Semi-Riemannian manifolds with a doubly warped structure

Rev. Mat. Iberoam. 28 (2012), no. 1, 1–24 doi 10.4171/rmi/664 c European Mathematical Society  Semi-Riemannian manifolds with a doubly warped structure Manuel Gutiérrez and Benjamı́n Olea Abstract. We investigate manifolds obtained as a quotient of a doubly warped product. We show that they are always covered by the product of two suitable leaves. This allows us to prove, under regularity hypothesis, that these manifolds are a doubly warped product up to a zero measure subset formed by an union of leaves. We also obtain a necessary and sufficient condition which ensures the decomposition of the whole manifold and use it to give sufficient conditions of geometrical nature. Finally, we study the uniqueness of direct product decomposition in the nonsimply connected case. 1. Introduction Let (Mi , gi ) be two semi-Riemannian manifolds and λi : M1 × M2 → R+ two positive functions, i = 1, 2. The doubly twisted product M1 ×(λ1 ,λ2 ) M2 is the manifold M1 × M2 furnished with the metric g = λ21 g1 + λ22 g2 . If λ1 ≡ 1 or λ2 ≡ 1, then it is called a twisted product. On the other hand, when λ1 only depends on the second factor and λ2 on the first one, it is called a doubly warped product (warped product if λ1 ≡ 1 or λ2 ≡ 1). The most simple metric, the direct product, corresponds to the case λ1 ≡ λ2 ≡ 1. Using the language of foliations, the classical De Rham–Wu Theorem says that two orthogonally, complementary and geodesic foliations (called a direct product structure) in a complete and simply connected semi-Riemannian manifold give rise to a global decomposition as a direct product of two leaves. This theorem can be generalized in two ways: one way is obtaining more general decompositions than direct products and the second one is removing the simply connectedness Mathematics Subject Classification (2010): Primary 53C50; Secondary 53C12. Keywords: Quotient manifold, doubly warped product, doubly warped structure, decomposition theorems. 2 M. Gutiérrez and B. Olea hypothesis. The most general theorems in the first direction were obtained in [15] and [20], where the authors showed that geometrical properties of the foliations determine the type of decomposition. In the second direction, P. Wang obtained that a complete semi-Riemannian manifold furnished with a direct product structure is covered by the direct product of two leaves, [25]. Moreover, if the manifold is Riemannian, using the theory of bundle-like metrics, he showed that a necessary and sufficient condition to obtain the global decomposition as a direct product is the existence of two regular leaves which intersect each other at only one point. There are other remarkable works avoiding the simply connectedness but they treat with codimension one foliations, see [10] and references therein. In this paper we study semi-Riemannian manifolds furnished with a doubly warped structure, that is, two complementary, orthogonal and umbilic foliations with closed mean curvature vector fields. If one of the mean curvature vector fields is identically null, then it is called a warped structure. Doubly warped and warped structures appear in different situations. For example, Codazzi tensors with exactly two eigenvalues, one of them constant [5], Killing tensors [9], semi-Riemann submersions with umbilic fibres and some additional hypotheses [8] and certain group actions [1], [16] lead to (doubly) warped structures. On the other hand, the translation of physical content into geometrical language gives rise to warped structures which, by a topological simplification, are supposed global products. In this way are constructed important spaces as Robertson–Walker, Schwarzschild, Kruskal, static spaces. . . Manifolds with a doubly warped structure are locally a doubly warped product and, under completeness hypothesis, they are a quotient of a global doubly warped product. This is why, after the preliminaries of section 2, we focus our attention on studying these quotients. The main tool in this paper is Theorem 3.4, which gives a normal semiRiemannian covering map with a doubly warped product of two leaves as domain. We use it to obtain a necessary and sufficient condition for a semi-Riemannian manifold with a doubly warped structure to be a global product, which extends the one given in [25] for direct products structures in the Riemannian setting. Other consequence of Theorem 3.4 is that any leaf is covered by a leaf without holonomy of the same foliation. We study the space of leaves obtaining that, under regularity hypothesis, a manifold with a doubly warped structure is a fiber bundle over the space of leaves. This allows us to compute the fundamental group of the space of leaves and to show that there is an open dense subset which is isometric to a doubly warped product. We also give a result involving the curvature that ensures the global decomposition and apply it to semi-Riemannian submersion with umbilic fibres. As a consequence of the De Rham–Wu Theorem, it can be ensured the uniqueness of the direct product decomposition of a simply connected manifold under nondegeneracy hypothesis. In the last section, we apply the decomposition results obtained to investigate the uniqueness of the decomposition without the simply connectedness assumption. 3 Doubly warped structures 2. Preliminaries Given a product manifold M1 × M2 and X ∈ X(Mi ), we will also denote X to its elevation to X(M1 ×M2 ) and Pi : T M1 ×T M2 → T Mi will be the canonical projection. Unless it is explicitly said, all manifolds are supposed to be semi-Riemannian. We write some formulaes about Levi-Civita connection and curvature, which are established in [20]. Lemma 2.1. Let M1 ×(λ1 ,λ2 ) M2 be a doubly twisted product and call ∇i the LeviCivita connection of (Mi , gi ). Given X, Y ∈ X(M1 ) and V, W ∈ X(M2 ) it holds 1. ∇X Y = ∇1X Y − g(X, Y )∇ ln λ1 + g(X, ∇ ln λ1 )Y + g(Y, ∇ ln λ1 )X. 2. ∇V W = ∇2V W − g(V, W )∇ ln λ2 + g(V, ∇ ln λ2 )W + g(W, ∇ ln λ2 )V . 3. ∇X V = ∇V X = g(∇ ln λ1 , V )X + g(∇ ln λ2 , X)V . It follows that canonical foliations are umbilic and the mean curvature vector field of the first canonical foliation is N1 = P2 (−∇ ln λ1 ) whereas that of the second is N2 = P1 (−∇ ln λ2 ). Lemma 2.2. Let M1 ×(λ1 ,λ2 ) M2 be a doubly twisted product and take Π1 = span(X, Y ), Π2 = span(V, W ) and Π3 = span(X, V ) nondegenerate planes where X, Y ∈ T M1 and V, W ∈ T M2 are unitary and orthogonal vectors. Then the sectional curvature is given by   1 1 1 ,∇λ1 ) 1. K(Π1 ) = K (Π1 )+g(∇λ ε − g(h (X), X) + ε g(h (Y ), Y ) . 2 X 1 Y 1 λ1 λ 1 2. K(Π2 ) = 2 K (Π2 )+g(∇λ2 ,∇λ2 ) λ22 − 3. K(Π3 ) = − ελV1 g(h1 (V ), V ) − 1 λ2 εX λ2   εV g(h2 (V ), V ) + εW g(h2 (W ), W ) . g(h2 (X), X) + g(∇λ1 ,∇λ2 ) , λ1 λ2 where K i is the sectional curvature of (Mi , gi ), hi is the hessian endomorphism of λi and εZ is the sign of g(Z, Z). A vector field is called closed if its metrically equivalent one form is closed. Lemma 2.3. Let M1 ×(λ1 ,λ2 ) M2 be a doubly twisted product. It is a doubly warped product if and only if Ni = P3−i (−∇ ln λi ) is closed, for i = 1, 2. Proof. Suppose dω1 = 0, where ω1 is the equivalent one form to N1 . If X ∈ X(M1 ) and V ∈ X(M2 ), then XV (ln λ1 ) = −Xω1 (V ) = −dω1 (X, V ) = 0. Thus there are functions f1 ∈ C ∞ (M1 ) and h1 ∈ C ∞ (M2 ) such that λ1 (x, y) = f1 (x)h1 (y) for all (x, y) ∈ M1 × M2 . Analogously, λ2 (x, y) = f2 (x)h2 (y) for certain functions f2 ∈ C ∞ (M1 ) and h2 ∈ C ∞ (M2 ). Hence, taking conformal metrics if necessary, M1 ×(λ1 ,λ2 ) M2 can be expressed as a doubly warped product. The only if part is trivial. ✷ We want to generalize the concept of doubly twisted or doubly warped product to manifold which are not necessarily a topological product. 4 M. Gutiérrez and B. Olea Definition 2.4. Two complementary, orthogonal and umbilic foliations (F1 , F2 ) in a semi-Riemannian manifold is called a doubly twisted structure. Moreover, if the mean curvature vectors of the foliations are closed, then it is called a doubly warped structure. Finally, we say that it is a warped structure if one mean curvature vector is closed and the other one is zero. Notice that this last case is equivalent to one of the foliations being totally geodesic and the other one spherical, see [20] for the definition. We call Ni the mean curvature vector field of Fi and ωi , which we call mean curvature form, to its metrically equivalent one-form. The leaf of Fi through x ∈ M is denoted by Fi (x) and Fi (x) will be the tangent plane of Fi (x) at the point x. If there is not confusion or if the point is not relevant, we simply write Fi . We always put the induced metric on the leaves. Remark 2.5. If M has a doubly twisted (warped) structure, then we can take around any point an adapted chart to both foliations. Lemma 2.3 and Proposition 3 of [20] show that M is locally isometric to a doubly twisted (warped) product. In the doubly warped structure case, the condition on the mean curvature vectors in Theorem 5.4 of [15] can be easily checked. So, if the leaves of one of the foliations are complete we can apply this theorem to obtain that M is a quotient of a global doubly warped product. Given a curve α : [0, 1] → M we call αt : [0, t] → M , 0 ≤ t ≤ 1, its restriction. Definition 2.6. Let M be a semi-Riemannian manifold with F1 and F2 two orthogonal and complementary foliations. Take x ∈ M , v ∈ F2 (x) and α : [0, 1] → F1 (x) a curve with α(0)= x. We define the adapted translation of v along αt as Aαt (v) = exp − αt ω2 W (t), where W is the normal parallel translation to F1 of v along α, [17]. In the same way we can define the adapted translation   of a vector of F1 (x) along a curve in F2 (x). Observe that |Aαt (v)| = |v| exp − αt ω2 . Lemma 2.7. Let M = M1 × M2 be a semi-Riemannian manifold such that the canonical foliations constitute a doubly twisted structure. Take α : [0, 1] → M1 a curve with α(0) = a and vb ∈ Tb M2 . The adapted translation of (0a , vb ) along the curve γ(t) = (α(t), b) is Aγt (0a , vb ) = (0α(t) , vb ). Proof. First we show that formula 3 of Lemma 2.1 is still true in this case. In fact, take X, Y ∈ X(M1 ) and V, W ∈ X(M2 ). Since [X, V ] = 0 we have g(∇X V, W ) = −g(X, ∇V W ) = −g(V, W )g(X, N2 ), and analogously g(∇X V, Y ) = −g(V, N1 )g(X, Y ). Therefore, it follows that ∇X V = −ω1 (V )X − ω2 (X)V for all X ∈ X(M1 ) and V ∈ X(M2 ).   Take V (t) = (0α(t) , vb ) and W (t) = λ(t)V (t), where λ(t) = exp γt ω2 . We only have to check that W (t) is the normal parallel translation of (0a , vb ) along γ. Doubly warped structures 5 But this is an immediate consequence of DW = λ′ V + λ(−ω1 (V ) γ ′ − ω2 (γ ′ )V ) = −λ ω1 (V ) γ ′ . dt ✷ Given a foliation F we call Hol(F ) the holonomy group of a leaf F (see [3] for definitions and properties). We say that F has no holonomy if its holonomy group is trivial. The foliation F has no holonomy if any leaf has no holonomy. Lemma 2.8. Let M be a semi-Riemannian manifold with (F1 , F2 ) a doubly twisted structure. Take x ∈ M and α : [0, 1] → F1 (x) a loop at x. If f ∈ Hol(F1 (x)) is the holonomy map associated to α, then f∗x (v) = Aα (v) for all v ∈ F2 (x). Proof. It is sufficient to show it locally. Take an open set of x isometric to a doubly twisted product U1 ×(λ1 ,λ2 ) U2 where Ui is an open set of Fi (x) with x ∈ Ui . If α(t) ∈ U1 , for 0 ≤ t ≤ t0 , then the holonomy map associated to this arc is f : {x} × U2 → {α(t0 )} × U2 given by f (x, y) = (α(t0 ), y) and clearly f∗x (0x , vx ) = ✷ (0α(t0 ) , vx ) = Aαt0 (vx ). Observe that, in the doubly warped structure case, holonomy maps are homotheties and thus they are determined by their derivative at a point. Therefore, a leaf F1 has no holonomy if and only if for any loop α in F1 it holds Aα = id. We finish this section relating two doubly twisted structure via a local isometry. Lemma 2.9. Let M and M be two semi-Riemannian manifold with a doubly twisted structure (F 1 , F 2 ) and (F1 , F2 ) respectively. Take f : M → M a local isometry which preserves the doubly twisted structure, that is, f∗x (F i (x)) = Fi (f (x)) for all x ∈ M and i = 1, 2. Then f also preserves (1) the leaves, f (F i (x)) ⊂ Fi (f (x)); (2) the mean curvature vector fields, f∗x (N i (x)) = Ni (f (x)); (3) the mean curvature forms, f ∗ (ωi ) = ωi ; (4) the adapted translation, that is, if γ : [0, 1] → F 1 (x) is a curve with γ(0) = x and v ∈ F 2 (x), then Af ◦γt (f∗x (v)) = f∗γ(t) (Aγt (v)) and analogously for a curve in F 2 (x). Proof. (1) It is immediate. (2) Take X, Y ∈ F 1 and call X = f∗ (X) and Y = f∗ (Y ) in a suitable open set. Since f is a local isometry, g(X, Y )N1 = P2 (∇X Y ) = f∗ (P 2 (∇X Y )) = g(X, Y )f∗ (N 1 ). Hence N1 = f∗ (N 1 ) and analogously for N 2 . (3) Immediate from point (2). (4) Take W (t) the parallel translation of v normal to F 1 along γ. Since f is a local isometry which preserves the foliations, f∗ (W (t)) is  the normal parallel translation of f∗ (v) along f ◦ γ. Using point (3), γt ω 2 = f ◦γt ω2 and therefore Af ◦γt (f∗x (v)) = f∗γ(t) (Aγt (v)). ✷ 6 M. Gutiérrez and B. Olea 3. Quotient of a doubly warped product From now on, M1 ×(λ1 ,λ2 ) M2 will be a doubly warped product and Γ a group of isometries such that: 1. Γ acts in a properly discontinuous manner. 2. It preserves the canonical foliations. This implies that if f ∈ Γ, then f = φ × ψ : M1 × M2 → M1 × M2 , where φ : M1 → M1 and ψ : M2 → M2 are homotheties with factor c21 and c22 respectively, such that λ1 ◦ ψ = c11 λ1 and λ2 ◦ φ = c12 λ2 .   The semi-Riemannian manifold M = M1 ×(λ1 ,λ2 ) M2 /Γ has a doubly warped structure, which, as always, we call (F1 , F2 ). We are going to work with F1 because all definitions and results are stated analogously for F2 . If we take the canonical projection p : M1 × M2 → M , which is a semiRiemannian covering map, applying Lemma 2.9 we have p(M1 × {b}) ⊂ F1 (p(a, b)) (a,b) for all (a, b) ∈ M1 × M2 . We call p1 : M1 × {b} → F1 (p(a, b)) the restriction of p.   Lemma 3.1. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product. We take p : M1 × M2 → M the canonical projection, (a, b) ∈ M1 × M2 (a,b) and x = p(a, b). Then, the restriction p1 : M1 × {b} → F1 (x) is a normal semiRiemannian covering map. (a,b) Proof. It is clear that p1 : M1 × {b} → F1 (x) is a local isometry. Let γ : [0, 1] → F1 (x) be a curve with γ(0) = x. Since p : M1 ×M2 → M is a covering map, there is a lift α : [0, 1] → M1 × M2 with α(0) = (a, b). But p∗ (α′ (t)) = γ ′ (t) ∈ F1 (γ(t)) and p preserves the foliations, so α(t) is a curve in M1 × {b}. Applying Theorem 28, pa(a,b) ge 201, of [17], we get that p1 is a covering map. Now we show that it is normal. (a,b) ′ ′ Take a ∈ M1 such that p1 (a , b) = x. Then, there exists f ∈ Γ with f (a′ , b) = (a, b) and since f preserves the canonical foliations, f (M1 × {b}) = M1 × {b}. So, (a,b) the restriction of f to M1 ×{b} is a deck transformation of the covering p1 which ′ sends (a , b) to (a, b). ✷ (a,b) (a,b) be the group of deck transformations of p1 : M1 ×{b} → F1 (p(a, b)). Let Γ1 When there is not confusion with the chosen point, we simply write p1 and Γ1 (a,b) (a,b) instead of p1 and Γ1 . If φ ∈ Γ1 , in general, it does not exist ψ ∈ Γ2 with φ × ψ ∈ Γ and it does not have to hold that λ2 ◦ φ = c12 λ2 , as it was said at the beginning of this section.   Lemma 3.2. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product. Fix (a0 , b0 ) ∈ M1 × M2 and x0 = p(a0 , b0 ) such that the leaf F1 (x0 ) has no holonomy. Then (a0 ,b0 ) (1) λ2 ◦ φ = λ2 for all φ ∈ Γ1 (a0 ,b0 ) (2) φ × id ∈ Γ for all φ ∈ Γ1 . (a0 ,b0 ) and Γ1 × {id} is a normal subgroup of Γ. Doubly warped structures 7 Proof. (1) The mean curvature forms ω 2 and ω2 of the foliations in M1 × M2 and M respectively are closed. Thus every point in M has an open neighborhood where ω2 = df2 for certain function f2 , and analogously every point in M1 × M2 has an open neighborhood where ω 2 = df 2 , being in this case f 2 = − ln λ2 . Since p1 ◦ φ = p1 , we have φ∗ (p∗ (ω2 )) = p∗ (ω2 ) and therefore f2 ◦ p ◦ φ = f2 ◦ p + k1 for certain constant k1 . On the other hand, using Lemma 2.9, we have f2 ◦ p = − ln λ2 + k2 for some constant k2 . Joining the last two equations, we get λ2 ◦ φ = cλ2 for certain constant c. This formula must be true in the whole M1 with the same constant c and, in fact, 1) c = λλ22 (a (a0 ) where a1 = φ(a0 ). Take α : [0, 1] → M1 × {b0 } with α(0) = (a0 , b0 ), α(1) = (a1 , b0 ) and w ∈ Tb0 M2 a non lightlike vector. Using Lemma 2.7 we have Aα (0a0 , w) = (0a1 , w) and since p preserves the adapted translation,   Ap◦α p∗(a0 ,b0 ) (0a0 , w) = p∗(a1 ,b0 ) (0a1 , w). Using that F1 (x0 ) has no holonomy and Lemma 2.8, we obtain p∗(a1 ,b0 ) (0a1 , w) = p∗(a0 ,b0 ) (0a0 , w), taking norms, we get c = 1. (2) Take φ ∈ Γ1 . Since λ2 ◦ φ = λ2 , it follows that φ × id is an isometry of M1 ×(λ1 ,λ2 ) M2 . Now, to show that p ◦ (φ × id) = p it is enough to check (p ◦ φ × id)∗(a0 ,b0 ) = p∗(a0 ,b0 ) . Using p1 ◦ φ = p1 , we see that (p ◦ (φ × id))∗(a0 ,b0 ) (v, 0b0 ) = p∗(a0 ,b0 ) (v, 0b0 ) for all v ∈ Ta0 M1 . Given w ∈ Tb0 M2 , we take α : [0, 1] → M1 × {b0 } a curve from (a0 , b0 ) to (a1 , b0 ), where φ(a0 ) = a1 . Then Aα (0a0 , w) = (0a1 , w) and   (p ◦ (φ × id))∗(a0 ,b0 ) (0a0 , w) = p∗(a1 ,b0 ) (0a1 , w) = p∗(a1 ,b0 ) Aα (0a0 , w)   = Ap◦α p∗(a0 ,b0 ) (0a0 , w) = p∗(a0 ,b0 ) (0a0 , w), where the last equality holds because F1 (x0 ) has no holonomy. Therefore φ×id ∈ Γ and Γ1 × {id} is a subgroup of Γ. Now to prove that it is a normal subgroup, we take φ ∈ Γ1 and f ∈ Γ and show that f −1 ◦ (φ × id) ◦ f ∈ Γ1 . Since f preserves the foliations, f −1 ◦ (φ × id) ◦ f takes M1 × {b0 } into M1 × {b0 } and therefore we can consider h = f −1 ◦ (φ × id) ◦ f M1 ×{b0 } ∈ Γ1 . But h × id coincides with f −1 ◦ (φ × id) ◦ f at a0 , thus f −1 ◦ (φ × id) ◦ f = h × id ∈ Γ1 × {id}. ✷ Remark 3.3. Observe that we have used that F1 (x0 ) has no holonomy only to ensure Ap◦α = id, thus we have a little bit more general result. Suppose that γ : [0, 1] → F1 (x) is a loop at x ∈ M such that its associated holonomy map is trivial. Take α : [0, 1] → M1 × {b} a lift through p1 : M1 × {b} → F1 (x) with basepoint (a, b) and suppose α(1) = (a′ , b). If φ ∈ Γ1 with φ(a) = a′ , then it can be proven, identically as in the above lemma, that for this deck transformation it holds λ2 ◦ φ = λ2 and φ × id ∈ Γ. This will be used in Theorem 3.10. 8 M. Gutiérrez and B. Olea Now, we can give the following theorem, which is the main tool of this paper (compare with Theorem 2 of [25] and Theorem 7 of [23]).   Theorem 3.4. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product. Fix (a0 , b0 ) ∈ M1 × M2 and x0 = p(a0 , b0 ). The leaf F1 (x0 ) has no holonomy if and only if there exists a semi-Riemannian normal covering map Φ : F1 (x0 ) ×(λ1 ,ρ2 ) M2 → M , where ρ2 : F1 (x0 ) → R+ . Moreover, the following diagram is commutative: M1 × M2 p /M s9 s s ss (a ,b ) p1 0 0 ×id ssΦ s  ss F1 (x0 ) × M2 In particular, Φ(x, b0 ) = x for all x ∈ F1 (x0 ). Proof. Suppose that F1 (x0 ) has no holonomy. Since Γ1 ×{id} is a normal subgroup of Γ, there exists a normal covering map   Φ : M1 ×(λ1 ,λ2 ) M2 / (Γ1 × {id}) → M.   But M1 ×(λ1 ,λ2 ) M2 / (Γ1 × {id}) is isometric to F1 (x0 ) ×(λ1 ,ρ2 ) M2 (x0 ) for certain function ρ2 with ρ2 ◦ p1 = λ2 and by construction Φ ◦ (p1 × id) = p. Conversely, we suppose the existence of such semi-Riemannian covering. Take α : [0, 1] → F1 (x0 ) a loop at x0 , w ∈ Tx0 F2 (x0 ) and v ∈ Tb0 M2 with p∗(a0 ,b0 ) (0, v) = w. Then   Φ∗(x0 ,b0 ) (0, v) = Φ∗(x0 ,b0 ) ( p1 × id)∗(a0 ,b0 ) (0, v) = p∗(a0 ,b0 ) (0, v) = w. Now, using Lemmas 2.7 and 2.9, and that the holonomy in F1 (x0 ) ×(λ1 ,ρ2 ) M2 is trivial,   Aα (w) = Φ∗(x0 ,b0 ) A(α,b0 ) (0, v) = Φ∗(x0 ,b0 ) (0, v) = w and therefore F1 (x0 ) has no holonomy. ✷ We say that x0 ∈ M has no holonomy if F1 (x0 ) and F2 (x0 ) have no holonomy.   Corollary 3.5. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product. Fix (a0 , b0 ) ∈ M1 × M2 and x0 = p(a0 , b0 ). The point x0 has no holonomy if and only if there is a semi-Riemannian normal covering map Φ : F1 (x0 ) ×(ρ1 ,ρ2 ) F2 (x0 ) → M, where ρ1 : F2 (x0 ) → R is commutative: + and ρ2 : F1 (x0 ) → R+ . Moreover, the following diagram M1 × M2 p / q8 M q q (a ,b ) (a ,b ) qqq p1 0 0 ×p2 0 0 qqΦ q q  q F1 (x0 ) × F2 (x0 ) In particular, Φ(x, x0 ) = x and Φ(x0 , y) = y for all x ∈ F1 (x0 ) and y ∈ F2 (x0 ). Doubly warped structures 9 Remark 3.6. It is known ([6]) that the set of leaves without holonomy is dense on M . Thus, we can always take a point x0 ∈ M without holonomy and apply Corollary 3.5.   Theorem 3.7. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product. If x0 ∈ M has no holonomy then card(F1 (x0 ) ∩ F2 (x0 )) = card(Φ−1 (x0 )), where Φ : F1 (x0 ) ×(ρ1 ,ρ2 ) F2 (x0 ) → M is the semi-Riemannian covering map of Corollary 3.5. Proof. Recall that, by construction, Φ(x, y) = p(a, b) where a ∈ M1 with p(a, b0 ) = x and b ∈ M2 with p(a0 , b) = y. Now we define Λ : Φ−1 (x0 ) → F1 (x0 ) ∩ F2 (x0 ) by Λ(x, y) = x. First we show that Λ is well defined. If (x, y) ∈ Φ−1 (x0 ) then p(a, b) = x0 , where p(a, b0 ) = x and p(a0 , b) = y. Hence p({a}×M2) ⊂ F2 (p(a, b)) = F2 (x0 ) and p(M1 × {b0 }) ⊂ F1 (p(a0 , b0 )) = F1 (x0 ), thus x = p(a, b0 ) = p(M1 × {b0 } ∩ {a} × M2 ) ∈ F1 (x0 ) ∩ F2 (x0 ). Now we check that Λ is onto. Take x ∈ F1 (x0 ) ∩ F2 (x0 ). Since p1 : M1 × {b0 } → F1 (x0 ) is a covering map there exists a ∈ M1 such that p(a, b0 ) = x. But (a,b ) p2 0 : {a} × M2 → F2 (x) = F2 (x0 ) is a covering map too, therefore there is (a, b) ∈ {a} × M2 such that p(a, b) = x0 . If we call y = p(a0 , b), then Φ(x, y) = x0 and Λ(x, y) = x. Finally, we show that Λ is injective. Take (x, y), (x, y ′ ) ∈ Φ−1 (x0 ) and a ∈ M1 , ′ b, b ∈ M2 such that p(a, b0 ) = x, p(a0 , b) = y and p(a0 , b′ ) = y ′ . Consider the cov(a,b ) (a,b ) (a,b ) ering p2 0 : {a}×M2 → F2 (x0 ). Since p2 0 (a, b) = p2 0 (a, b′ ) and this covering (a,b ) is normal, there exists a deck transformation ψ ∈ Γ2 0 such that ψ(a, b) = (a, b′ ). But F2 (x0 ) has no holonomy, so Lemma 3.2 assures that id × ψ ∈ Γ. Now, (id × ψ)(a0 , b) = (a0 , b′ ) and thus y = y ′ . ✷ Now we give a necessary and sufficient condition for a doubly warped structure to be a global doubly warped product, which extends the one given in [25] for direct products and Riemannian manifolds.   Corollary 3.8. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product and x0 ∈ M . Then M is isometric to the doubly warped product F1 (x0 )×(ρ1 ,ρ2 ) F2 (x0 ) if and only if x0 has no holonomy and F1 (x0 ) ∩ F2 (x0 ) = {x0 }. Condition F1 (x0 )∩F2 (x0 ) = {x0 } alone is not sufficient to split M as a product F1 (x0 ) × F2 (x0 ), as intuition perhaps suggests. The Möbius trip illustrates this point.   Theorem 3.9. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product. If x0 has no holonomy then card(F1 (x) ∩ F2 (x)) ≤ card(F1 (x0 ) ∩ F2 (x0 )) for all x ∈ M . 10 M. Gutiérrez and B. Olea Proof. Take (a0 , b0 ) ∈ M1 × M2 such that p(a0 , b0 ) = x0 . First suppose that x ∈ F1 (x0 ) and F1 (x) ∩ F2 (x) = {xi : i ∈ I}. If we take a ∈ M1 such that (a,b ) p(a, b0 ) = x, then we know that p2 0 : {a} × M2 → F2 (x) is a covering map, so (a ,b ) (a,b ) we can take bi ∈ M2 with p2 0 (a, bi ) = xi . If we call yi = p2 0 0 (a0 , bi ), then both xi , yi ∈ p(M1 ×{bi }) = F1 (p(a, bi )) = F1 (x0 ), and moreover, since yi ∈ F2 (x0 ) we have yi ∈ F1 (x0 ) ∩ F2 (x0 ). Now, we show that the map Λ : F1 (x) ∩ F2 (x) → F1 (x0 ) ∩ F2 (x0 ) given by (a ,b ) (a ,b ) Λ(xi ) = yi is injective. If yi = p2 0 0 (a0 , bi ) = p2 0 0 (a0 , bj ) = yj for i = j then (a ,b ) there is ψ ∈ Γ2 0 0 such that ψ(a0 , bi ) = (a0 , bj ). Since F2 (x0 ) has no holonomy (Lemma 3.2), id × ψ ∈ Γ and it sends (a, bi ) to (a, bj ). Therefore xi = xj . This shows that card(F1 (x) ∩ F2 (x)) ≤ card(F1 (x0 ) ∩ F2 (x0 )) when x ∈ F1 (x0 ). Take now an arbitrary point x ∈ M and (a, b) ∈ M1 × M2 with p(a, b) = x. We have that F2 (x) intersects F1 (x0 ) at some point z = p(a, b0 ). In the same way as above, using that F1 (x0 ) has no holonomy, we can show that card(F1 (x)∩F2 (x)) ≤ card(F1 (z) ∩ F2 (z)), but we have already proven that card(F1 (z) ∩ F2 (z)) ≤ ✷ card(F1 (x0 ) ∩ F2 (x0 )). Take x0 = p(a0 , b0 ) ∈ M such that F1 (x0 ) has no holonomy and let Φ : F1 (x0 ) ×(λ1 ,ρ2 ) M2 → M be the semi-Riemannian covering map constructed in Theorem 3.4, which has  (a ,b )  Ω = Γ/ Γ1 0 0 × {id} as deck transformation group. Take x ∈ F2 (x0 ) and a point b ∈ M2 with Φ(x0 , b) = x. Applying Lemma 3.1, (x0 ,b) Φ1 : F1 (x0 ) × {b} → F1 (x), (x0 ,b) the restriction of Φ, is a normal semi-Riemannian covering map. Call Ω1 deck transformations group. its Theorem 3.10. In the above situation, the following sequence is exact (x ,b) Φ1#0 H 0 −→ π1 (F1 (x0 ), x0 ) −−−−→ π1 (F1 (x), x) −→ Hol (F1 (x)) −→ 0, where H : π1 (F1 (x), x) −→ Hol (F1 (x)) is the usual holonomy homomorphism. In (x ,b) particular we have Ω1 0 = Hol (F1 (x)). Proof. It is clear that Φ1# is injective and H is onto, so we only prove that Ker H = Im Φ1# . Take [γ] ∈ π1 (F1 (x), x) such that H([γ]) = 1, i.e., fγ = id, where fγ is the associated holonomy map. Take α a lift of γ in F1 (x0 ) × {b} with basepoint (x0 , b) (x ,b) and φ ∈ Ω1 0 such that φ(x0 , b) = α(1). Since fγ = id it follows that ρ2 ◦ φ = ρ2 and φ × id ∈ Ω, see Remark 3.3. Therefore, taking into account that Φ(x, b0 ) = x for all x ∈ F1 (x0 ), we get x0 = Φ1 (x0 , b0 ) = Φ1 (φ(x0 ), b0 ) = φ(x0 ). Hence α is a loop at x0 which holds Φ1# ([α]) = [γ]. This shows that Ker H ⊂ Im Φ1# . The other inclusion is trivial because the holonomy of the first canonical foliation in the product F1 (x0 ) × M2 is trivial and Φ preserves the foliations. ✷ Doubly warped structures 11 Summarizing, we obtain   Corollary 3.11. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product and take x0 ∈ M such that F1 (x0 ) has no holonomy. (1) For any leaf F1 there exists a normal semi-Riemannian covering map Φ : F1 (x0 ) → F1 with deck transformation group Hol(F1 ). (2) All leaves without holonomy are homothetic. Proof. For the first point, notice that given any leaf F1 it always exists x ∈ F2 (x0 ) such that F1 = F1 (x). For the second statement, just notice that F1 (x0 ) and F1 (x0 ) ×{b} are homothetic. ✷   Corollary 3.12. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product. If there is a noncompact leaf, then any compact leaf has nontrivial holonomy.   Corollary 3.13. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product and x0 = p(a0 , b0 ) ∈ M . If F1 (x0 ) has no holonomy, then π1 (F1 (x0 ), x0 ) is a normal subgroup of π1 (M, x0 ). Proof. From Theorem 3.4 it is immediate that π1 (F1 (x0 ), x0 ) is a subgroup of π1 (M, x0 ). Take [α] ∈ π1 (F1 (x0 ), x0 ) and [γ] ∈ π1 (M, x0 ). We show that [γ ·α·γ −1 ] is homotopic to a loop in F1 (x0 ). Take the covering map Φ : F1 (x0 ) × M2 → M and γ̃ = (γ̃1 , γ̃2 ) a lift of γ starting at (x0 , b0 ). Since Φ(γ̃(1)) = x0 , using the above corollary we have Φ : F1 (x0 ) × {γ̃2 (1)} → F1 (x0 ) is an isometry, thus we can lift the loop α to a loop α̃ starting at γ̃(1). Therefore, the lift of γ · α · γ −1 to F1 (x0 ) × M2 starting at (x0 , b0 ) is γ̃ · α  · γ̃ −1 . But it is clear that this last loop is homotopic to a loop in F1 (x0 ) × {b0 } and so γ · α · γ −1 is homotopic to a loop in F1 (x0 ). ✷ Example 3.14. Lemma 3.2, and thus the results that depend on it, does not hold if we consider more general products than doubly warped product, as the following example shows. First, we are going to construct a function λ : R2 → R+ step by step. Take h : R → R, such that h = id in a neighborhood of 0 and h′ (x) > 0 for all x ∈ R, and λ : (−ε, ε) × R → R+ any C ∞ function for ε < 21 . We extend λ to the trip (1 − ε, 1 + ε)× R defining λ(x, y) = λ(x− 1, h(y))h′ (y) for every (x, y) ∈ (−ε, ε)× R. Now extend it again to [ε, 1 − ε] × R in any way such that λ : (−ε, 1 + ε) × R → R+ is C ∞ . Thus, we have a function with λ(x, y) = λ(x − 1, h(y))h′ (y) for all (x, y) ∈ (1 − ε, 1 + ε) × R, or equivalently, λ(x, y) = λ(x + 1, f (y))f ′ (y) for all (x, y) ∈ (−ε, ε) × R, where f is the inverse of h. Now, we define λ in [1 + ε, ∞) recursively by λ(x, y) = λ(x − 1, h(y))h′ (y), and in (−∞, −ε] by λ(x, y) = λ(x + 1, f (y))f ′ (y). It is easy to show that λ : R2 → R+ is C ∞ . Take R2 endowed with the twisted metric dx2 + λ(x, y)2 dy 2 and Γ the group generated by the isometry φ(x, y) = (x + 1, f (y)), which preserves the canonical foliations and acts in a properly discontinuous manner. Take p : R2 → R2 /Γ = M the projection. The leaf of the first foliation through p(0, 0) is diffeomorphic to S1 12 M. Gutiérrez and B. Olea and has no holonomy. But Theorem 3.4 does not hold because if Φ : S1 × R → M were a covering map, then S1 would be a covering of all leaves of the first foliation (Corollary 3.11). But this is impossible because for a suitable choice of h, there are leaves diffeomorphic to R. We finish this section with a cohomological obstruction to the existence of a quotient of a doubly warped product with compact leaves. If M1 and M2 are n-dimensional, compact and oriented manifolds, Künneth formula implies that the n-th Betti number of the product M1 × M2 is greater or equal than 2. The following theorem shows that the same is true for any oriented quotient of a doubly warped product with n-dimensional compact leaves.   Theorem 3.15. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be an oriented quotient of a doubly warped product such that the leaves of both foliations on M are n-dimensional and compact submanifolds of M . Then, the n-th Betti number of M satisfies bn ≥ 2. Proof. Take a point (a0 , b0 ) ∈ M1 × M2 such that x0 = p(a0 , b0 ) has no holonomy and Φ : F1 (x0 ) ×(ρ1 ,ρ2 ) F2 (x0 ) −→ M the covering map given in Corollary 3.5. Since M is oriented, M1 and M2 are orientable and Γ preserves the orientation. (a ,b ) But Γi 0 0 is a normal subgroup of Γ and therefore it preserves the orientation (a ,b ) of Mi . Thus Fi (x0 ) = Mi /Γi 0 0 is orientable. Let [̟1 ], [̟2 ] ∈ H n (M ) be the Poincaré dual of F1 (x0 ) and F2 (x0 ) respectively. The submanifolds Si = Φ−1 (Fi (x0 )) are closed in F1 (x0 ) × F2 (x0 ) and therefore they are compact. With the appropriate orientation, they have Poincaré duals [σi ] = Φ∗ ([̟i ]), [2]. Call πi : F1 (x0 ) × F2 (x0 ) → Fi (x0 ) the canonical projection, Φi : Si → Fi (x0 ) the restriction of Φ to Si , and ij : Sj → F1 (x0 ) × F2 (x0 ) the canonical inclusion. Consider the following commutative diagram: Φj Sj / Fj (x0 ) ♦7 ♦ ♦ ♦♦ ij ♦♦♦ ♦ ♦ ♦  F1 (x0 ) × F2 (x0 ) πj If Θ1 is a volume form of F1 (x0 ), then Φ∗1 (Θ1 ) = i∗1 (π1∗ (Θ1 )) is a volume form in S1 . Therefore 0= S1 i∗1 π1∗ (Θ1 ) = F1 ×F2 π1∗ (Θ1 ) ∧ σ1 , thus [σ1 ] is not null. In the same way we can show that [σ2 ] is not null. Now if σ1 − cσ2 = dτ for some 0 = c ∈ R and τ ∈ Λn−1 (M ), then F1 ×F2 π1∗ (Θ1 ) ∧ σ1 = c F1 ×F2 π1∗ (Θ1 ) ∧ σ2 = c S2 i∗2 π1∗ (Θ1 ) = 0, which is a contradiction. Therefore [σ1 ] and [σ2 ] are linearly independent, so the same is true for [̟1 ] and [̟2 ]. ✷ Doubly warped structures 13 Observe that if the dimension of the foliations are n and m with n = m, then we can only conclude that the n-th and m-th Betti numbers of M satisfy bn , bm ≥ 1. In the category of four dimensional Lorentzian manifolds we have the following result. Corollary 3.16. In the conditions of the above theorem, if M is a four dimensional Lorentzian manifold, then its first and second Betti numbers satisfies b1 , b2 ≥ 2. Proof. It is clear that M is compact (Corollary 3.5). The existence of a Lorentz metric implies that the Euler characteristic is null, thus 2b1 = 2 + b2 . ✷ 4. Space of leaves Given a foliation F on a manifold M , a point x is called regular if it exists an adapted chart (U, ϕ) to F , with x ∈ U , such that each leaf of the foliation intersects U in an unique slice. The open set U is also called a regular neighborhood of x. If all points are regular (i.e., F is a regular foliation), then the space of leaves L of F is a manifold except for the Hausdorffness, and the canonical projection η : M → L is an open map, [14], [19].   Given M = M1 ×(λ1 ,λ2 ) M2 /Γ a quotient of a doubly warped product, we call Li the space of leaves of the induced foliations Fi on M . Take x0 ∈ M without holonomy and the normal covering map Φ : F1 (x0 ) ×(ρ1 ,ρ2 ) F2 (x0 ) → M , whose group of deck transformation is Ψ = Γ/(Γ1 × Γ2 ). The set Σx0 formed by those maps ψ ∈ Diff(F2 (x0 )) such that there exists φ ∈ Diff(F1 (x0 )) with φ × ψ ∈ Ψ is a group of homotheties of F2 (x0 ).   Lemma 4.1. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product. Suppose that the foliation F1 has no holonomy and take x0 ∈ M such that F2 (x0 ) has no holonomy. Then the action of Σx0 on F2 (x0 ) is free. Proof. Take ψ ∈ Σx0 and suppose that it has a fixed point x ∈ F2 (x0 ). If φ ∈ Diff(F1 (x0 )) with φ × ψ ∈ Ψ, then Φ(z, x) = Φ(φ(z), ψ(x)) = Φ(φ(z), x) for all z ∈ F1 (x0 ), but since F1 has no holonomy, applying Corollary 3.11, Φ : F1 (x0 ) × {x} → F1 (x) is an isometry. Therefore φ = id and hence ψ = id. ✷   Theorem 4.2. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product such that F1 is a regular foliation. If F2 (x0 ) has no holonomy then (1) The group Σx0 acts in a properly discontinuous manner (in the topological sense) on F2 (x0 ). (2) The restriction ηx0 = η|F2 (x0 ) : F2 (x0 ) → L1 is a normal covering map with Σx0 as deck transformation group. Proof. (1) Suppose that Σx0 does not act in a properly discontinuous manner. Then, there exists x ∈ F2 (x0 ) such that for all neighborhood U of x in F2 (x0 ) there is ψ ∈ Σx0 , ψ = id, with U ∩ ψ(U ) = ∅. 14 M. Gutiérrez and B. Olea Take V ⊂ M a regular neighborhood of x adapted to F1 . Since Φ(x0 , x) = x, we can lift V through the covering Φ : F1 (x0 ) × F2 (x0 ) → M and suppose that there are Ui ⊂ Fi (x0 ) open sets with x0 ∈ U1 , x ∈ U2 and Φ : U1 × U2 → V an isometry. Using that Σx0 does not act in a properly discontinuous manner, there is ψ ∈ Σx0 , ψ = id, with y = ψ(z) for certain y, z ∈ U2 . Moreover, z = y since ψ does not have fixed points (Lemma 4.1). If we take φ with φ × ψ ∈ Ψ, then z = Φ(x0 , z) = Φ(φ(x0 ), y) and thus F1 (z) = F1 (y). Now, Φ(U1 × {y}) and Φ(U1 × {z}) are two different slices of F1 in V which belong to the same leaf F1 (z). Contradiction. (2) It is quite easy to show that F2 (x0 )/Σx0 = L1 , where the identification is given by [x] ←→ F1 (x). ✷   Corollary 4.3. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product such that F1 is a regular foliation. If the space of leaves L1 is simply connected, then M is isometric to a global doubly warped product F1 ×(ρ1 ,ρ2 ) F2 . We give some conditions for L1 to be a true manifold. Recall that given (a, b) ∈ (a,b) F1 (x0 ) × F2 (x0 ) with Φ(a, b) = x we denote Ψ1 the deck transformation group (a,b) of the restriction Φ1 : F1 (x0 ) × {b} → F1 (x) of the covering map Φ : F1 (x0 ) × F2 (x0 ) → M .   Theorem 4.4. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product. If M2 is a complete Riemann manifold and F1 a regular foliation, then the space of leaves L1 is a Riemannian manifold. Proof. Take x0 ∈ M without holonomy. We show that Σx0 is a group of isometries. Take ψ ∈ Σx0 and φ : F1 (x0 ) → F1 (x0 ) such that f = φ × ψ ∈ Ψ. As we already said, there exists a constant c such that ψ ∗ (g2 ) = c2 g2 and ρ2 = cρ2 ◦ φ. Suppose c = 1. Taking the inverse of ψ if it were necessary, we can suppose c < 1. Then ψ : F2 (x0 ) → F2 (x0 ) is a contractive map and it is assured the existence of a fixed point b ∈ F2 (x0 ). Therefore f (F1 (x0 ) × {b}) = F1 (x0 ) × {b} (a,b) and f |F1 (x0 )×{b} ∈ Ψ1 , where a ∈ F1 (x0 ) is some point. Using Lemma 3.2, c = 1 and we get a contradiction. Using the above theorem, Σx0 acts in a properly and discontinuously manner on F2 (x0 ) in the topological sense, but since F2 (x0 ) is Riemannian and Σx0 a group of isometries, it actually acts in a proper and discontinuous manner in the differentiable sense, i.e., points in different orbits have open neighborhood with ✷ disjoint orbits. Thus, L1 is a Riemannian manifold. Remark 4.5. Given a nondegenerate foliation F , it is called semi-Riemannian (or metric) when, locally, the leaves coincide with the fibers of a semi-Riemannian submersion, [21], [24]. If the orthogonal distribution is integrable, then F is a semi-Riemannian foliation if and only if F ⊥ is totally geodesic, [13]. In the case of a doubly warped product F1 (x0 ) ×(ρ1 ,ρ2 ) F2 (x0 ), the first canonical foliation is  2 semi-Riemannian for the conformal metric ρρ21 g1 + g2 . Doubly warped structures 15 In the hypotheses of the above theorem, ρ2 is invariant under Ψ and thus there exists a function σ2 : M → R+ such that σ2 ◦Φ = ρ2 . In this case, it is easy to show that F1 is semi-Riemannian for the conformal metric σ12 g, where g is the induced 2 metric on M . Observe that in the Riemannian case, under regularity hypothesis, it is known that the space of leave of a Riemannian foliation is a true manifold and, moreover, the manifold is a fiber bundle over it [21], but there is not an analogous in the semi-Riemannian case.   Corollary 4.6. Let M = M1 ×(λ1 ,1) M2 /Γ be a quotient of a warped product, where M2 is a complete Riemannian manifold. If F1 is a regular foliation, then the projection η : M → L1 is a semi-Riemannian submersion. Proof. We already know that L1 is a Riemannian manifold and ηx0 : F2 (x0 ) → L1 a local isometry, where x0 has no holonomy. Given x ∈ F1 (x0 ), the following diagram is commutative: Φ / F2 (x) {x} × F2 (x0 ) q q q ηx0 ◦pr2 qqηqx q q  xqqq L1 Since λ2 = 1, the map Φ : {x} × F2 (x0 ) → F2 (x) is a local isometry for all x ∈ F1 (x0 ). Thus, ηx : F2 (x) → L1 is a local isometry for all x ∈ M and therefore, η : M → L1 is a semi-Riemannian submersion. ✷ Observe that in the corollary, the fibres are the leaves of a warped structure, thus they are automatically umbilic.   Theorem 4.7. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product such that F1 is a regular foliation. Then (1) The projection η : M → L1 is a fiber bundle. Moreover, we have π1 (L1 , F1 ) = π1 (M, x)/π1 (F1 , x) where x ∈ F1 ∈ L1 . (2) There exists an open dense subset W ⊂ M globally isometric to a doubly warped product. Proof. (1) Take F1 ∈ L1 and x0 ∈ F1 a point without holonomy. Since ηx0 : F2 (x0 ) → L1 is a covering map, there are open sets U ⊂ F2 (x0 ) and V ⊂ L1 with x0 ∈ U and F1 ∈ V such that ηx0 : U → V is a diffeomorphism. Now we show that Φ(F1 (x0 ) × U ) = η −1 (V ). If (a, b) ∈ F1 (x0 ) × U , then η(Φ(a, b)) = η(Φ(x0 , b)) = ηx0 (b) ∈ V . Given x ∈ η −1 (V ), if we call b = (η(x)) ∈ U , then η(x) = ηx0 (b) and Φ : F1 (x0 ) × {b} → F1 (x) is an isomeηx−1 0 try because F1 has no holonomy (Corollary 3.11). Thus, there exists a ∈ F1 (x0 ) with Φ(a, b) = x. The map Φ : F1 (x0 )×U → η −1 (V ) is injective (and therefore a diffeomorphism). In fact, if (a, b), (a′ , b′ ) ∈ F1 (x0 ) × U with Φ(a, b) = Φ(a′ , b′ ) then ηx0 (b) = ηx0 (Φ(x0 , b)) = ηx0 (Φ(a, b)) = ηx0 (Φ(a′ , b′ )) = ηx0 (Φ(x0 , b′ )) = ηx0 (b′ ). 16 M. Gutiérrez and B. Olea But since b, b′ ∈ U , we get that b = b′ . Now, using that Φ : F1 (x0 )×{b} → F1 (b) is an isometry, we deduce that a = a′ . The map hV that makes commutative the following diagram F1 (x0 ) × U ❖❖❖ ❖❖id×η ❖❖❖ x0 Φ ❖❖❖  ' h V / F1 (x0 ) × V η −1 (V ) shows that M is locally trivial. Finally, using Theorem 4.41 of [12], η# : π1 (M, F1 , x0 ) → π1 (L1 , F1 ) is an isomorphism. But π1 (F1 , x0 ) is a normal subgroup of π1 (M, x0 ) (Corollary 3.13), hence π1 (M, F1 , x0 ) = π1 (M, x0 )/π1 (F1 , x0 ). (2) Since ηx0 : F2 (x0 ) → L1 is a covering map, we can take an open dense set Θ ⊂ L1 and an open set U ⊂ F2 (x0 ) such that ηx0 : U → Θ is a diffeomorphism. (F1 ) ∈ F1 and thus the restriction (F1 )) = ηx−1 Given F1 ∈ Θ we have Φ(x0 , ηx−1 0 0 −1 Φ : F1 (x0 ) × {ηx0 (F1 )} → F1 is an isometry. Now, since Θ is dense, W = η −1 (Θ) is dense in M , and taking V = Θ and W = η −1 (Θ) in the above proof we get the result. ✷ Recall that this open set W is obtained removing a suitable set of leaves of F1 from M . This is false for more general product, as twisted products (see example 3.14). Remark 4.8. In [22] a notion of local warped product on a manifold is given as follows. Take a fiber bundle Π : M → B with fibre F , where M , B and F are semi-Riemannian manifolds. Suppose that there is a function λ : B → R+ such that we can take a covering {Ui : i ∈ I} of trivializing open sets of B with (Π−1 (Ui ), g) → (Ui × F, gB + λ2 gF ) an isometry for all i ∈ I. Then it is said that M is a local warped product. It follows that the orthogonal distribution to the fibre is integrable and using that M is locally isometric to a warped product, it is easy to show that these two foliations constitute a warped structure in the sense of definition 2.4. But not all warped structures arise in this way, since the foliation induced by the fibres of a fibre bundle has no holonomy (in fact, it is a regular foliation). 5. Global decomposition Given a product manifold M1 × M2 , a plane Π = span(X, V ), where X ∈ T M1 and V ∈ T M2 , is called a mixed plane. In this section, we show how the sign of sectional curvature of this kind of planes determines the global decomposition of a doubly warped structure. Lemma 5.1. Let M be a complete semi-Riemannian manifold of index ν. Take λ : M → R+ a smooth function and hλ its hessian endomorphism. If ν < dim M and g(hλ (X), X) ≤ 0 for all spacelike vector X (or 0 < ν and g(hλ (X), X) ≤ 0 for all timelike vector X), then λ is constant. Doubly warped structures 17 Proof. Suppose ν < dim M . Take x ∈ M and V ∋ x a normal convex neighborhood. Call S(x) the set formed by the points y ∈ V such that there exists a nonconstant spacelike geodesic inside V joining x with y. It is obvious that S(x) is an open set for all x ∈ M and it does not contain x. Let γ : R → M be a spacelike geodesic with γ(0) = x. If we call y(t) = λ(γ(t)) then y ′′ (t) ≤ 0 and y(t) > 0 for all t ∈ R, which implies that λ is constant in S(x). Take x1 ∈ S(x). In the same way, λ is constant in S(x1 ), which is an open neighborhood of x. Since x is arbitrary, λ is constant. The case 0 < ν is similar taking timelike geodesics. ✷ Proposition 5.2. Let M1 ×(λ1 ,λ2 ) M2 be a doubly warped product with M1 and M2 complete semi-Riemannian manifold of index νi < dim Mi . If K(Π) ≥ 0 for all spacelike mixed plane Π, then λ1 and λ2 are constant. Proof. First note that if f ∈ C ∞ (M1 ), then g(hf (X), X) = g1 (h1f (X), X) for all X ∈ X(M1 ), where hf is the hessian respect to the doubly warped metric g and h1f respect to g1 . Suppose there exists a point p ∈ M1 and a spacelike vector Xp ∈ Tp M1 such that 0 ≤ g(h2 (X), X). Given an arbitrary spacelike vector Vq ∈ Tq M2 , we have 0 ≤ K(X, V ) + λ12 g(h2 (X), X) = − λ11 g(h1 (V ), V ) and applying the above lemma, λ1 is constant. Therefore, 0 ≤ − λ12 g(h2 (X), X) for all spacelike vector X and applying the above lemma again, λ2 is constant too. Suppose now the contrary case: for all spacelike vector X ∈ T M1 we have g(h2 (X), X) < 0. Then, the above lemma gives us that λ2 is constant. Thus 0 ≤ − λ11 g(h1 (V ), V ) for all spacelike vector V and the above lemma ensures that λ1 is constant too. ✷   Theorem 5.3. Let M = M1 ×(λ1 ,λ2 ) M2 /Γ be a quotient of a doubly warped product, being M1 a complete Riemannian manifold and M2 a semi-Riemannian manifold with 0 < ν2 . Suppose that F2 has no holonomy, K(Π) < 0 for all mixed nondegenerate plane Π and λ2 has some critical point. Then M is globally a doubly warped product. Proof. Suppose that there is a nonlightlike vector V ∈ T M2 with εV g(h1 (V ), V ) ≤ 0. Given an arbitrary non zero vector X ∈ T M1 , span{X, V } is a nondegenerate plane, thus − εV 1 g(h2 (X), X) − g(h1 (V ), V ) = K(X, V ) < 0 λ2 λ1 and therefore 0 < g(h2 (X), X) for all X ∈ T M1, X = 0. In the opposite case, 0 < εV g(h1 (V ), V ) for all non lightlike vector V ∈ T M2 . Applying Lemma 5.1, we get that λ1 is constant, and therefore g(h2 (X), X) = −λ2 K(X, V ) > 0 for all X ∈ T M1, X = 0. In any case h2 is positive definite and so λ2 has exactly one critical point. Take x0 ∈ M without holonomy and the associated covering map Φ : F1 (x0 ) ×(ρ1 ,ρ2 ) F2 (x0 ) → M . 18 M. Gutiérrez and B. Olea Let x1 ∈ F1 (x0 ) be the only critical point of ρ2 . If φ × ψ is a deck transformation of this covering, then ρ2 ◦ φ = cρ2 for some constant c, and it follows that φ(x1 ) ∈ F1 (x0 ) is a critical point of ρ2 too. Thus φ(x1 ) = x1 , but since F2 has no holonomy, ✷ applying Lemma 4.1, we get φ × ψ = id. Thus Φ is an isometry. Observe that in the conditions of the above theorem we can prove that M = M1 ×(ρ1 ,λ2 ) (M2 /Γ2 ). In fact, let (a0 , b0 ) ∈ M1 ×M2 such that p(a0 , b0 ) = x0 . Since the points of the fibre p−1 1 (x1 ) are critical points of λ2 , being p1 : M1 × {b0 } → F1 (x0 ) the covering map given in Lemma 3.1, and λ2 has only one critical point, it follows that p1 is an isometry. Example 5.4. The Kruskal space has warping function with exactly one critical point. Thus, the last part of the above proof shows that any quotient without holonomy is a global warped product. Now we apply the above results to semi-Riemannian submersions. We denote H and V the horizontal and vertical spaces and E v (resp. E h ) will be the vertical (resp. horizontal) projections of a vector E. Lemma 5.5. Let π : M → B be a semi-Riemannian submersion with umbilic fibres and T and A the O’Neill tensors of π. Then for arbitrary E, F ∈ X(M ) and X ∈ H, it holds (1) T (E, F ) = g(E v , F v )N − g(N, F )E v , (2) (∇X T )(E, F ) = g(F, A(X, E ∗ ))N − g(N, F )A(X, E ∗ ) + g(E v , F v )∇X N − g(∇X N, F )E v , where N is the mean curvature vector field of the fibres and E ∗ = E v − E h . Proof. The first point is immediate. For the second, we have (∇X T )(E, F ) = ∇X T (E, F ) − T (∇X E, F ) − T (E, ∇X F ). We compute each term ∇X T (E, F ) = ∇X (g(E v , F v )N − g(N, F )E v )   = g(∇X E v , F v ) + g(E v , ∇X F v ) N + g(E v , F v )∇X N   − g(∇X N, F ) + g(N, ∇X F ) E v − g(N, F )∇X E v , T (∇X E, F ) = g((∇X E)v , F v )N − g(N, F )(∇X E)v , T (E, ∇X F ) = g(E v , (∇X F )v )N − g(N, ∇X F )E v . Rearranging terms and using that ∇X E v − (∇X E)v = A(X, E ∗ ), we obtain   (∇X T )(E, F ) = g(A(X, E ∗ ), F v ) + g(E v , A(X, F ∗ )) N − g(N, F )A(X, E ∗ ) + g(E v , F v )∇X N − g(∇X N, F )E v . But g(A(X, E ∗ ), F v ) + g(E v , A(X, F ∗ )) = −g(A(X, E h ), F v ) − g(E v , A(X, F h )) = −g(A(X, E h ), F v ) + g(A(X, E v ), F h ) = g(A(X, −E h ), F ) + g(A(X, E v ), F ) = g(A(X, E ∗ ), F ). And we obtain the result. ✷ 19 Doubly warped structures We need to introduce the lightlike curvature of a degenerate plane in a Lorentzian manifold (M, g), [11]. Fix a timelike and unitary vector field ξ and take a degenerate plane Π = span(u, v), where u is the unique lightlike vector in Π with g(u, ξ) = 1. We define the lightlike sectional curvature of Π as Kξ (Π) = g(R(v, u, u), v) . g(v, v) This sectional curvature depends on the choice of the unitary timelike vector field ξ, but its sign does not change if we choose another vector field. Thus, it makes sense to say positive lightlike sectional curvature or negative lightlike sectional curvature. Lemma 5.6. Let (M, g) and (B, h) be a Lorentzian and a Riemannian manifold respectively and π : M → B a semi-Riemannian submersion with umbilic fibres. If ξ ∈ V is an unitary timelike vector field and Π = span(u, X) is a degenerate plane with u ∈ V, X ∈ H, g(u, u) = 0 and g(u, ξ) = 1, then Kξ (Π) = g(A(X, u), A(X, u)) . g(X, X) Proof. Using the formulaes of [18], we have g(X, X)Kξ (Π) = g((∇X T )(u, u), X) − g(T (u, X), T (u, X)) + g(A(X, u), A(X, u)). Since u is lightlike, the first two terms are null by the above lemma. ✷ Given a warped product M1 ×(1,λ2 ) M2 , the projection π : M1 × M2 → M1 is a semi-Riemannian submersion with umbilic fibres. The following theorem assures the converse fact. Theorem 5.7. Let M be a complete Lorentzian manifold, B a Riemannian manifold and π : M → B a semi-Riemannian submersion with umbilic fibres of dimension greater than one and mean curvature vector N . If K(Π) < 0 for all mixed spacelike plane of M , and N is closed with some zero, then M is globally a warped product. Proof. By continuity, it follows that M has nonpositive lightlike curvature for all mixed degenerated plane and thus, applying the above lemma, A(X, u) = 0 for all X ∈ H and all lightlike u ∈ V. Therefore A ≡ 0 and H is integrable and necessarily totally geodesic (see [18]), which gives rise to a warped structure, since N is closed. But being M complete M = (M1 ×(1,λ2 ) M2 )/Γ (see Remark 2.5). Now, using the formulaes of Lemma 2.2 we can easily check that the curvature of a mixed plane Π = span(X, V ) is independent of the vertical vector V and thus K(Π) < 0 for all mixed nondegenerate plane. Finally, since N has some zero, λ2 has some critical ✷ point and applying Theorem 5.3 we get the result. 20 M. Gutiérrez and B. Olea 6. Uniqueness of product decompositions In [7], the uniqueness of direct product decompositions of a non necessarily simply connected Riemannian manifold is studied, where the uniqueness is understood in the following sense: a decomposition is unique if the corresponding foliations are uniquely determined. The authors use a short generating set of the fundamental group in the sense of Gromov, which is based in the Riemannian distance. So, the techniques employed can not be used directly in the semi-Riemannian case. In this section we apply the results of this paper to study the uniqueness problem in the semi-Riemannian setting. Proposition 6.1. Let M = F1 ×· · ·×Fk be a semi-Riemannian direct product and F1 , . . . , Fk the canonical foliations. Take S an umbilic/geodesic submanifold of M and suppose that there exists i ∈ {1, . . . , k} such that Fi (x)∩Tx S is a nondegenerate subspace with constant dimension for all x ∈ S. Then the distributions T1 and T2 on S determined by T1 (x) = Fi (x) ∩ Tx S and T2 (x) = T1⊥ (x) ∩ Tx S are integrable. Moreover, T1 is a regular and umbilic/geodesic foliation and T2 is a geodesic one. Proof. It is clear that T1 is integrable. We show that T2 is integrable and geodesic in S. Consider the tensor J given by J(v1 , . . . , vi , . . . , vk ) = (−v1 , . . . , vi , . . . , −vk ), where (v1 , . . . , vk ) ∈ T F1 × · · · × T Fk , and take X, V, W ∈ X(S) with Xx ∈ T1 (x) and Vx , Wx ∈ T2 (x) for all x ∈ S. Since ∇J = 0, we have 0 = (∇V J)(X) = ∇V X−J(∇V X), which means that ∇Vx X ∈ Fi (x) since ∇V X is invariant under J. Using that S is umbilical and X, V are orthogonal, we have ∇Vx X = ∇SVx X ∈ Tx S. Therefore ∇Vx X ∈ T1 (x) for all x ∈ S. Now, we have g(∇SV W, X) = g(∇V W, X) = −g(W, ∇V X) = 0. Thus, ∇SV W ∈ T2 for all V, W ∈ T2 which means that T2 is integrable and geodesic in S. To see that T1 is umbilic, take X, Y ∈ X(S) with g(X, Y ) = 0 and Xx , Yx ∈ T1 (x) for all x ∈ S. It is easy to show that ∇Xx Y ∈ Fi (x) and since S is umbilic, we have ∇Xx Y = ∇SXx Y ∈ Fi (x) ∩ Tx S = T1 (x) for all x ∈ S. Therefore the second fundamental form of the leaves of T1 inside M satisfies I(X, Y ) = 0 for every couple of orthogonal vectors X, Y ∈ T1 , which is equivalent to be umbilic submanifolds of M . The same argument with the second fundamental form of T1 as a foliation of S shows that T1 is an umbilic foliation of S. Observe that if S is geodesic it is clear that T1 is also geodesic. Finally, we show that T1 is a regular foliation. Take the map P : F1 ×· · ·×Fk → F1 ×· · ·×Fi−1 ×Fi+1 ×· · ·×Fk given by (x1 , . . . , xk ) → (x1 , . . . , xi−1 , xi+1 , . . . , xk ) and i : T2 (p) → F1 × · · · × Fk the canonical inclusion where p ∈ S is a fixed point. The map P ◦ i is locally injective, since Ker(P ◦ i)∗x = Fi (x) ∩ T2 (x) = 0 for all x ∈ T2 (p). Therefore, we can take a neighborhood U ⊂ S of p adapted to both foliations T1 and T2 such that (P ◦ i)|V is injective, being V the slice of T2 (p) in U through p. Since P is constant through the leaves of T1 , it follows that U is a regular neighborhood of p. ✷ Doubly warped structures 21 Remark 6.2. Observe that if S is geodesic then dim Tx S ∩ Fi (x) is constant for all x ∈ S. We say that a semi-Riemannian manifold is decomposable if it can be expressed globally as a direct product. In the contrary case it is indecomposable. Lemma 6.3. Let M = F1 ×· · ·×Fk be a complete semi-Riemannian direct product and F1 , . . . , Fk the canonical foliations. Suppose S is a nondegenerate foliation of dimension greater than one and invariant by parallel translation such that Fi (p) ∩ S(p) = {0} for all i ∈ {1, . . . , k} and some p ∈ M . Then the leaves of S are flat and decomposable. Proof. Being all foliations invariant by parallel translation, the property supposed at p is in fact true at any other point of M . Take x = (x1 , . . . , xk ) ∈ F1 × · · · × Fk and suppose there is a loop αi : [0, 1] → Fi at xi and v ∈ S(x) such that Pγ (v) = v, where γ(t) = (x1 , . . . , αi (t), . . . , xk ). If we decompose v = kj=1 vj ∈ kj=1 Fj (x), k then Pγ (v) = Pγ (vi ) + j=i vj and so 0 = v − Pγ (v) = vi − Pγ (vi ) ∈ S(x) ∩ Fi (x), which is a contradiction. Therefore, Pγ (v) = v for all v ∈ S(x) and all loops γ of the form γ(t) = (x1 , . . . , αi (t), . . . , xk ). Since M = F1 × · · · × Fk has the direct product metric, Pγ (v) = v for all v ∈ S(x) and an arbitrary loop γ at x. In particular, the parallel translation along any loop of a leaf S is trivial. But this ✷ implies that it splits as a product of factors of the form R or S1 . Given a curve γ : [0, 1] → M we define vγ : [0, 1] → Tγ(0) M by −1 vγ (t) = Pγ,γ(0),γ(t) (γ ′ (t)), where P is the parallel translation. We will denote ΩM p (t1 , . . . , tm ) the set of broken geodesics in M which start at p and with breaks at ti , where 0 < t1 < · · · < tm < 1. If γ ∈ ΩM p (t1 , . . . , tm ) then vγ is a piecewise constant function, ⎧ ⎪ ⎨ v0 if 0 ≤ t ≤ t1 .. vγ (t) = . ⎪ ⎩ vm if tm ≤ t ≤ 1 which we will denote by (v0 , . . . , vm ). On the other hand, if M is complete, given (v0 , . . . , vm ) ∈ (Tp M )m+1 we can construct a broken geodesic γ ∈ ΩM p (t1 , . . . , tm ) with vγ ≡ (v0 , . . . , vm ). Now, suppose that a semi-Riemannian manifold M splits as a direct product in two different manners, M = F1 × · · · × Fk = S1 × · · · × Sk′ . We call F1 , . . . , Fk and S1 , . . . , Sk′ the canonical foliations of each decomposition and πi : M → Fi , σi : M → Si will be the canonical projections. Observe that given a point p ∈ M , the leaf of Fi through p is Fi (p) = {π1 (p)} × · · · × Fi × · · · × {πk (p)}. We will denote by Πpi the projection Πpi : M → Fi (p) given by Πpi (x) = (π1 (p), . . . , πi (x), . . . , πk (p)). Analogously, Σpi : M → Si (p) is given by Σpi (x) = (σ1 (p), . . . , σi (x), . . . , σk′ (p)). 22 M. Gutiérrez and B. Olea Theorem 6.4. Let M = F0 × · · · × Fk be a complete semi-Riemannian direct product with F0 a maximal semi-euclidean factor and each Fi indecomposable for i > 0. If M = S0 × · · · × Sk′ is another decomposition with S0 a maximal semieuclidean factor and each Sj indecomposable for j > 0 such that Fi (p) ∩ Sj (p) is zero or a nondegenerate space for some p ∈ M and all i, j, then k = k ′ and, after rearranging, Fi = Si for all i ∈ {0, . . . , k}. Proof. Fix x ∈ M and suppose that dimS1 (x) > 1 and S1 (x) = Fi (x) for all i ∈ {0, . . . , k}. Using the above lemma we have that S1 (x) ∩ Fi (x) = 0 for some i ∈ {0, . . . , k}. Moreover, since S1 (x) = Fi (x) it holds S1 (x) ∩ Fi (x) = S1 (x) or S1 (x) ∩ Fi (x) = Fi (x). We suppose the first one (the second case is similar). Proposition 6.1 ensures that T1 = Fi ∩ S1 is a regular foliation and, since S1 (x) is a geodesic submanifold, T1 and T2 = T1⊥ ∩S1 are two geodesic and nondegenerate foliations in S1 (x). We can choose p ∈ S1 (x) such that the leaf T2 (p) of T2 has no holonomy. We want to show that T1 (p) ∩ T2 (p) = {p} and apply Corollary 3.8. For this, fix an orthonormal basis in Tp M and take a definite positive metric such that this basis is orthonormal too. Denote by | · | its associated norm. Given m γ ∈ ΩM p (t1 , . . . , tm ) with vγ ≡ (v0 , . . . , vm ) we call |γ| = j=0 |vj |. Suppose there is q ∈ T1 (p) ∩ T2 (p) with p = q. Then it exists a curve in T (p) Ωp 2 (t1 , . . . , tm ) joining p and q for certain 0 < t1 < · · · < tm < 1 and so we can define r = inf{|γ| : γ ∈ ΩTp 2 (p) (t1 , . . . , tm ) and γ(1) = q}. We have that T (p) • r > 0. In fact, if r = 0 then it exists γ ∈ Ωp 2 (t1 , . . . , tm ) with γ(1) = q which lays in a neighborhood of p adapted to both foliations T1 and T2 and regular for T1 . But since T1 (p) = T1 (q) and γ is a curve is T2 (p), the only possibility is p = q, which is a contradiction. • r is a minimum. Take a sequence γn ∈ Ωp 2 (t1 , . . . , tm ) with vγn ≡ n (v0n , . . . , vm ), γn (1) = q and |γn | → r. Then we can extract a convergent subT (p) n sequence of (v0n , . . . , vm ) to, say, (v0 , . . . , vm ). Take γ0 ∈ Ωp 2 (t1 , . . . , tm ) with vγ0 ≡ (v0 , . . . , vm ). Using the differentiable dependence of the solution respect to the initial conditions and the parameters of an ordinary differential equation (see Appendix I of [14]), it is easy to show that γ0 (1) = limn→∞ γn (1) = q. Since |γ0 | = r, the infimum is reached. T (p) Now, take the map η = Σp1 ◦ Πpi : M → S1 (p), which holds • η(T2 (p)) ⊂ T2 (p), since η takes geodesics into geodesics and η∗p (T2 (p)) = T2 (p). • η(p) = p and η(q) = q. • |η∗p (v)| ≤ |v| and the equality holds if and only if v ∈ T1 (p). T (p) Consider the broken geodesic α = η ◦ γ0 ∈ Ωp 2 (t1 , . . . , tm ). Then, using that η commutes with the parallel translation along any curve, we have vα (t) = η∗p (vγ0 (t)) ≡ (η∗p (v0 ), . . . , η∗p (vm )), and so |α| < |γ0 |. Since α(1) = q we get a contradiction. Doubly warped structures 23 Therefore T1 (p) ∩ T2 (p) = {p} and S1 (p) can be decomposed as T1 (p) × T2 (p), which is a contradiction because S1 is indecomposable. The contradiction comes from supposing that S1 (x) = Fi (x) for all i ∈ {0, . . . , k}, thus it has to hold that S1 (x) = F1 (x) for example. But this means S1 = F1 . Applying repeatedly the above reasoning we can eliminate the factors with dimension greater than one, except S0 , in the decomposition S0 ×· · ·×Sk′ , reducing the problem to prove the uniqueness of the decomposition of a semi-Riemannian direct product S0 × S1 × · · · × S1 , where S0 is semi-euclidean. But, in this product, we can trivially change the metric to obtain a Riemannian direct product where ✷ we can apply [7]. Observe that the nondegeneracy hypothesis is redundant in the Riemannian case. On the contrary, in the semi-Riemannian case it is necessary as the following example shows. Example 6.5. Take L a complete and simply connected Lorentzian manifold with a parallel lightlike vector field U , but such that L can not be decomposed as a direct product, (for example a plane fronted wave, [4]). Take M = L×R with the product metric and X = U +∂t . Then X is a spacelike and parallel vector field and since M is complete and simply connected, M splits as a direct product with the integral curves of X as a factor. Thus M admits two different decomposition as direct product, although L is indecomposable. References [1] Arouche, A., Deffaf, M. and Zeghib, A.: On Lorentz dynamics: from group actions to warped products via homogeneous spaces. Trans. Amer. Math. Soc. 359 (2007), no. 3, 1253–1263. [2] Bott, R. and Tu, L. W.: Differential forms in algebraic topology. Graduate Texts in Mathematics 82, Springer-Verlag, New York-Berlin, 1982. [3] Camacho, C. and Lins Neto, A.: Geometric theory of foliations. Birkhäuser, Boston, MA, 1985. [4] Candela, A. M., Flores, J. L. and Sánchez, M.: On general plane fronted waves. Geodesics. Gen. Relativity Gravitation 35 (2003), no. 4, 631–649. [5] Derdziński, A.: Classification of certain compact Riemannian manifolds with harmonic curvature and nonparallel Ricci tensor. Math. Z. 172 (1980), no. 3, 273–280. [6] Epstein, D. B. A., Millett, K. C. and Tischler, D.: Leaves without holonomy. J. London Math. Soc. (2) 16 (1977), no. 3, 548–552. [7] Eschenburg, J.-H. and Heintze, E.: Unique decomposition of Riemannian manifolds. Proc. Amer. Math. Soc. 126 (1998), 3075–3078. [8] Escobales, R. H., Jr. and Parker, P. E.: Geometric consequences of the normal curvature cohomology class in umbilic foliations. Indiana Univ. Math. J. 37 (1988), no. 2, 389–408. [9] Jelonek, W.: Killing tensors and warped products. Ann. Polon. Math. 75 (2000), no. 1, 15–33. 24 M. Gutiérrez and B. Olea [10] Gutiérrez, M. and Olea, B.: Global decomposition of a Lorentzian manifold as a generalized Robertson–Walker space. Differential Geom. Appl. 27 (2009), no. 1, 146–156. [11] Harris, S. G.: A triangle comparison theorem for Lorentz manifolds. Indiana Univ. Math. J. 31 (1982), no. 3, 289–308. [12] Hatcher, A.: Algebraic topology. Cambridge University Press, Cambridge, 2002. http://www.math.cornell.edu/~hatcher/AT/ATpage.html. [13] Johnson, D. L. and Whitt, L. B.: Totally geodesic foliations. J. Differential Geom. 15 (1980), no. 2, 225–235. [14] Kobayashi, S. and Nomizu, K.: Foundations of differential geometry. Vol. I. Interscience Publishers, New York 1963. [15] Koike, N.: Totally umbilic foliations and decomposition theorems. Saitama Math. J. 8 (1990), 1–18. [16] Lappas, D.: Locally warped products arising from certain group actions. Japan J. Math. 20 (1994), no. 2, 365–371. [17] O’Neill, B.: Semi-Riemannian geometry. With applications to relativity. Pure and Applied Mathematics, 103, Academic Press, New York, 1983. [18] O’Neill, B.: The fundamental equations of a submersion. Michigan Math. J. 13 (1966), 459–469. [19] Palais, R.: A global formulation of the Lie theory of transformation groups. Mem. Amer. Math. Soc. 22, 1957. [20] Ponge, R. and Reckziegel, H.: Twisted product in pseudo-Riemannian geometry. Geom. Dedicata 48 (1993), no. 1, 15–25. [21] Reinhart, B. L.: Foliated manifolds with bundle-like metrics. Ann. of Math (2) 69 (1959), 119–132. [22] Romero, A. and Sánchez, M.: On completeness of certain families of semiRiemannian manifolds. Geom. Dedicata 53 (1994), no. 1, 103–107. [23] Shapiro, Y. L., Zhukova, N. I. and Igoshin, V. A.: Fibering on some classes of Riemannian manifolds. Izv. Vyssh. Uchebn. Zaved. Mat. 7 (1979), 93–96. [24] Walschap, G.: Spacelike metric foliations. J. Geom. Phys. 32 (1999), no. 2, 97–101. [25] Wang, P.: Decomposition theorems of Riemannian manifolds. Trans. Amer. Math. Soc. 184 (1973), 327–341. Received March 19, 2010. Manuel Gutiérrez: Departamento de Álgebra, Geometrı́a y Topologı́a, Facultad de Ciencias, Campus de Teatinos, Universidad de Málaga, 29071-Málaga, Spain. E-mail: [email protected] Benjamı́n Olea: Departamento de Álgebra, Geometrı́a y Topologı́a, Facultad de Ciencias, Campus de Teatinos, Universidad de Málaga, 29071-Málaga, Spain. E-mail: [email protected] This paper was supported in part by MEYC-FEDER Grant MTM2010-18089.