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.


Introduction.
One of the most extensively studied objects in mathematics and physics are Einstein manifolds (see for example [1]), i.e. manifolds whose Ricci tensor is a constant multiple of the metric tensor.In his work [2] 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.
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 [2] the first example of such strict A-manifold, which was the sphere S 3 with appropriately defined homogeneous metric.
The first example of a non-homogeneous A-manifold was given in [3].This example is a S 1 -bundle over some Kähler-Einstein manifold.Recently Z. Tang and W. Yan in [12] obtained some new examples of A-manifolds as focal sets of isoparametric hypersurfaces in spheres.
The result from [3] was generalized in [4] 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 [5].

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 where ∇ is the Levi-Civita connection of g.The above condition is clearly equivalent to the following for all X, Y, Z ∈ Γ(T M ) where C X,Y,Z denotes the cyclic sum over X, Y, Z.
It is easy to prove that the 1-form P is given by where X ∈ Γ(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.
A conformal Killing form or a twistor form is a differential p-form ϕ on (M, g) satisfying the following equation An extensive description of conformal Killing forms can be found in a series of articles by Semmelmann and Moroianu ([11], [7]).
It is known that if ϕ is a co-closed conformal Killing form (also called a Killing form), then the (0, 2)-tensor field K ϕ defined by 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 ϕ and ψ be conformal Killing p-forms.Then the tensor field K ϕ,ψ defined by is a conformal Killing tensor field.
Proof.The proof is straightforward.Let X be any vector field and ϕ, ψ conformal Killing p-forms.We will check that K ϕ,ψ as defined above satisfies (1).
From the fact that ϕ satisfies (3) we have The same is valid for ψ with Hence we have 3. Torus bundles.Let (M, h) be a Riemannian manifold and suppose that β i are closed 2-forms on M for i = 1, . . ., r such that their cohomology classes [β i ] are integral.In [6] it was proven that to each such cohomology class there corresponds a principal circle bundle p i : P i → M with a connection form θ i such that ( 4) Taking the Whitney sum of bundles (p i , P i , M ), we obtain a principal rtorus bundle p : P → M classified by cohomology classes of β i , i = 1, . . ., r.The connection form θ is a vector valued 1-form with coefficients θ i , where θ i are as before.For each connection form θ i we define a vector field ξ i by θ i (ξ i ) = 1.This vector field is just the fundamental vector field for θ 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) 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 [8]).Lemma 3.Each vector field ξ i for i = 1, . . ., r is Killing with respect to the metric g.Moreover, define a tensor field T i of type (1, 1) by T i X = ∇ X ξ i for X ∈ Γ(T P ), where ∇ is the Levi-Civita connection of g.Then we have Proof.To prove that ξ s is a Killing vector field for s = 1, . . ., r observe that Hence we only have to check that L ξ s θ i = 0 for any i, s = 1, . . ., r.Using Cartan's magic formula for Lie derivative we have and it is immediate that the first term is zero, since θ i (ξ s ) = δ s i , where δ s i is the Kronecker delta.For the second term we have (6) dθ 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 [ξ s , X] is a horizontal vector field, since ξ s is a fundamental vector field on P .This gives us the vanishing of ξ s ⌟ dθ i on horizontal vector fields.
When X is vertical we can take it to be just ξ k and we immediately see that [ξ s , ξ k ] = 0 since the fields ξ j come from the action of a torus on P .
For the second part of the lemma observe that g(ξ i , ξ j ) is constant.For any vector field X this gives us Recall that for any Killing vector field we have where X and Y are arbitrary vector fields.In our situation we have which ends the proof.
Hence tensor fields T i are horizontal, i.e. for each i there exists a tensor field Ti on M such that p * • T i = Ti • p * .
We now compute the O'Neill tensors ( [8]) 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) where b ij are the coefficients of the inverse matrix of [b ij ] r i,j=1 and E, F ∈ Γ(T P ).
Observe that from the fact that θ i (ξ i ) = 1 for E ∈ Γ(T P ) we get that Taking the exterior differential, we get (8) where E, F ∈ Γ(T P ).
Using formulae from [1] 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 Here E i is an element of the orthonormal basis of the horizontal distribution H, Ric M 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 As for the value of Ric(X, U ) we compute the covariant derivative where we used the fact that g(ξ s , ∇ E i X) = −g(T s E i , X) which follows from A X being anti-symmetric with respect to g for any horizontal vector field X.Now since tensors T s are anti-symmetric with respect to g so is ∇ X T s , hence As a result we have Ric(X, U ) = r t=1 δdθ t (X)g(ξ 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 = −id T 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 ω the so-called Kähler form which is a 2-form defined by ω(X, Y ) = g(JX, Y ).If ω is closed we call the (M, g, J) an almost Kähler manifold.Moreover, one can prove that in this case the Kähler form is also co-closed.If additionally J is integrable, then (M, g, J) is a Kähler manifold.In [14] the author constructed examples of A-manifolds over a base which is a product of Kähler-Einstein manifolds.In particular 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 Kähler A-manifold has parallel Ricci tensor by a result of Sekigawa and Vanhecke [10].
In the more general situation of almost Kähler metrics the situation is different.In [5] Jelonek constructed a strictly almost Kähler A-manifold with non-parallel Ricci tensor.Moreover, the Kähler form of such a manifold 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 in H 2 (M ; Z).An almost Kähler manifold whose Kähler form satisfies this condition is called an almost Hodge manifold.
Returning to our construction, suppose that (M i , g i , J i ), i = 1, . . ., n are almost Hodge manifolds such that Kähler forms ω i are constant multiples of 2-forms α i and their cohomology classes are integral, i.e. [α i ] ∈ H 2 (M i ; Z).Denote by (M, g, J) the product manifold with the product metric and product almost complex structure and let pr i 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 β 1 , . . ., β r given by where [a ji ] is some r × n matrix with integer coefficients.By (4) the coefficients θ j of the connection form of (p, P, M ) satisfy for every j = 1, . . ., r.
Since α i 's and Kähler forms ω i of (M i , g i , J i ) are connected by ω i = c i α i for some constants c i , i = 1, . . ., n we have ( 14) where by ω * i we denote the 2-form obtained from lifting ω i to P .Comparing this with (8), we get a formula for each tensor field Ti where J * k is the almost complex structure tensor of (M k , g k , J k ) 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 We used the fact that for k = h images of J k and J h are orthogonal.It is easy to see that 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 Kähler form of each almost Hodge manifold (M k , g k , J k ) 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 Ric M is the Ricci tensor of the product metric h = g 1 + . . .+ g n and Ricci tensors Ric k are J k -invariant Killing tensors.We have Theorem 5. Let K i be a Killing tensor on (M i , g i , J i ) for i = 1, . . ., n.Then the lift K * of K = K 1 + . . .+ K n to P is a Killing tensor iff each K i is J i -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 ∇ ξ i ξ 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 = ξ i and X, Y be basic horizontal vector fields.We compute where the next to last equality is due to the fact that X and Y are basic (see [8]) and the last one follows from the definition of the O'Neill tensor A. Next we have Summing up, we have Now we use the formula (15) for the tensor Ti Since each J i projects vector fields on T M k , we see from the definition of 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 g k , k = 1, . . ., n. From the above theorem we infer that, since a Riemannian metric is a Killing tensor and each g k is J k -invariant, the tensor field K(X, Y ) = r s,t=1 b st g(T s X, T t 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 (M k , g k , J k ), 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 Remark.Observe that if at least one of the manifolds (M k , g k ) 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.

b 2 rb 2 ra jk a lk c 2 k
st g(T s X, T t Y ) = π Since J k and J l are orthogonal for different k, l = 1, . . ., n, st g(T s X, T t Y ) = π g k (X, Y ).

a jk a lk c 2 k
) we can write it asRic(E, F ) = π 2 r s,l=1 g(ξ s , E)g(ξ l , F ) g k (E, F ) 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.