Riemannian foliation with dense leaves on a compact manifold Cyrille Dadi, Adolphe Codjia To cite this version: Cyrille Dadi, Adolphe Codjia. Riemannian foliation with dense leaves on a compact manifold. 2016. <hal-01337400v2> HAL Id: hal-01337400 https://hal.archives-ouvertes.fr/hal-01337400v2 Submitted on 29 Aug 2016 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Riemannian foliation with dense leaves on a compact manifold Cyrille Dadi 1 and Adolphe Codjia 2 Fundamental Mathematics Laboratory, University Felix Houphouet-Boigny, ENS 08 PO Box 10 Abidjan Ivory Coast email 1 :cyriledadi@yahoo.fr, email 2 :ad_wolf2000@yahoo.fr August 29, 2016 Abstract The purpose of this article is to generalize the following results existing respectively in [6], [8] and in [7] to Riemannian foliations with dense leaves on a compact manifold: i If F is a Lie G-foliation with dense leaves on a compact manifold, then there exists a biunivocal correspondence between the Lie subalgebras of G = Lie G and F extensions, ii if F H is an extension of a Lie G-foliation with dense leaves on a compact manifold corresponding to a subalgebra H of G and lm, F H the Lie algebra of F H foliated transverse vectors fields then we have lm, F H = {u H / [u, h] = 0 for every h H}. Let G be a Lie simply connected group and let G = Lie G be the structural Lie algebra of a Riemannian foliation F with dense leaves on a compact manifold. This study show us that there exists a discreet group H 0 and a representation ρ : H 0 Diff V where V is an open of G such as: i there exists a biunivocal correspondence between the Lie subalgebras of G = Lie G invariant by Ad ρa v 1.v for every a, v H 0 V and extensions of F, ii if F H is an extension of F corresponding to a subalgebra H of G then we have lm, F H = {u H / h, a, v H H 0 V, [u, h] = 0 and Ad ρa v 1.v u = u}. Keywords: Lie foliation, Riemannian foliation, foliation with dense leaves, extension of a foliation. 1
1 Introduction The purpose of this article is to generalize the following results existing respectively in [6], [8] and in [7] to Riemannian foliations with dense leaves on a compact manifold: i If F is a Lie G-foliation with dense leaves on a compact manifold, so it exists a biunivocal correspondence between the Lie subalgebras of G = lie G and F extensions, ii if F H is an extension of a Lie G-foliation with dense leaves on a compact manifold corresponding to a subalgebra H of G and lm, F H the Lie algebra of F H foliated transverse vectors fields then we have lm, F H = {u H / [u, h] = 0 for every h H}. To achieve there, we first established that the closure of a leaf F of lifted foliation F on the orthonormal transverse frame bundle M of a Riemannian foliation F with dense leaves on a compact manifold is a covering of the manifold M. We note that the fact that F is a covering of the manifold M entails that the dimension of the Lie structural algebra of a Riemannian foliation F on a compact manifold which is less than or equal to the codimension of the foliation F. And this allows us to say that we should limit the classification of Molino in codimension 1, 2 and 3 of Riemannian foliation on a compact manifold [12] to only a few cases of this classification. We note by H 0 the structure group of covering φ : F M. And, H 0 will be called the discreet goup of Riemannian foliation F on a compact manifold with dense leaves. That said, we show in this paper that if G is the Lie structural algebra of a Riemannian foliation with dense leaves M, F on a compact manifold M there exists a representation ρ : H 0 Diff V where V is an open of G such as: i there exists a biunivocal correspondence between the Lie subalgebras of G = Lie G invariant by Ad ρa v 1.v for every a, v H 0 V and F extensions, ii an extension is a Lie foliation if the subalgebra corresponding is an ideal of G, iii every extension F of F is a Riemannian foliation and there exists a common bundle-like metric for the foliations F and F, iv if F H is an extension of F corresponding to a subalgebra H of G then to isomorphism nearly of Lie algebras we have lm, F H = {u H / h, a, v H H 0 V, [u, h] = 0 and Ad ρa v 1.v u = u }. In particular lm, F = {u G/ a, v H 0 V, Ad ρa v 1.v u = u}. Our paper is divided into two parts: 2
-the first part is devoted to reminders on Riemannian foliations and on extensions of foliations, - the second part is devoted to the establishment of the already stated primary outcome. In all that follows, the manifolds considered are supposed connected and differentiability is C. 2 Definitions and Reminders In this section, in the direction that is helpful, we reformulate some existing definitions and theorems in [4], [6], [7], [8], [10] [11], [12], [13]. Definition 2.1 Let M be a manifold. An extension of a codimension q foliation M, F is a codimension q foliation M, F such that 0 < q < q and M, F leaves are M, F leaves meetings it is noted F F. We show that if M, F is a simple extension of a simple foliation M, F and if M, F and M, F are defined respectively by submersions π : M T and π : M T, then there exists a submersion θ : T T such that π = θ π. We say that the submersion θ is a bond between the foliation M, F and its extension foliation M, F. It is shown in [6] that if the foliation M, F and its extension M, F are defined respectively by the cocycles U i, f i, T, γ ij i I and U i, f, T, γ i ij i I then we have f i = θ i f i and γ θ ij j = θ i γ ij where θ s is a bond between the foliation U s, F and its extension foliation U s, F. Definition 2.2 Let F q be a codimension q foliation on a manifold M. A flag of extensions of F q is a sequence D k = F Fq q 1, F q 2,..., F k of foliations on the manifold M such that F q F q 1 F q 2... F k and each foliation F s is a codimension s foliation. For k = 1, the flag of extensions D k will be called complete and will be Fq denoted D Fq. If each foliation F s is a Riemannian foliation, the flag of extensions D k Fq will be called flag of Riemannian extensions of F q. The following theorem is the biunivocal correspondence theorem between Lie subalgebras of G =Lie G and the extensions of a Lie G foliation with dense leaves existing in [8]. Theorem 2.3 [8] Let M, F be a Lie G-foliation with dense leaves on compact connected manifold and let G be the Lie algebra of G. Then: 3
1-There exists a biunivocal correspondence between the Lie subalgebras of G or if you prefer between the connected Lie subgroups of G and extensions of F. 2- An extension of F is a Riemannian G H-foliation having trivial normal bundle and defined by a 1-form with values in G H. 3- An extension of F is transversely homogeneous resp. Lie if and only if the Lie subgroup of G corresponding is a closed subgroup resp. Normal subgroup in G. In [7] was calculated lm, F H where F H is the extension of a Lie G foliation with dense leaves on a compact manifold corresponding to a Lie subalgebra H of G. This calculation gave us the following result: Theorem 2.4 [7] Let H be a Lie subalgebra of Lie algebra G of a Lie foliation F with dense leaves on a compact manifold, let ω be a 1-form of Fedida defining F and let F H be the extension of F corresponding to H. Then: i lm, F H = lm, F <h> h H where F <h> is the extension of F corresponding to the Lie subalgebra < h > of G generated by h. ii ω lm, F H H where H is the ortho-complementary of H in G by the transverse metric associated with F. iii For h G, ω lm, F <h> = {u < h > / [u, h] = 0 }. iv ω lm, F H = {u H / [u, h] = 0 for every h H}. The theorem and the three following proposals allow us to give a generalization of the previous two theorems for Riemannian foliations with dense leaves on a compact manifold. Theorem 2.5 [7] Let F be a Lie G foliation with dense leaves on a compact manifold M, let λ be a metric on M which is bundle-like for F and which admits λ T as its associated transverse metric and let X be a F transverse foliated vectors field. Then: i for every point a M there exist an open F-distinguished V a of M containing the point a such that the restriction X Va of X at V a is a local F-transverse Killing vectors field, ii any vectors field left invariant of G is a Killing vectors field for the F- transverse metric λ T left invariant, 4
iii in the case where G is connected, the right translation R a associated with the element a G is an isometry for the metric λ T left invariant. Proposition 2.6 [7] Let F be an extension of a Riemannian foliation F on a manifold M. Then lm, F lm, F. Proposition 2.7 [12] Let F be a codimension q Riemannian foliation on a compact connected manifold M, let F be the closure of a leaf F of lifted foliation F of F on the orthonormal transverse frame bundle M, let φ:m M be the projection which to a frame at x associates x. Them: i φ F is a leaf of F and φ F = φ F, ii the map φ : F φ F is a locally trivial fibration. Proposition 2.8 [12] Let M, λ be a Riemannian connected manifold and let K M, λ be a Lie algebra of Killing vectors field on M, λ. Then the orbits of K M, λ having maximal dimension form an open dense in M. 3 Riemannian foliation with dense leaves on a compact manifold In what follows G is a Lie group of Lie algebra G, X r resp. X l is the vectors field on G right resp. Left invariant obtained from X G and L a resp. R a is the left resp. right translation associated with a G. Proposition 3.1 Let K M, λ be a Lie algebra of Killing vectors fields on a Riemannian connected manifold M, λ, let i x = {X K M, λ /X x = 0} be the isotropy of K M, λ at the point x M, λ, let O x be the orbit at the point x of K M, λ and let O x0,x be the orbit at the point x M, λ of the isotropy i x0 where x 0 M, λ. If for x M, λ, all orbits O x have the same dimension then O x0,x = {x} for every x 0, x M, λ 2. Proof. Let x M, λ. We have dim i x = dim K M, λ dim O x. As all orbits O x have the same dimension then dim i x = dim i y for every y M, λ. We note before continuing that i x is a Lie subalgebra of Killing vectors fields of K M, λ. Let x 0 M, λ and let i x0,x be the isotropy at the point x of i x0. We know [12] there exists an open neighborhood U x0 of x 0 such as if x U x0 then i x0,x = i x that is to say the isotropy at the point x of i x0 is equal to the isotropy at the point x of K M, λ for every x U x0. For proof we assume 5
that k 1, k 2,..., k r, k 1, k 2,..., k s is a base of K M, λ such as k 1, k 2,..., k r is a base of i x0. For every j {1,..., s}, k j x 0 0. Hence there exists an open U x0 containing x 0 such as for every x U x0 and for every j {1,..., s}, k j x 0. Let x U x0 and let Y i x. there exists two finite sequences y j 1 j r and r s y j 1 j s of real numbers such that Y = y j.k j + y j.k j. We have r y j.k j x + j=1 j=1 j=1 s y j.k j x = 0 because Y i x. j=1 Quits to reorder the base of vectors fields k 1, k 2,..., k r of i x0 we can assume that there exists r {0, 1,..., r} such as k j x 0 for j r and k j x = 0 for r + 1 j r. r s Thus, the fact that Y i x implies that y j.k j x + y j.k j x = 0. As k j x 0 for j r and k j x 0 and k 1, k 2,..., k r, k 1, k 2,..., k s is a free system of vectors fields of K M, λ then k 1 x,..., k r x, k 1 x,..., k s x form a free system. Therefore the equality shows that y j = 0 for j r and y j = 0 for every j. r s It follows from this that the equality Y = y j.k j + y j.k j implies that Y = r j=r +1 j=1 j=1 y j.k j. Therefore Y i x0 and i x i x0. The constancy of the dimension of i y for all y M, λ implies that i x = i x0. Thus for any x U x0, i x0,x = i x = i x0 and the orbit O x0,x of i x0 at the point x is the point x. We know that [14], [12] the orbits of Lie algebra of Killing vectors fields on Riemannian connected manifold V having maximal dimension form an open dense in V. From where, the fact that the maximal dimension of the orbits of i x0 in the open U x0 is zero implies that O x0,x = {x} for all x M, λ. We note that this proposition means that when a Lie algebra of Killing vectors fields K M, λ on a connected Riemannian manifold M, λ has his orbits having the same dimension then the isotropy i x of K M, λ at any point x M, λ induces on M, λ a null Lie algebra of Killing vectors fields. That said, the previous proposition allows us to establish the following result: Proposition 3.2 Let H be a closed Lie subgroup of a connected Lie group G, let λ be a metric on G left invariant, let G =LieG and let H =LieH. If λ is invariant by right translations obtained from the elements of H then the Lie subalgebra H is an ideal of G. Proof. If the Lie group H is discreet, then H = {0}. And in this case H is an ideal of G. In what follows we assume that H is not discreet. j=1 j=1 6
The Lie subgroup H is closed in the Lie group G. From where π : G G H is a principal fibration having H for structure group. We note in passing that π is a Riemannian submersion because the metric λ left invariant on G is invariant by right translations obtained from the elements of H. Let α be the left Maurer Cartan form of G, let p H : G H be the orthogonal projection on H and let α H = p H α. It is easy to see that α H is a differential form on G with values in H. Let a H, let X T G and let p H : G H be the orthogonal projection on H where H is the ortho-complementary of H in G. We have R aα H X = α H R a X = p H α R a X = p H Ad a 1 α X = p H Ad a 1 p H α X + p H α X = p H Ad a 1 p H α X + p H Ad a 1 p H α X. We signal that for all a H, H and H are invariant by Ad a 1 because the left invariant metric λ is right invariant by translations obtained from elements of H. It follows from this that p H Ad a 1 p H α X = Ad a 1 p H α X and p H Ad a 1 p H α X = 0. Thus, we get that for all a H and for all X T G, R aα H X = Ad a 1 p H α X = Ad a 1 α H X. And, this means that α H is a connection on the principal bundle π : G G H. Let F G,H be the foliation obtained by the left translations of H and let T F G,H be the orthogonal bundle of T F G,H. It is clear to see that T F G,H ker α H. But dim ker α H = dim G H = dim T F G,H so ker α H = T F G,H. In what follows the transverse bundle V F G,H is identified to T F G,H. Above all, we note that H is not discreet then foliation F G,H is not a foliation by points. Let G r be the Lie algebra of right invariant vectors fields of G and let H r be the Lie subalgebra of right invariant vectors fields obtained from the vectors of H. Any vectors field u r G r associated with the vector u G commutes with every vectors field left invariant. From where u r is a F G,H foliated vectors field. As the submersion π : G G H defined the foliation F G,H then u r is projected by π on G H following a vectors field notes that ur. Let X G H be the Lie algebra of vectors fields tangent to G H, let X G be the Lie algebra of vectors fields tangent to G, let w X G H, let X X G, 7
let w be the horizontal lift of w, let X h be the horizontal component of X and let X v be the vertical component of X. We have [1], [13] for w 1 X G H and w2 X G H, π [ w 1, w 2 ] = π [ w 1, w 2 ] h = π [w1, w 2 ] = [w 1, w 2 ] = [π w 1, π w 2 ]. Furthermore, for [ all for all u 1, u 2 G 2 as u r 1 and u r 2 are F G,H foliated vectors fields then u r 1 h, u r 2 v] is a section of T F G,H because u r 1 v and u r 2 v are sections of T F G,H and [ u r 1 h, u r 2 v] = [u r 1, u r 2 v ] [u r 1 v, u r 2 v ]. It follows from the foregoing that π [u r 1, u r 2] = [π u r 1, π u r 2] for all u 1, u 2 G 2 because π [u r 1, u r 2] = π [ u r 1 v + u r 1 h, u r 2 v + u r 2 h] [ = π [u r 1 v, u r 2] + π u r 1 h, u r 2 v] [ + π u r 1 h, u r 2 h] [ = π u r 1 h, u r 2 h] [ = π π u r 1, π ] u r 2 [ ] = π π u r 1, π π u r 2 = [π u r 1, π u r 2] The equality π [u r 1, u r 2] = [π u r 1, π u r 2] for all u 1, u 2 G 2 show that G r = π G r is a Lie algebra because G r is a Lie algebra. As the right invariant vectors fields u r for all u G are Killing fields for the metric λ, then the fact that π is a Riemannian submersion implies that the vectors fields u r are also Killings vectors fields. Thus G r is a Lie algebra of Killing fields on G H. We note that u r h is the F G,H foliated transverse vectors field associated with u r since ker α H = T F G,H and we have identified T F G,H and the transverse bundle V F G,H. We note also that the horizontal lift ũ r of u r = π u r checks ũ r = u r h. Let u 1, u 2...u s, u s+1,..., u q be an orthonormal base of G such as u s+1,..., u q is a base of H and u r i the right invariant vectors field obtained from u i. 8
It results from the equality ũ r i = u r i h that u r i h is invariant by right translations obtained from the elements of H. Therefore, these right translations being isometric, we have for all i s and for all a H, u r 1 h a, u r 2 h a,..., u r s h a is an orthonormal base of T a H and ũ r i a = ur i h a = u r i a. Thus the fact that π is a riemannian submersion implies that e e e u r 1, u r 2,..., u r G s is an orthonormal base of T e H where e is the class of the identity element e of G in G H. And, this entails that Gr is a Lie algebra of Killing fields on G H having dim G H for maximal dimension of its orbits. We know that [14], [12] the orbits of maximal dimension of a Lie algebra of Killing fields on a connected manifold V form an open dense in V. From where there is an open U r of G H dense in G H and containing e on which the orbits of G r are the dimension of G H. We note that there exists an open U r U r such as U r is connected and e U r. We note also that G H is connected because G is connected and the map π : G G H is continuous. According to the Proposition 3.1 the dimension of any orbit O e, x of isotropy i e of G r at every point x of U r is null. Now the the orbits of maximal dimension of the Lie algebra of Killing fields i e on the manifold G H form a dense open { x } of G H. From where the fact that O e, x = for every x U r implies that { x } O e, x = for every x G H. It is easy to see that H r = π H r is a Lie subalgebra of i e. Therefore the { x } fact that O e, x = for every x G H shows us that Hr = π H r is null. This means that any vector field right invariant obtained from the vectors of H is tangent to the foliation F G,H. Thus for every a G, we have ah = Ha. In other words H is an ideal of G. The following result is another consequence of the proposition 3.1. This result is the basis of the generalization that we do in this paper. It allows us to look Riemannian foliations with dense leaves on a compact manifold with a new look. With this result we can for example associated with a Riemannian foliation F with dense leaves on a compact manifold a finished group whose properties depend on the nature of F. Proposition 3.3 Let F be a riemannian foliation with dense leaves on a compact connected manifold M and let F be the closure of a leaf F of lifted foliation F of F on the orthonormal transverse frame bundle M. Then F is a compact covering of M. 9
Proof. Let F be a riemannian foliation with dense leaves on a compact connected manifold M, let φ : F M be the projection which to a frame at x associates x, let f : U U be a Riemannian submersion defining Riemannian foliation F on a distinguished connected open U and let f : φ 1 U E U be the projection of φ 1 U on the orthonormal frame bundle of the local F- quotient manifold U. We know that [12] φ : F M is a principal fibration and the submersion f defined the lifted foliation F on φ 1 U and there exists a submersion φ : E U U making the diagram φ 1 U φ U f E U f commutative because φ sends the fibers of f on the fibers of f. According to Molino [12] φ : E U U is the orthonormal transverse frame bundle above the local F-quotient manifold U. Let X be a Killing vector fields on U. We recall that if ϕ X t is the local 1 parameter group associated to X t <ε then the local 1 parameter group ϕ X t x where x E U defined a vectors field X on E U that we call the lifted vectors field of X on E U [12]. This lifted vectors field commutes with the canonical parallelism [12] of E U. We also recall that [12] any vectors field Y of E U coincides in a neighborhood of each point of E U with the lift of a local Killing vectors field on U if and only if Y commutes with the canonical parallelism of E U. That said, as F is a Riemannian foliation with dense leaves then the Lie algebra G r of right invariant vectors fields obtained from the structural Lie algebra G of F is a Lie algebra of Killing vectors fields operating transitively [12] respectively on each connected component of E U and on U. To be specific it is the Lie algebra of Killing fields φ G r isomorphic [12], [11] to G r which operates transitively on U and the lifted Lie algebra of φ G r on E U is G r. Let x 0 U. According to Proposition 3.1 the dimension of any orbit O x0,x of the isotropy i x0 of φ G r at every point x of U is null. Thus, the isotropy i x of φ G r for every x U is null. As dim i x = dim φ G r dim O x = dim φ G r dim U φ where O x is the orbit at the point x U of φ G r then dim G = dim G r = dim φ G r = dim U = co dim F. Let H be the structure group of the principal bundle φ : F M. Using the fact that the restriction F of F at F is a Lie G-foliation with F dense leaves is obtained that: dim M + dim H = dim F = dim F+ dim G = dim F+co dim F = dim M. U 10
And, equality dim M + dim H = dim M show that dim H = 0. Which means that H is discreet. It follows from the foregoing that the principal bundle φ : F M is a covering. Before concluding, we signal that H is finished because it is a structure group of a compact covering. We note that this proposition allows us to say that the dimension of the Lie structural algebra of a Riemannian foliation on a compact connected manifold is lower or equal to the codimension of this foliation. Corollary 3.4 All Riemannian foliation F with dense leaves on a compact connected simply connected manifold M is an abelian G-foliation with dense leaves. Proof. Let F be a Riemannian foliation with dense leaves on a compact connected simply connected manifold M and let F be the closure of a leaf F of lifted foliation F of F on the orthonormal transverse frame bundle M. According to the previous proposition F is a covering of M. As M is a simply connected manifold then M is diffeomorphic to F. And, that implies that F is a Lie G-foliation with dense leaves. As the Lie structural algebra of a Riemannian foliation on a simply connected compact connected manifold is an abelian Lie algebra [12] then we obtain that F is an abelian Lie G-foliation with dense leaves. The following theorem is the main result of this article. This is a generalization of theorems 2.3 and 2.4 to Riemannian foliations with dense leaves on a compact manifold. In what follows the structure group of covering φ : F M will be noted H 0 and we will call it the discreet goup of Riemannian foliation F with dense leaves. Theorem 3.5 Let G = Lie G be the Lie structural algebra of a Riemannian foliation with dense leaves M, F on a compact manifold M, let H 0 be the discreet group of F and let λ be a metric on M which is bundle-like for F and which admits λ T as its associated transverse metric. Then there exists a representation ρ : H 0 Diff V where V is an open of G such as: i there exists a biunivocal correspondence between the Lie subalgebras of G = Lie G invariant by Ad ρa v 1.v for every a, v H 0 V and F extensions, ii an extension is a Lie foliation if the subalgebra corresponding is an ideal of G, iii every extension F of F is a Riemannian foliation and there exists a common bundle-like metric for the foliations F and F, iv if F H is an extension of F corresponding to a subalgebra H of G then to Lie algebra isomorphism nearly we have lm, F H = {u H / h, a, v H H 0 V, [u, h] = 0 and Ad ρa v 1.v u = u }. 11
In particular lm, F = {u G/ a, v H 0 V, Ad ρa v 1.v u = u }. Proof. Let F be a Riemannian foliation on a compact connected manifold M, let F be the closure of a leaf F of lifted foliation F of F on the orthonormal transverse frame bundle M, let φ:f M be the projection which to a frame at x associates x and let F F be the restriction of F at F. We now that φ : F M is a covering having H 0 for structure goup cf. Proposition 3.3. In what follows U i, f i, T, γ ij i I denotes a foliated cocycle defining the Riemannian foliation F such as open U i are open of local trivialization of covering φ : F M and f : φ 1 U i i E U i denotes the projection of φ 1 U i on the orthonormal transverse frame bundle φ i : E U i U i above the local F-quotient manifold U i of U i. We note that the fact that φ : F M is a covering of M implies that for every open U i of foliated cocycle U i, f i, T, γ ij i I of F there are open U of ia F where a H 0 such that: - φ 1 U i = U and φ : U U is a local difféomorphism, ia ia i a H 0 - for every a, b H 0 H 0, R b U ia = U where R iab b is the right translation on F associated to b. And, U is an open cover of F ia i,a I H. 0 According to Molino [12], each submersion f : φ 1 U i i E U i defined the Lie G foliation F on F φ 1 U i. Thus E U i G. We note also that according to Molino [12] for all a H 0 and all x φ 1 U i we have f R i a x = ρ a f i x where ρ a is a diffeomorphism on the open V = E U i of G induce by R a because F is invariant by F R a. We easily verify that ρ : H 0 Diff V is a representation and ρ H 0 H 0. Let λ be a F bundle-like metric on M. As φ : F M is a covering and F = φ F then the metric λ = φ λ is F a F bundle-like metric on φ : F M and the right translation R F a on F associated to a H 0 is an isometry for the metric λ. The equality f i R a = ρ a f i implies that ρ a is an isometry because R a is an isometry and f i is a riemannian submersion defining F F on φ 1 U i. Before finishing these remarks we signal that [12] for all i I the diagram φ 1 U i φ U i f i E U i f i φ i U i 12
is commutative and the f R i a = ρ a f implies again that φ i i : E U i U i is a covering having ρ H 0 H 0 for structure group. That said, as F is a Riemannian foliation with dense leaves then the Lie algebra G r of right invariant vectors fields obtained from the structural Lie algebra G of F is a Lie algebra of Killing vectors fields operating transitively [12] on each connected component of E U and for each X r G r, φ i Xr is a vector fields on U then for all v, a E U i H0 and for all X r G r, ρ a X r v = X r ρ a v = R v 1.ρ a v X r v. We note that the fact G r operate transitively [12] on each connected component of E U implies that for all v, X v E U i Tv E U there exists X r G r such as X v = Xv. r Thus for all v, x, a E U i φ 1 U i H 0 and for all X v, Y x T v E U T x φ 1 U i, ρ a X v = R v 1.ρ a v X r v = R v 1.ρ a v X v and f R i a Y x = ρ a f Y x = R i v 1.ρ a v f Y x i for v = f i x. F F In what follows we will designated by ω the 1-form of Fedida of Lie G foliation whose restriction ω i at each open U i = φ 1 U i is such that ω i = f i αi where α i is the restriction of left Maurer Cartan form α of G at E U i. For all x, a φ 1 U i H 0 and for all Y T x z φ 1 U i we have R a ω Y x = ω i Ra Y x f = α i R i a Y x f = α i ρ a Y x i f i = α i R v 1.ρ a v = Ad ρa v 1.v ω Y. x Y x for v = f i x i Let F be an extension of F and let φ F be the inverse image of F. We easily verify that F φ F. This implies that cf. theoreme 2.3 there F exists a Lie subalgebra of G =Lie G corresponding to φ F. It will be noted G F. 13
We know that [6] φ F is defined by the differential system P defined on F by P x = T x F ev F x G F where G F is the Lie algebra of F foliated transverse vectors fields associated F to G F and ev x X = X for every X G x F. Let a H 0. The foliation φ F is invariant by the right translation Ra on F associated to a. Hence the differential system x P x is invariant by Ra. As T F and the differential system x P x are invariant by the F isometry Ra and as the ortho-complementary of T x F in P x is ev x F G F then for all x φ 1 U i, R a ev x G F = ev R a x G F. But [6] for all x φ 1 U i ev x G F = G F so for all v = f i ω i x E U i we have G F = ω i ev R a x G F R = ω i a ev x G F = f i αi Ra ev x G F f = α i R a ev x G F = α R v 1.ρ a v f i ev x G F f = Ad ρa v 1.v α ev i x G F f = Ad ρa v 1.v i αi ev x G F = Ad ρa v 1.v ω i ev x G F = Ad ρa v 1.v G F Reciprocally, suppose there is a Lie subalgebra G of G such as Ad ρa v 1.v G = G for all v = f i x E U i and for all a H0. As Ad ρa v 1.v G = G for all a H 0 and for all v = f i x E U i then the differential system S defined on F by S x = ev x G where G is the Lie algebra of F foliated transverse vectors fields associated to G, is F invariant by R a for all a H 0. 14
Indeed for a H 0 we have: ω i ev R a x G = G = Ad ρa v 1.v G for v = f i = Ad ρa v 1.v evx G = Ra ω i evx G = ω i R a evx G. x The fact that ω i T : R a x F G is an isomorphism, because F F G Lie foliation with dense leaves, implies that R a evx x G = ev R a x G F is a for all a H 0 and x F. It follows from the foregoing that the extension F of F corresponding to G F G is invariant by the right translation of the elements of H 0 because [6], [8] T x F = T G x F ev F x G for all x F and T F F is invariante by R a for all a H 0. Thus F is projected by φ into an extension F of F. G ii Let F be an extension of Lie of Riemannian foliation M, F with dense leaves on a compact manifold M, let φ F be the lifted foliation of F on the covering φ : F M, let lm, F be the Lie algebra of F foliated transverse vectors fields and let l M, F be the lifted of lm, F on the covering F. For all X l M, F and for all Y X F, φ F where X F, φ F is the Lie algebra of vectors fields tangent to φ F, we have i, a I H 0, [X φ, Y ] ] = [φ /U X, φ ia /U Y. ia /U ia ] We note that i, a I H 0, [φ X, φ /U Y is tangent to F ia /U ia because φ X is F foliated and φ /U Y is tangent to F. It follows ia /U [X ia from this that φ, Y ] is tangent to F for all i, a I H /U 0. Which ia causes that [ X, Y ] is tangent to φ F. Thus all vectors fields of l M, F is φ F foliated. Using the fact that for any i, a I H 0, φ : U ia U i is an isometry relatively to the metrics λ and λ, it is easily verified that all vectors field of l M, F is φ F tranverse since we identify the orthogonal bundle T φ F of T φ F and the transverse bundle V φ F. 15
We note that l M, F is stable for the Lie bracket of two vectors fields. Indeed for all X l M, F and Y l M, F, there exists X lm, F and Y lm, F such as φ X = X and φ Y = Y. The lifted vectors fields X and Y are invariant by the right translations obtained from the elements of H 0. It follows from this that [ X, Y ] is invariant by the right translations obtained from elements of H 0. Thus [ X, Y ] is projected by φ along a vectors field on M. And we have for all i, a I H 0, [ φ X, Y ] [X = φ /Ui, Y ] /U ia ] = [φ X, φ /U Y ia /U ia [ φ = X, φ /Ui Y ] /U i = [ φ X, φ Y ] /U i = [X, Y ] /Ui. [ The above shows that φ X, Y ] = [X, Y ]. As [X, Y ] lm, F then [ X, Y ] l M, F. Which means that l M, F is stable for the Lie bracket. It follows from the above that l M, F is a Lie algebra of φ F foliated transverse vectors fields. We note that l M, F has the same dimension as lm, F. The fact that φ F is an extension of the Lie foliation F ensures by Proposition 2.6 that l M, F is a Lie algebra of vectors fields F foliated transverse. F F Let G F be the Lie algebra of F foliated transverse vectors field associated F to the Lie subalgebra G F corresponding of the extension φ F of F. F We signal that l M, F is ortho- complementary to G F in lf, F F. Indeed dim l M, F = dim lm, F = co dim φ F and [6] dim G F = co dim F F co dim φ F = dim lf, F dim F l M, F and l M, F is orthogonal to G F because l M, F is a Lie algebra of φ F foliated transverse vectors fields. For X l M, F and Y G F we have [ X, Y ] lf, F and the F equality show us that [ X, Y ] is tangent to φ F that is to say [ X, Y ] X F, φ F. Thus for all X l M, F and Y G F we have [ X, Y ] G F because [6] lf, F F X F, φ F = G F. 16
It results from the above that G F is an ideal of lf, F. And this implies F that G F is an ideal of G. iii Let F be an extension of F and let λ be a F-bundle-like metric on M. Quits to reduce the size of opens U i, we can assume distinguished both for F and F each open U i. The local isometry φ : U ia U i sends the leaves of U ia, F F on leaves of U i, F and leaves of U ia, φ F on leaves of U i, F for all i, a I H 0. As the metric λ = φ λ is bundle-like for U ia, F F and U ia, φ F then the metric λ is bundle-like for U i, F and U i, F. And, this implies that the metric λ is a common bundle-like metric for the foliations F and F. iv Let F H be the extension of F corresponding to a Lie subalgebra H of G, let F H be the extension of F corresponding to H, let lf, F F H be the Lie algebra of F H foliated transverse vectors fields and let l invf, F H be the Lie subalgebra of lf, F H of invariant vectors fields by the right translations R a for all a H 0. One checks easily through the equality F H = φ F H and thanks to the fact that i, a I H 0, φ : U ia U i is a local isometry that φ : l inv F, F H lm, F H is an isomorphism of Lie algebras. Using the theorem 2.4 and the fact that for all a H 0 and Y x T x φ 1 U i R a ω Y x = Ad ρa v 1.v ω Y x one easily gets equality between ω l inv F, F H and for v = f i x {u H / h H, [u, h] = 0 and a, v H 0 V, Ad ρa v 1.v u = u }. But ω : lf, F G is an isomorphism of Lie algebras because the Lie F foliation F has leaves that are dense, so ω φ 1 is an isomorphism of Lie F algebras between lm, F H and {u H / h H, [u, h] = 0 and a, v H 0 V, Ad ρa v 1.v u = u }. Corollary 3.6 Let G be the structural Lie algebra of a Riemannian foliation with dense leaves F on a compact manifold M. Then F admits a complete flag of extensions if and only if F is an abelian Lie G foliation with dense leaves. 17
Proof. Let q be the codimension of F and let λ be a F-bundle-like metric on M. We suppose that F admits a complete flag of extensions D F = F q 1, F q 2,..., F 1. It is easily verified for i {1, 2,..., q } that the dimension of the bundle T F i T F i 1 is 1. We denote by X i the unified field that orients T F i T F i 1. We note in passing that the foliation F 0 has one leaf and this leaf is the manifold M. We note also that T F i 1 = T F i T F i T F i 1 = T F i < X i > That said, we can say that X i is a F i foliated transverse vectors field. Indeed F is a Riemannian foliation with dense leaves then according to the theorem 3.7, D F is a complete flag of Riemannian extensions and the metric λ is a common bundle-like metric for the foliations F i. As X i is tangent to T F i then we have for all Y X F i, 0 = Y λ X i, X i = 2λ [Y, X i ], X i. Therefore [Y, X i ] is orthogonal to X i. But [Y, X i ] X F i 1 and T F i 1 = T F i < X i > so [Y, X i ] is tangent to T F i. Which implies that X i lm, F i because X i is tangent to T F i. In what follows we denote F by F q. According to the proposition 2.6 for all i and j such as 1 i < j q we have lm, F i lm, F j. It follows that for all i j we have X i lm, F j. The fact that the unified vectors fields X i are orthogonal two by two implies that F j is a Lie G j foliation for all j and lm, F j =< X j, X j 1,...X 1 >. Indeed for all j the codimension of F j is j and the leaves of F j are dense. Let U be a F q distinguished open, let f : U U be a Riemannian submersion defining F q on U, let ω q be a Fedida 1-forme of F q such as ω q/u = f α q where α q is the left Maurer Cartan form of G q, let ω q X i = X i and let G q = Lie G q. We note in passing that ω q :lm, F q G q is an isomorphism of Lie algebra because F q is a Lie G q foliation with dense leaves. It is easily verified using the equality ω q/u = f α q and the fact that the unified vectors fields X i are orthogonal two by two that X q, X q 1,...X 1 is an orthonormal base for G q. Let H j be the Lie subalgebra of G q corresponding to the extension F j of F q. 18
We note [6] that dim H j = q j. And that implies that dim H j = j. According to the Theorem 2.4 we have ω q lm, F j H j. Consequently the fact that dim H j = j and ω q lm, F j = ω q < X j, X j 1,...X 1 > =< X j, X j 1,...X 1 > implies that Thus we have As H j =< X j, X j 1,...X 1 >. H j =< X q, X q 1,...X j+1 >. ω q lm, F j =< X j, X j 1,...X 1 > and H j =< X q, X q 1,...X j+1 > then according to the theorem 2.4 for all i and k such as 1 i j < k q we have [ ] X i, X k = 0. Is thus obtained for all i and k such that i < k, [ ] X i, X k = 0. And this implies that G q is an abelian Lie algebra. Consequently F is an abelian Lie G foliation because G = G q. Reciprocally we suppose that F is an abelian Lie G foliation. In this case the theorem 2.3 show us that F admits a complete flag of extensions. References [1] A.Abouqateb et D.Lehmann, 2010. Classes caractéristiques en Géométrie diff érentielle ISBN 978-2-7298-6083-7 Ellipses Edition Marketing S.A., 2010. 32 rue Bargue 75740 Paris cedex 15. [2] R.Almeida et P.Molino, 1986. Flot riemanniens sur les 4-variétés compactes Tôhoku Mathematical Journal, The Second Series, Vol. 38, no. 2, pp. 313-326. [3] B.Bossoto and H.Diallo, 2002, Sur les drapeaux de feuilletages riemanniens. JP Journal of Geometry and Topology, 2,3, 281-288. [4] R.A. Blumental and J.Hebda, 1983, De Rham decomposition theorem for foliated manifolds. 33 2, 133-198. [5] Y.Carrière, Flots Riemanniens. In Structures transverses des feuilletages, Astérisque, 116 1984, 31-52. [6] C.Dadi, 2008. Sur les extensions des feuilletages. Thèse unique, Université de Cocody, Abidjan. 19
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