situation when the Ricci tensor of the metric g is a Killing tensor. We call such a manifold an A-manifold. In the more general situation, when the Ricci tensor is a conformal tensor we call (M, g) a AC -manifold. We will use the following easy property of conformal Killing tensors. Proposition 1. Suppose that (M, g) is a Riemannian product of (Mi , gi ), i = 1, 2. Moreover, let Ki be conformal tensors on (Mi , gi ). Then K = K1 + K2 is a conformal tensor for (M, g). A conformal Killing form or a twistor form is a differential p-form on (M, g) satisfying the following equation (3) X = 1 1 X d - X . p+1 n-p+1 An extensive description of conformal Killing forms can be found in a series of articles by Semmelmann and Moroianu (,). It is known that if is a co-closed conformal Killing form (also called a Killing form), then the (0, 2)-tensor field K defined by K (X, Y ) = g(X , Y ) is a Killing tensor. The following theorem generalizes the above observation. After proving this fact the author found that it was known in physics literature. Theorem 2. Let and be conformal Killing p-forms. Then the tensor field K, defined by K, = g(X , Y ) + g(Y , X ) is a conformal Killing tensor field. Proof. The proof is straightforward. Let X be any vector field and , conformal Killing p-forms. We will check that K, as defined above satisfies (1). X K, (X, X) = 2X (g(X , X )) - 2g(X X , X ) - 2g(X X , X ) = 2g(X (X ), X ) + 2g(X , X (X )) - 2g(X X , X ) - 2g(X X , X ) = 2g(X X , X ) + 2g(X , X X ). From the fact that satisfies (3) we have g(X X , X ) = - 1 g(X (X d), X ) p+1 1 g(X (X ), X ) n-p+1 1 = - (g(X, X)g(, X ) - g(X (X ), X )) n-p+1 1 g(X, X)g(, X ). = - n-p+1 The same is valid for with g(X X , X ) = - Hence we have X K, (X, X) = - 2 g(X, X) (g(, X ) + g(, X )) . n-p+1 1 g(X, X)g(, X ). n-p+1 3. Torus bundles. Let (M, h) be a Riemannian manifold and suppose that i are closed 2-forms on M for i = 1, . . . , r such that their cohomology classes [i ] are integral. In  it was proven that to each such cohomology class there corresponds a principal circle bundle pi : Pi M with a connection form i such that (4) di = 2p i . i Taking the Whitney sum of bundles (pi , Pi , M ), we obtain a principal rtorus bundle p : P M classified by cohomology classes of i , i = 1, . . . , r. The connection form is a vector valued 1-form with coefficients i , where i are as before. For each connection form i we define a vector field i by i ( i ) = 1. This vector field is just the fundamental vector field for i corresponding to 1 in the Lie algebra of i-th S 1 -factor of the bundle (p, P, M ). It is easy to check that the tensor field g given by (5) g= bij i j + p h is a Riemannian metric on P if [bij ]r is some symmetric, positive definite r × r matrix with real coefficients. This Riemannian metric makes the projection p : (P, g) (M, h) a Riemannian submersion (see ). Lemma 3. Each vector field i for i = 1, . . . , r is Killing with respect to the metric g. Moreover, define a tensor field Ti of type (1, 1) by Ti X = X i for X (T P ), where is the Levi-Civita connection of g. Then we have Ti j = 0, for i = j. Proof. To prove that s is a Killing vector field for s = 1, . . . , r observe that Li Tj = 0, L s g = bij ((Ls i ) j + i (Ls j )) . Hence we only have to check that Ls i = 0 for any i, s = 1, . . . , r. Using Cartan's magic formula for Lie derivative we have Ls i = d (i ( s )) + s di s s and it is immediate that the first term is zero, since i ( s ) = i , where i is the Kronecker delta. For the second term we have (6) di ( s , X) = s (i (X)) - X (i ( s )) - i ([ s , X]) , where X is arbitrary. We will consider two cases, namely when X is a horizontal or vertical vector field. In both cases the first two components vanish, hence we only have to look at the third. In the first case we notice that [ s , X] is a horizontal vector field, since s is a fundamental vector field on P . This gives us the vanishing of s di on horizontal vector fields. When X is vertical we can take it to be just k and we immediately see that [ s , k ] = 0 since the fields j come from the action of a torus on P . For the second part of the lemma observe that g( i , j ) is constant. For any vector field X this gives us 0 = Xg( i , j ) = g(X i , j ) + g( i , X j ) = -g(X, j i ) - g(i j , X). Now, since [ i , j ] = 0 we have i j = j i which proves that Ti j = 0. Recall that for any Killing vector field we have L X Y = L X Y + X (L Y ), where X and Y are arbitrary vector fields. In our situation we have (Li Tj )X = Li (Tj X) - Tj (Li X) = [i ,X] j + X [ i , j ] - [i ,X] j = 0, which ends the proof. Hence tensor fields Ti are horizontal, i.e. for each i there exists a tensor ~ ~ field Ti on M such that p Ti = Ti p . We now compute the O'Neill tensors () of the Riemannian submersion p : P M. Proposition 4. The O'Neill tensor T is zero. The O'Neill tensor A is given by (7) AE F = bij g(E, Ti F ) j + g( i , F )Tj E , where bij are the coefficients of the inverse matrix of [bij ]r and E, F (T P ). Observe that from the fact that i ( i ) = 1 for E (T P ) we get that g( i , E) = hence j (E) = r bij j (E) j=1 bji g( i , E). Taking the exterior differential, we get (8) where E, F (T P ). dj (E, F ) = 2 bji g(Ti E, F ), Using formulae from  Chapter 9 and the fact that the fibre of the Riemannian submersion (p, P, M ) is totally geodesic and flat, we see that the Ricci tensor on the total space of Riemannian submersion is given by (9) (10) (11) Ric(U, V ) = Ric(X, U ) = - g(AEi U, AEi V ), m g ((Ei A)Ei X, U ) , m Ric(X, Y ) = RicM (X, Y ) - 2 g(AX Ei , AY Ei ). Here Ei is an element of the orthonormal basis of the horizontal distribution H, RicM is a lift of the Ricci tensor of the base (M, h), X, Y are horizontal vector fields and U, V any vertical vector fields. Using the formula (7) for the O'Neill tensor A we can compute all components of the Ricci tensor Ric. We obtain (12) Ric(U, V ) = b g( , U )Tt Ei , k,l=1 st bkl g( k , V )Tl Ei , (13) Ric(X, Y ) = RicM (X, Y ) - r bst g(Ts X, Tt Y ). As for the value of Ric(X, U ) we compute the covariant derivative (Ei A)Ei X = Ei r r bst g (Ei , Ts X) t - bst g (Ei Ei , Ts X) t - = bst g (Ei , Ts Ei X) t + g ( s , Ei X) Tt Ei r st b g (Ei , (Ei Ts ) X) t , where we used the fact that g( s , Ei X) = -g(Ts Ei , X) which follows from AX being anti-symmetric with respect to g for any horizontal vector field X. Now since tensors Ts are anti-symmetric with respect to g so is X Ts , hence (Ei A)Ei X = - b g ((Ei Ts ) Ei , X) = t=1 st dt (X) t . As a result we have Ric(X, U ) = t=1 dt (X)g( t , U ). 4. Torus bundle over a product of almost Hodge manifolds. Let (M, g, J) be an almost Hermitian manifold, where J denotes the almost complex structure, i.e. a type (1, 1) tensor field such that J 2 = -idT M and g is any compatible metric satisfying g(X, Y ) = g(JX, JY ) for any vector fields X and Y on M . We denote by the so-called K¨hler form which is a a 2-form defined by (X, Y ) = g(JX, Y ). If is closed we call the (M, g, J) an almost K¨hler manifold. Moreover, one can prove that in this case the a K¨hler form is also co-closed. If additionally J is integrable, then (M, g, J) is a a K¨hler manifold. In  the author constructed examples of A-manifolds a over a base which is a product of K¨hler­Einstein manifolds. In particular a it has parallel Ricci tensor and is a degenerate case of an A-manifold, so this paper is a generalization of the former. Moreover, any K¨hler A-manifold a has parallel Ricci tensor by a result of Sekigawa and Vanhecke . In the more general situation of almost K¨hler metrics the situation is a different. In  Jelonek constructed a strictly almost K¨hler A-manifold a with non-parallel Ricci tensor. Moreover, the K¨hler form of such a mana ifold has a useful property. It is a constant multiple of some differential 2-form that belongs to an integral cohomology class i.e. a differential form a a in H 2 (M ; Z). An almost K¨hler manifold whose K¨hler form satisfies this condition is called an almost Hodge manifold. Returning to our construction, suppose that (Mi , gi , Ji ), i = 1, . . . , n are almost Hodge manifolds such that K¨hler forms i are constant multiples of a 2-forms i and their cohomology classes are integral, i.e. [i ] H 2 (Mi ; Z). Denote by (M, g, J) the product manifold with the product metric and product almost complex structure and let pri be the projection on the i-th factor. From our earlier discussion we know that there exists a principal r-torus bundle classified by the forms 1 , . . . , r given by j = aji pri i , where [aji ] is some r × n matrix with integer coefficients. By (4) the coefficients j of the connection form of (p, P, M ) satisfy dj = 2p j = 2 for every j = 1, . . . , r. n aji p (pri i ) Since i 's and K¨hler forms i of (Mi , gi , Ji ) are connected by i = ci i a for some constants ci , i = 1, . . . , n we have (14) dj = 2 aji , ci i we denote the 2-form obtained from lifting i to P . Comparing where by ~ this with (8), we get a formula for each tensor field Ti r n (15) ~ Ti X = j=1 bij k=1 ajk J X ck k where Jk is the almost complex structure tensor of (Mk , gk , Jk ) lifted to the product manifold M . We will now compute the Ricci tensor of (P, g) using (9)­(11), computations that follows those formulas and above observations. We begin with Ric(U, V ) = (16) r s,l=1 r s,l=1 s=1 g( , U ) = 2 = 2 g( s , U )g( l , V ) g( s , U )g( l , V ) k=1 m ask J Ei , ck k r l=1 g( , U ) ask J Ei , ck k h=1 n h=1 alh J Ei ch h m k=1 n alh J Ei ch h gk k=1 ask alk J k Ei , J k Ei . ck ck We used the fact that for k = h images of Jk and Jh are orthogonal. It is easy to see that m n ask alk gk J k Ei , J k Ei ck ck k=1 are constants for each s, l = 1, . . . , r. Hence the Ricci tensor of (P, g) on vertical vector fields is a symmetrized product of Killing vector fields. Next, since the K¨hler form of each almost Hodge manifold (Mk , gk , Jk ) a is co-closed we see from (14) that (17) Ric(X, U ) = 0 for any horizontal vector field X and vertical vector field U . The last component of the Ricci tensor of (P, g) is the horizontal one. First observe that RicM is the Ricci tensor of the product metric h = g1 + . . . + gn and Ricci tensors Rick are Jk -invariant Killing tensors. We have Theorem 5. Let Ki be a Killing tensor on (Mi , gi , Ji ) for i = 1, . . . , n. Then the lift K of K = K1 + . . . + Kn to P is a Killing tensor iff each Ki is Ji -invariant. Proof. We need to check the cyclic sum condition (2) for different choices of vector fields. It is easy to see that if all three vector fields are vertical, then each component of the cyclic sum vanishes, since K is non-vanishing only on horizontal vector fields. If only two of the vector fields are vertical, then again all components vanish, since i j = 0. For three horizontal vector fields we again see that the cyclic sum vanish, since the covariant derivative of K with respect to metric g on P is the same as that of K with respect to the product metric h on M . By Proposition 1, K is a Killing tensor for (M, h). The remaining case is when only one vector field is vertical. Let us put Z = i and X, Y be basic horizontal vector fields. We compute i K (X, Y ) = -K (i X, Y ) - K (X, i Y ) = -K (AX i , Y ) - K (X, AY i ) = -K (X i , Y ) - K (X, Y i ), where the next to last equality is due to the fact that X and Y are basic (see ) and the last one follows from the definition of the O'Neill tensor A. Next we have X K ( i , Y ) = -K (X i , Y ). Summing up, we have Ci ,X,Y i K (X, Y ) = -2 K (X i , Y ) + K (X, Y i ) ~ ~ = -2 K(Ti X, Y ) + K(X, Ti Y ) . ~ Now we use the formula (15) for the tensor Ti Ci ,X,Y i K (X, Y ) = -2 r n bij j=1 k=1 ajk (K(Jk X, Y ) + K(X, Jk Y )) . ck Since each Ji projects vector fields on T Mk , we see from the definition of K that K(Jk X, Y ) + K(X, Jk Y ) = Kk (Jk X, Y ) + Kk (X, Jk Y ). By Jk -invariance of Kk for k = 1, . . . , n we have completed the proof. Remark. It is worth noting, that we cannot lift in that way a conformal Killing tensor with non-vanishing P . In fact taking three vertical vector fields, we see that P vanishes on vertical distribution. On the other hand, for two vertical vector fields U, V and one horizontal vector field X the lefthand side of (2) vanishes and the right-hand side reads P (X)g(U, V ), hence P has to vanish also on the horizontal distribution. Corollary 1. An r-torus bundle with metric defined by (5) can not be an AC -manifold. Especially there are no AC structures on K-contact and Sasakian manifolds. Next we show that the second component of the horizontal part of the Ricci tensor (13) is just a sum of lifts of metrics gk , k = 1, . . . , n. r n n r r r ajk ail bst g(Ts X, Tt Y ) = 2 h bsj J X, bti J Y . ck k cl l j=1 k=1 l=1 Since Jk and Jl are orthogonal for different k, l = 1, . . . , n, we obtain r r r r n ajk aik bst g(Ts X, Tt Y ) = 2 h bsj Jk X, bti J Y ck ck k k=1 j=1 (18) r r ajk alk bjl gk (X, Y ). = 2 c2 k j,l=1 k=1 From the above theorem we infer that, since a Riemannian metric is a Killing tensor and each gk is Jk -invariant, the tensor field K(X, Y ) = r st b g(Ts X, Tt Y ) is a Killing tensor field. Now we can prove the following theorem. Theorem 6. Let P be a r-torus bundle over a Riemannian product (M, h) of almost Hodge A-manifolds (Mk , gk , Jk ), k = 1, . . . n with metric g defined by (5). Then (P, g) is itself an A-manifold. Proof. Since distributions H and V are orthogonal with respect to the Ricci tensor Ric of (P, g) by (17) we can write it as Ric(E, F ) = 2 r s,l=1 g( s , E)g( l , F ) gk k=1 r r ask alk J k Ei , J k Ei ck ck ajk alk gk (E, F ) c2 k 1 + RicM (E, F ) - 2 2 bjl j,l=1 k=1 using (18) and (16).The first component is a Killing tensor as a symmetrized product of Killing vector fields by Theorem 2. The second and third components are Killing tensors by Theorem 5. Since a sum of Killing tensors with constant coefficients is again a Killing tensor we have proved the theorem. Remark. Observe that if at least one of the manifolds (Mk , gk ) has nonparallel Ricci tensor, then the Ricci tensor Ric of (P, g) is also non-parallel with respect to the metric g. Thus we have constructed a large number of strict A-manifolds. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales UMCS, Mathematica de Gruyter

# A-manifolds on a principal torus bundle over an almost Hodge A-manifold base

, Volume 69 (1) – Jun 1, 2015
11 pages      /lp/de-gruyter/a-manifolds-on-a-principal-torus-bundle-over-an-almost-hodge-a-PgrZ0vHTYg
Publisher
de Gruyter
ISSN
2083-7402
eISSN
2083-7402
DOI
10.1515/umcsmath-2015-0016
Publisher site
See Article on Publisher Site

### Abstract

An A-manifold is a manifold whose Ricci tensor is cyclic-parallel, equivalently it satisfies X Ric(X, X) = 0. This condition generalizes the Einstein condition. We construct new examples of A-manifolds on r-torus bundles over a base which is a product of almost Hodge A-manifolds. 1. Introduction. One of the most extensively studied objects in mathematics and physics are Einstein manifolds (see for example ), i.e. manifolds whose Ricci tensor is a constant multiple of the metric tensor. In his work  A. Gray defined a condition which generalizes the concept of an Einsten manifold. This condition states that the Ricci tensor Ric of the Riemannian manifold (M, g) is cyclic parallel, i.e. X Ric(Y, Z) + Y Ric(Z, X) + Z Ric(X, Y ) = 0, where denotes the Levi-Civita connection of the metric g and X, Y, Z are arbitrary vector fields on M . A Riemannian manifold satisfying this condition is called an A-manifold. It is obvious that if the Ricci tensor of (M, g) is parallel, then it satisfies the above condition. On the other hand, if Ric is cyclic-parallel, but not parallel, then we call (M, g) a strict A-manifold. A. Gray gave in  the first example of such strict A-manifold, which was the sphere S 3 with appropriately defined homogeneous metric. 2010 Mathematics Subject Classification. Primary 53C25. Key words and phrases. A-manifold, cyclic parallel Ricci, torus bundle, Einstein-like manifold, Killing tensor. The author would like to thank Prof. W. Jelonek for helping with the present paper. The first example of a non-homogeneous A-manifold was given in . This a example is a S 1 -bundle over some K¨hler­Einstein manifold. Recently Z. Tang and W. Yan in  obtained some new examples of A-manifolds as focal sets of isoparametric hypersurfaces in spheres. The result from  was generalized in  to K-contact manifolds. Namely, over every almost Hodge A-manifold with J-invariant Ricci tensor we can construct a Riemannian metric such that the total space of the bundle is an A-manifold. In the present paper we take a next step in the generalization process and we prove that there exists an A-manifold structure on every r-torus bundle over product of almost Hodge A-manifolds. Our result and that of Jelonek are based on the existence of almost Hodge A-manifolds, which was proven in . 2. Conformal Killing tensors. Let (M, g) be any Riemannian manifold. We call a symmetric tensor field of type (0, 2) on M a conformal Killing tensor field iff there exists a 1-form P such that for any X (T M ) (1) X K(X, X) = P (X)g(X, X), where is the Levi-Civita connection of g. The above condition is clearly equivalent to the following (2) CX,Y,Z X K(Y, Z) = CX,Y,Z P (X)g(Y, Z) for all X, Y, Z (T M ) where CX,Y,Z denotes the cyclic sum over X, Y, Z. It is easy to prove that the 1-form P is given by P (X) = 1 (2divK(X) + d tr K(X)) , n+2 where X (T M ) and divS and tr S are the divergence and trace of the tensor field S with respect to g. If the 1-form P vanishes, then we call K a Killing tensor. Of particular interest in this work is a situation when the Ricci tensor of the metric g is a Killing tensor. We call such a manifold an A-manifold. In the more general situation, when the Ricci tensor is a conformal tensor we call (M, g) a AC -manifold. We will use the following easy property of conformal Killing tensors. Proposition 1. Suppose that (M, g) is a Riemannian product of (Mi , gi ), i = 1, 2. Moreover, let Ki be conformal tensors on (Mi , gi ). Then K = K1 + K2 is a conformal tensor for (M, g). A conformal Killing form or a twistor form is a differential p-form on (M, g) satisfying the following equation (3) X = 1 1 X d - X . p+1 n-p+1 An extensive description of conformal Killing forms can be found in a series of articles by Semmelmann and Moroianu (,). It is known that if is a co-closed conformal Killing form (also called a Killing form), then the (0, 2)-tensor field K defined by K (X, Y ) = g(X , Y ) is a Killing tensor. The following theorem generalizes the above observation. After proving this fact the author found that it was known in physics literature. Theorem 2. Let and be conformal Killing p-forms. Then the tensor field K, defined by K, = g(X , Y ) + g(Y , X ) is a conformal Killing tensor field. Proof. The proof is straightforward. Let X be any vector field and , conformal Killing p-forms. We will check that K, as defined above satisfies (1). X K, (X, X) = 2X (g(X , X )) - 2g(X X , X ) - 2g(X X , X ) = 2g(X (X ), X ) + 2g(X , X (X )) - 2g(X X , X ) - 2g(X X , X ) = 2g(X X , X ) + 2g(X , X X ). From the fact that satisfies (3) we have g(X X , X ) = - 1 g(X (X d), X ) p+1 1 g(X (X ), X ) n-p+1 1 = - (g(X, X)g(, X ) - g(X (X ), X )) n-p+1 1 g(X, X)g(, X ). = - n-p+1 The same is valid for with g(X X , X ) = - Hence we have X K, (X, X) = - 2 g(X, X) (g(, X ) + g(, X )) . n-p+1 1 g(X, X)g(, X ). n-p+1 3. Torus bundles. Let (M, h) be a Riemannian manifold and suppose that i are closed 2-forms on M for i = 1, . . . , r such that their cohomology classes [i ] are integral. In  it was proven that to each such cohomology class there corresponds a principal circle bundle pi : Pi M with a connection form i such that (4) di = 2p i . i Taking the Whitney sum of bundles (pi , Pi , M ), we obtain a principal rtorus bundle p : P M classified by cohomology classes of i , i = 1, . . . , r. The connection form is a vector valued 1-form with coefficients i , where i are as before. For each connection form i we define a vector field i by i ( i ) = 1. This vector field is just the fundamental vector field for i corresponding to 1 in the Lie algebra of i-th S 1 -factor of the bundle (p, P, M ). It is easy to check that the tensor field g given by (5) g= bij i j + p h is a Riemannian metric on P if [bij ]r is some symmetric, positive definite r × r matrix with real coefficients. This Riemannian metric makes the projection p : (P, g) (M, h) a Riemannian submersion (see ). Lemma 3. Each vector field i for i = 1, . . . , r is Killing with respect to the metric g. Moreover, define a tensor field Ti of type (1, 1) by Ti X = X i for X (T P ), where is the Levi-Civita connection of g. Then we have Ti j = 0, for i = j. Proof. To prove that s is a Killing vector field for s = 1, . . . , r observe that Li Tj = 0, L s g = bij ((Ls i ) j + i (Ls j )) . Hence we only have to check that Ls i = 0 for any i, s = 1, . . . , r. Using Cartan's magic formula for Lie derivative we have Ls i = d (i ( s )) + s di s s and it is immediate that the first term is zero, since i ( s ) = i , where i is the Kronecker delta. For the second term we have (6) di ( s , X) = s (i (X)) - X (i ( s )) - i ([ s , X]) , where X is arbitrary. We will consider two cases, namely when X is a horizontal or vertical vector field. In both cases the first two components vanish, hence we only have to look at the third. In the first case we notice that [ s , X] is a horizontal vector field, since s is a fundamental vector field on P . This gives us the vanishing of s di on horizontal vector fields. When X is vertical we can take it to be just k and we immediately see that [ s , k ] = 0 since the fields j come from the action of a torus on P . For the second part of the lemma observe that g( i , j ) is constant. For any vector field X this gives us 0 = Xg( i , j ) = g(X i , j ) + g( i , X j ) = -g(X, j i ) - g(i j , X). Now, since [ i , j ] = 0 we have i j = j i which proves that Ti j = 0. Recall that for any Killing vector field we have L X Y = L X Y + X (L Y ), where X and Y are arbitrary vector fields. In our situation we have (Li Tj )X = Li (Tj X) - Tj (Li X) = [i ,X] j + X [ i , j ] - [i ,X] j = 0, which ends the proof. Hence tensor fields Ti are horizontal, i.e. for each i there exists a tensor ~ ~ field Ti on M such that p Ti = Ti p . We now compute the O'Neill tensors () of the Riemannian submersion p : P M. Proposition 4. The O'Neill tensor T is zero. The O'Neill tensor A is given by (7) AE F = bij g(E, Ti F ) j + g( i , F )Tj E , where bij are the coefficients of the inverse matrix of [bij ]r and E, F (T P ). Observe that from the fact that i ( i ) = 1 for E (T P ) we get that g( i , E) = hence j (E) = r bij j (E) j=1 bji g( i , E). Taking the exterior differential, we get (8) where E, F (T P ). dj (E, F ) = 2 bji g(Ti E, F ), Using formulae from  Chapter 9 and the fact that the fibre of the Riemannian submersion (p, P, M ) is totally geodesic and flat, we see that the Ricci tensor on the total space of Riemannian submersion is given by (9) (10) (11) Ric(U, V ) = Ric(X, U ) = - g(AEi U, AEi V ), m g ((Ei A)Ei X, U ) , m Ric(X, Y ) = RicM (X, Y ) - 2 g(AX Ei , AY Ei ). Here Ei is an element of the orthonormal basis of the horizontal distribution H, RicM is a lift of the Ricci tensor of the base (M, h), X, Y are horizontal vector fields and U, V any vertical vector fields. Using the formula (7) for the O'Neill tensor A we can compute all components of the Ricci tensor Ric. We obtain (12) Ric(U, V ) = b g( , U )Tt Ei , k,l=1 st bkl g( k , V )Tl Ei , (13) Ric(X, Y ) = RicM (X, Y ) - r bst g(Ts X, Tt Y ). As for the value of Ric(X, U ) we compute the covariant derivative (Ei A)Ei X = Ei r r bst g (Ei , Ts X) t - bst g (Ei Ei , Ts X) t - = bst g (Ei , Ts Ei X) t + g ( s , Ei X) Tt Ei r st b g (Ei , (Ei Ts ) X) t , where we used the fact that g( s , Ei X) = -g(Ts Ei , X) which follows from AX being anti-symmetric with respect to g for any horizontal vector field X. Now since tensors Ts are anti-symmetric with respect to g so is X Ts , hence (Ei A)Ei X = - b g ((Ei Ts ) Ei , X) = t=1 st dt (X) t . As a result we have Ric(X, U ) = t=1 dt (X)g( t , U ). 4. Torus bundle over a product of almost Hodge manifolds. Let (M, g, J) be an almost Hermitian manifold, where J denotes the almost complex structure, i.e. a type (1, 1) tensor field such that J 2 = -idT M and g is any compatible metric satisfying g(X, Y ) = g(JX, JY ) for any vector fields X and Y on M . We denote by the so-called K¨hler form which is a a 2-form defined by (X, Y ) = g(JX, Y ). If is closed we call the (M, g, J) an almost K¨hler manifold. Moreover, one can prove that in this case the a K¨hler form is also co-closed. If additionally J is integrable, then (M, g, J) is a a K¨hler manifold. In  the author constructed examples of A-manifolds a over a base which is a product of K¨hler­Einstein manifolds. In particular a it has parallel Ricci tensor and is a degenerate case of an A-manifold, so this paper is a generalization of the former. Moreover, any K¨hler A-manifold a has parallel Ricci tensor by a result of Sekigawa and Vanhecke . In the more general situation of almost K¨hler metrics the situation is a different. In  Jelonek constructed a strictly almost K¨hler A-manifold a with non-parallel Ricci tensor. Moreover, the K¨hler form of such a mana ifold has a useful property. It is a constant multiple of some differential 2-form that belongs to an integral cohomology class i.e. a differential form a a in H 2 (M ; Z). An almost K¨hler manifold whose K¨hler form satisfies this condition is called an almost Hodge manifold. Returning to our construction, suppose that (Mi , gi , Ji ), i = 1, . . . , n are almost Hodge manifolds such that K¨hler forms i are constant multiples of a 2-forms i and their cohomology classes are integral, i.e. [i ] H 2 (Mi ; Z). Denote by (M, g, J) the product manifold with the product metric and product almost complex structure and let pri be the projection on the i-th factor. From our earlier discussion we know that there exists a principal r-torus bundle classified by the forms 1 , . . . , r given by j = aji pri i , where [aji ] is some r × n matrix with integer coefficients. By (4) the coefficients j of the connection form of (p, P, M ) satisfy dj = 2p j = 2 for every j = 1, . . . , r. n aji p (pri i ) Since i 's and K¨hler forms i of (Mi , gi , Ji ) are connected by i = ci i a for some constants ci , i = 1, . . . , n we have (14) dj = 2 aji , ci i we denote the 2-form obtained from lifting i to P . Comparing where by ~ this with (8), we get a formula for each tensor field Ti r n (15) ~ Ti X = j=1 bij k=1 ajk J X ck k where Jk is the almost complex structure tensor of (Mk , gk , Jk ) lifted to the product manifold M . We will now compute the Ricci tensor of (P, g) using (9)­(11), computations that follows those formulas and above observations. We begin with Ric(U, V ) = (16) r s,l=1 r s,l=1 s=1 g( , U ) = 2 = 2 g( s , U )g( l , V ) g( s , U )g( l , V ) k=1 m ask J Ei , ck k r l=1 g( , U ) ask J Ei , ck k h=1 n h=1 alh J Ei ch h m k=1 n alh J Ei ch h gk k=1 ask alk J k Ei , J k Ei . ck ck We used the fact that for k = h images of Jk and Jh are orthogonal. It is easy to see that m n ask alk gk J k Ei , J k Ei ck ck k=1 are constants for each s, l = 1, . . . , r. Hence the Ricci tensor of (P, g) on vertical vector fields is a symmetrized product of Killing vector fields. Next, since the K¨hler form of each almost Hodge manifold (Mk , gk , Jk ) a is co-closed we see from (14) that (17) Ric(X, U ) = 0 for any horizontal vector field X and vertical vector field U . The last component of the Ricci tensor of (P, g) is the horizontal one. First observe that RicM is the Ricci tensor of the product metric h = g1 + . . . + gn and Ricci tensors Rick are Jk -invariant Killing tensors. We have Theorem 5. Let Ki be a Killing tensor on (Mi , gi , Ji ) for i = 1, . . . , n. Then the lift K of K = K1 + . . . + Kn to P is a Killing tensor iff each Ki is Ji -invariant. Proof. We need to check the cyclic sum condition (2) for different choices of vector fields. It is easy to see that if all three vector fields are vertical, then each component of the cyclic sum vanishes, since K is non-vanishing only on horizontal vector fields. If only two of the vector fields are vertical, then again all components vanish, since i j = 0. For three horizontal vector fields we again see that the cyclic sum vanish, since the covariant derivative of K with respect to metric g on P is the same as that of K with respect to the product metric h on M . By Proposition 1, K is a Killing tensor for (M, h). The remaining case is when only one vector field is vertical. Let us put Z = i and X, Y be basic horizontal vector fields. We compute i K (X, Y ) = -K (i X, Y ) - K (X, i Y ) = -K (AX i , Y ) - K (X, AY i ) = -K (X i , Y ) - K (X, Y i ), where the next to last equality is due to the fact that X and Y are basic (see ) and the last one follows from the definition of the O'Neill tensor A. Next we have X K ( i , Y ) = -K (X i , Y ). Summing up, we have Ci ,X,Y i K (X, Y ) = -2 K (X i , Y ) + K (X, Y i ) ~ ~ = -2 K(Ti X, Y ) + K(X, Ti Y ) . ~ Now we use the formula (15) for the tensor Ti Ci ,X,Y i K (X, Y ) = -2 r n bij j=1 k=1 ajk (K(Jk X, Y ) + K(X, Jk Y )) . ck Since each Ji projects vector fields on T Mk , we see from the definition of K that K(Jk X, Y ) + K(X, Jk Y ) = Kk (Jk X, Y ) + Kk (X, Jk Y ). By Jk -invariance of Kk for k = 1, . . . , n we have completed the proof. Remark. It is worth noting, that we cannot lift in that way a conformal Killing tensor with non-vanishing P . In fact taking three vertical vector fields, we see that P vanishes on vertical distribution. On the other hand, for two vertical vector fields U, V and one horizontal vector field X the lefthand side of (2) vanishes and the right-hand side reads P (X)g(U, V ), hence P has to vanish also on the horizontal distribution. Corollary 1. An r-torus bundle with metric defined by (5) can not be an AC -manifold. Especially there are no AC structures on K-contact and Sasakian manifolds. Next we show that the second component of the horizontal part of the Ricci tensor (13) is just a sum of lifts of metrics gk , k = 1, . . . , n. r n n r r r ajk ail bst g(Ts X, Tt Y ) = 2 h bsj J X, bti J Y . ck k cl l j=1 k=1 l=1 Since Jk and Jl are orthogonal for different k, l = 1, . . . , n, we obtain r r r r n ajk aik bst g(Ts X, Tt Y ) = 2 h bsj Jk X, bti J Y ck ck k k=1 j=1 (18) r r ajk alk bjl gk (X, Y ). = 2 c2 k j,l=1 k=1 From the above theorem we infer that, since a Riemannian metric is a Killing tensor and each gk is Jk -invariant, the tensor field K(X, Y ) = r st b g(Ts X, Tt Y ) is a Killing tensor field. Now we can prove the following theorem. Theorem 6. Let P be a r-torus bundle over a Riemannian product (M, h) of almost Hodge A-manifolds (Mk , gk , Jk ), k = 1, . . . n with metric g defined by (5). Then (P, g) is itself an A-manifold. Proof. Since distributions H and V are orthogonal with respect to the Ricci tensor Ric of (P, g) by (17) we can write it as Ric(E, F ) = 2 r s,l=1 g( s , E)g( l , F ) gk k=1 r r ask alk J k Ei , J k Ei ck ck ajk alk gk (E, F ) c2 k 1 + RicM (E, F ) - 2 2 bjl j,l=1 k=1 using (18) and (16).The first component is a Killing tensor as a symmetrized product of Killing vector fields by Theorem 2. The second and third components are Killing tensors by Theorem 5. Since a sum of Killing tensors with constant coefficients is again a Killing tensor we have proved the theorem. Remark. Observe that if at least one of the manifolds (Mk , gk ) has nonparallel Ricci tensor, then the Ricci tensor Ric of (P, g) is also non-parallel with respect to the metric g. Thus we have constructed a large number of strict A-manifolds.