Foliate fiield of an extension of a minimal Lie foliation on a compact manifold

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1 Foliate fiield of an extension of a minimal Lie foliation on a compact manifold Cyrille Dadi, Adolphe Codjia To cite this version: Cyrille Dadi, Adolphe Codjia Foliate fiield of an extension of a minimal Lie foliation on a compact manifold 2017 <hal > AL Id: hal Submitted on 1 Jan 2017 AL 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 AL, 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

2 Foliate field of an extension of a minimal Lie foliation on a compact manifold Cyrille Dadi 1 and Adolphe Codjia 2 Fundamental Mathematics Laboratory, University Felix ouphouet-boigny, ENS 08 PO Box 10 Abidjan Ivory Coast 1 :cyriledadi@yahoofr, 2 :ad_wolf2000@yahoofr December 31, 2016 Abstract 1 Is shown in [3], [4] as any extension of a Lie minimal G-foliation F on a compact manifold M corresponds a Lie subalgebra of G We denote by F the extension corresponds to the subalgebra In this paper we seek to determine the Lie algebra lm, F of transverse foliated fields of the extension F This study show us that 1 Introduction lm, F = {u / [u, h] = 0 for every h } Is shown in [3], [4] as any extension of a minimal Lie G-foliation F on a compact manifold M corresponds a Lie subalgebra of G We denote by F the extension corresponding to subalgebra The purpose of this article is to compute the Lie algebra lm, F of F - foliate transverse vector fields To achieve this objectif, the following results are first established: i If F is a minimal Lie G-foliation on a compact manifold M then there exists an open neighborhood V e of the neutral element e of G such as the restriction X V e to V e of a left invariant vectors field X on G is a Killing vector field for all left invariant metric on G Mathematics Subject Classification: 53C05, 53C12, 58A30 Keywords and phrases : foliation, foliation with dense leaves, extension of a foliation riemannian foliation, Lie foliation, Lie foliation with dense leaves, foliate vector field, foliate vector field of a extension 1

3 ii If F is an extension of Riemannian foliation F on a compact manifold M then lm, F lm, F That said, we show in this paper that all informations about lm, F are contained in the Lie algebra G of G In fact, it is established to an isomorphism of Lie algebras nearly that: lm, F = {u / [u, h] = 0 for all h } This paper is articulated in three parts: - the first part is devoted to reminders about extensions of foliation and G -foliations, - the second part is devoted to left invariant vector fields on the Lie group G of a minimal Lie G-foliation on a compact manifold M, - the third part is devoted to computation of lm, F 2 Reminders In this section, are formulated in the helpful sense some definitions and theorems which are in [3], [4], [6] Definition 21 An extension of a q codimensional foliation M, F is a q codimensional foliation M, F such as 0 < q < q and the leaves of M, F are union of leaves of M, F we note 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 respectively defined by submersions π : M T and π : M T, then there exists a submersion θ : T T such as π = θ π We say that the submersion θ is a bond between the foliation M, F and its extension M, F It is shown in [3] that if the foliation M, F and its extension M, F are respectively defined 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 U s, F In what follows G is the Lie algebra of a q dimensional Lie group G, is a q dimensional Lie subalgebra of G, e 1,, e q is a base of G such as e q,, e +1 q is a base of and ω is a 1-form on a manifold M taking values in G Relatively to q e 1,, e q we have ω = ω i e i What is written ω = ω 1,, ω q If dω [ω, ω] = 0 and ω1,, ω q are linearly independent at any point of M then the differential system ω 1 = = ω q = 0 defines a q codimensional foliation F i=1 2

4 Definition 22 The foliation F defined is called G -foliation defined by the 1-form ω The following proposition wase established by El KACIMI, G GUASP and M NICOLAU in [5] And, this proposition is the basis of this article Proposition 23 Let F be a G -foliation defined by a 1-form ω and let F = π F be the lifted foliation of the F -foliation on the universal covering π : M M of M Then, there exists a differentiable map D : M G transverse for the foliation F G, obtained by the left translations of that is to say that for every x M, T D x G = T x D T x M + T D x F G, and a homomorphism ρ : π 1 M G such as: i D is ρ-equivariant that is to say that D γ x = ρ γ D x, ii π ω = D α that is to say F = D F G, where α is the Maurer-Cartan form over G We say that D is a developing map of the G -foliation F on M In all that follows, ω will be the 1-form of Fedida of a Lie G foliation F In this case, any developping map of F associated to ω is also a developping map of F and we have F F We note that we have the following diagram: M D G π M Moreover, for all a M there exists an open neighborhood V a of a contained in an open localized trivialization of M and a Riemannian submersion f : V a G defining F on V a and checking: i f α = ω and f a = e where e is the neutral element of G, ii the following diagram is commutative Ṽ π V a D f V e where Ṽ is an open set of M such as π : Ṽ V a is a local diffeomorphism and V e is an open of G containing e In fact, V a, f is a primitive of ω at point a, e In other words Df = ω in V a containing a and f a = e we note also that as dω + [ω, ω] = 0 then the equation Dϕ = ω is completely integrable, which means that for any a 0, g 0 M G there exists a couple V, j such as: - V is an open neighborhood of a 0, - j:v G is a differentiable map, 3

5 - Dj = ω dans V, - j a 0 = g 0 That said, subject to reducing the size of V a, it can be assumed to be distinguished both for F and F so that there exists a submersion f definingf on V a We denote by V the local quotient manifold of F on V a and by θ a bond between V a, F and V a, F We have the following commutative diagram: D V e Ṽ π f θ V a f V Before finishing this part of recall, we give the theorem of the biunivocal correspondence between Lie subalgebras of G =Lie G and the extensions of a G-minimal Lie foliation existing in [4] Theorem 24 [4] 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: 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 -foliation having trivial normal bundle and defined by a 1-form with values in G 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 We end this part with a remark that we will use in the establishment of theorem 43 which is the main result of this paper Remark 25 We consider the preceding commutative diagram We have : i ω x X x = 1 L fx f x X x for every X x T x U that is to say that the Darboux s differential of f is ω ii f x T x F = L fx = T x F θ for every x U that is to say that f F = F U, = F θ where F U, is the restriction to U of F G, iii X x T x F ω x X x for X x T x M iv ω [X, Y ] = [ω X, ω Y ] for every F transverse foliate vector fields X and Y because F have dense leaves v X F = X F A 1 M where X F respx F is the Lie algebra of tangent vector fields of F respf, A 1 M is the space of differentiable fonctions on M and is the Lie algebra of F-transverse foliate vectors fields obtained from [3], [4] 4

6 3 Left invariant vector field on the Lie group G of a minimal Lie G foliation on a compact manifold Let G be the Lie algebra of a Lie group G and let h T be a left-invariant metric on G Is denoted by X d resp X g the right invariant resp left invariant vector fields on G obtained from X G and by L g resp R g the left translation resp right translation associated to g G Using Cauchy-Lipschitz s theorem about existence and uniqueness of the maximal solution of Cauchy s problem for a vector field Y with initial condition 0, a, we establish the following theorem: Theorem 31 Let F be a Lie G foliation with dense leaves on a compact manifold M, let h be a metric on M which is bundle-like for F and which admits h T as its associated transverse metric and let X d be the left invariant vector field on G associated to a F transverse foliate vector field X Then: i there exists an open neighborhood V e of the neutral element e of G such as the restriction X g V e of X g at V e is a Killing vector field on V e, ii for every point a M there exists 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 F-transverse Killing vectors field on V a Proof In what follows, h is a F-bundle-like metric on M having h T for F-transverse metric and h is the lifted metric of h on the universal covering π : M M of M i Let ω be a 1-form of Fedida defining the Lie G-foliation F and let X be a F-transverse foliate vector field As the leaves of F are dense then ω X G It is clear that if ω X = 0 then X = 0 And in this case, X is a F-transverse Killing field We assume in what follows that ω X 0 and we pose ω X = X We denote by < X > the Lie algebra generated by the vector X, by F g G,X the flow of the letf-invariant vector field X g and by F X the leaf of F G,X g containing e the neutral of G We note before continuing that for z, z F X F X there are two numbers s and s such that z = exp sx and z = exp s X and [9] exp sx exp s X = exp s + s X = exp s + s X = exp s X exp sx The equality zz = z z for z, z F X F X shows that X g z = X d z for every z F X Thus, Cauchy-Lipschitz s theorem about existence and uniqueness of the maximal solution of Cauchy s problem for a vector field Y with initial condition 5

7 0, z shows that along F g X,{e} the respective flows ϕ Xg t e and ϕ Xd t e of X g and X d are equal These flows form a one-parameter group of isometries because X d is a right invariant vector field and the metric h T considered on G is left invariant That said, we consider in what follow that the one-parameter group of flow along F g X,{e} is a one-parameter group of isometries Let D : M G be the developping map of F associated to ω, let F be the lifted foliation of F on M, let F be the lifted foliation of F on M, and let F X be the lifted foliation of F X on M where F X is the extension of F corresponding to the subalgebra < X > of G In fact F X is the foliation on M defines transversally to F by the orbits of X [3], [4] According to the proposition 23 we have: F X = D F G,X g Which means that each leaf of F X is projected by the surjective submersion D on a leaf of F G,X g in other words D FX = F G,X g We note that D F X is the leaf of F X projecting by D on F X because DD F X = F X We note also that π D F g,{e} X = F X,{e} is a leaf of F X And, this leaf is dense in M because the foliation F X have dense leaves since it s the extension of foliation having dense leaves Let a M and let V a be the open of the preceding commutative diagram which we recall for the circumstance D Ṽ V e π f θ V a f V The set V a F X,{e} is non-empty because F X,{e} is dense in M Furthermore V a F X,{e} is dense in V a and V a F X,{e} = r V a F X,{e} P Va X,{r},{e} where P Va X,{r},{e} is the plaque of F X,{e} contained in V a and containing the point r of V a F X,{e} The foliations V a, F X and Ṽ, FX are diffeomorphic and simple Therefore: 6

8 1 π P 1 Va X,{r},{e} where π : Ṽ V a is a plaque of Ṽ, FX and P Ṽe,X = π 1 P Va r V X,{r},{e} a F X,{e} is dense in Ṽ, 2 D π 1 P Va X,{r},{e} is a plaque of flow F g G,X of X g in V e and P Ve e,x = D π 1 P Va r V X,{r},{e} a F X,{e} is dense in V e We note that π is a local isometry relatively to the metrics h and h We note also that D : M G is a Riemannian submersion That said, transversely to the lifted foliation F of Fthe leaf D F X of F X is defined by the flow ϕ Xg t e formed of groups of one parameter of isometries since it is projected by the Riemannian submersion D on the flow ϕ Xg t e of X g formed of groups of one parameter of isometries along F g X,{e} Furthermore, the fact that π is a local isometry relatively to metrics h and h implies that transversally to F the leaf F X,{e} is defined by the flow ϕ Xg t e projected of ϕ Xg t e by π on M And, this flow is formed of groups of one parameter of isometries It follows from the foregoing that the plaques P Va X,{r},{e} of F X,{e} contained in V a are defined transversally to F by the flow ϕ Xg t e formed of groups of one parameter of isometries As in the diagram the map f : V a V e is a Riemannian submersion and as according to remark 25 f F X = F V e,x g = D π 1 F /Ṽ X where F V e,x g is the restriction of F G,X g on V e then on a set of orbits of F V e,x g dense in V e the flow F V e,x g is defined by groups of one parameter of isometries Thus, the continuity of the flow ϕ Xg : I V e V e t, g ϕ Xg t g of X g on V e where I is an open interval implies that it is formed of groups of one parameter of isometries And, this means that the restriction X g V e of X g at V e is a F-transverse Killing vector fieldfor the metric F-transverse h T ii The Riemannian Submersion f : V a V e defining the restriction of F at V a and the equality f F X = F V e,x g asssure that the restriction X V a of X at V a is a F-transverse Killing vector field 7

9 For any point x M we determine V x in the same way that V a and thereafter it is shown in the same way as X Va that the restriction X Vx of X at V x is a F-transverse Killing vector field We note that the fact that the restriction X g V e of X g at V e is a local Killing field for the F-transverse metric h T is a very important fact for the Lie groups G for which there exists a Lie G foliation having dense leaves It should not be believed that R exptx is an isometry for all t R On the other hand, we can say that there exists an open interval I 0 centered in zero for which R exptx is an isometry for the left invariant metric h T for all t I 0 It can therefore be deduced that the left invariant fields of a Lie group G for which there exists a Lie G foliation having dense leaves are local Killing vector fields for all metric on G left invariant Proposition 32 Let G be the structural Lie algebra of a Lie G-foliation having dense leaves, let λ be a metric left invariant on G, let be a Lie subalgebra of G and let be the ortho-complementary of in G Then h and u we have [h, u] Proof Let be a connected Lie subgroup of a connected Lie group G, let =Lie and let v There exists an open interval I 0 centered in zero for which R exptx is an isometry for the left invariant metric λ for all t I 0 Comme la translation L exptv est une isométrie pour λ alors Ad exptv est une isométrie pour tout t I 0 But for all a we have Ad a which leaves invariant so Ad exptv u for all u and for all t I 0 Which means that L exptv R exp tv u for all u and for all t I 0 We note that L exptv R exp tv u = ϕ vg t u g exptv where ϕ vg t is the flow of v g Thus u we have ϕ vg t u g exptv for all t I 0 Therefore u we have for all t I 0 \ {0} and all v, ϕ vg 0 = λ v t u g exptv u, t 8

10 As λ is continuous then making tender t to zero in the previous equality we get λ v, [v, u] = 0 for all v and for all u It follows from the foregoing that v and u we have [v, u] The preceding proposition allows us to see that for every Lie subalgebra of the tructural Lie algebra G of a Lie G-foliation having dense leaves, the following assertions are equivalent: i [u, h] = 0 for all h, u ii [u, h] for all h, u We are finishing this paragraph with the following proposition which will be useful to us in the following Proposition 33 Let G be the structural Lie algebra of a Lie G-foliation having dense leaves, let λ be a metric left invariant on G, let be a Lie subalgebra of G and let be the ortho-complementary of in G relatively to λ Then {u / [u, h] = 0 for all h } is a Lie subalgebra of G Proof We consider two vectors u and v of {u / [u, h] = 0 h } We have for k, for all 0 = [[u, v], k] + [[v, k], u] + [[k, u], v] = [[u, v], k] + [0, u] + [0, v] = [[u, v], k] moreover, according to theorem 31 there exists an open neighborhood V e of the neutral element e of G such as for all b G the restriction b g of b g at V V e e is a Killing vector field on V e relatively to λ From where, we have 0 = u g λ v gv, k gv V e e e ] ] = λ [u gv, v gv, k g + λ v g, [u gv, k gv e e V e V e e e ] = λ [u gv, v gv, k g + λ v g, 0 e e V e V e ] = λ [u gv, v gv, k g e e V e = λ [u, v], k The equalities [[u, v], k] = 0 and λ [u, v], k = 0 for every k ensures that [u, v] is a vector of {u / [u, h] = 0 fo every h } It follows that {u / [u, h] = 0 for all h } is a Lie subalgebra of G 4 Foliate field of an extension of a minimal Lie foliation on a compact manifold In the following if there exists a metric λ on a manifold M, we will identify for every foliation F on M, the normal bundle V F of F and the orthogonal bundle T F of T F 9

11 The purpose of this section is to calculate the Lie algebra lm, F of F - transverse foliate vector fields where F is an extension of a minimal Lie G- foliation F on a compact connected manifold M corresponding to the subalgebra of G = Lie G Before stating the next proposal, we recall some remarks on the transversally parallelizable foliations on a compact manifold Let F be a transversally parallelizable foliation on a compact manifold M, let be F b the basic foliation of F and let π : M W be the basic fibration of F It is known [8] that π defines F b and we have the following exact sequence: 0 ε w lm, F X W 0 where ε w is the Lie algebra of F -transverse foliate vector fields which projection by π is zero and X W is the Lie algebra of vector fields of the basic manifold W As F b is a simple foliation defined by π, we have X W = lm, F b Whence the previous exact sequence becomes: 0 ε w lm, F lm, F b 0 ence lm, F b lm, F This inducted us asking about the nature of relations existing between lm, F and lm, F for any extension F of riemannian foliation F Proposition 41 Let F be an extension of a Riemannian foliation F on a manifold M Then lm, F lm, F Proof Let X be a F -transverse foliate vector field, let λ be a F -bundlelike metric having λ T for F -transverse metric and let U be an open set of M distinguished both for F and F There are three submersions f : U U, f : U U and θ : U U such that θ is a bond between U, F and U, F and the restriction foliations U, F and U, F are defined respectively by f and f We then have the following diagram which is commutative: U f f U θ U The submersion f is Riemannian submersion since it defines the Riemannian foliation U, F Now considering Px,F U and P x,f U plaques respectively of F and F in U passing by x U Let z Px,F U and let F θ be the foliation in U defined by the submersion θ Let us show that T z F T z F = T f xf θ f z 10

12 We have f = θ f Whence f z T z F T z F ker θ f x The vector spaces f z T z F T z F and T z F T z F have the same dimension because f z : T z F T f xu is an isometry But dim T z F T z F = dim T z F + dim T z F dim < T z F T z F > = dim F dim F = co dim F co dim F = dim ker θ f x so the inclusion f z T z F T z F ker θ f x ensures that f z T z F T z F = ker θ f x = T f xf θ That said, X being F -transverse foliate, we have f X is a vector field on U It is noted X Furthermore X is valued in T F because we identify the normal bundle V F of F and the orthogonal bundle T F of T F As T F = T F T F T F and f z : T z F T f xu is an isometry then for any z P U x,f any Y z T z F T z F we have and for 0 = h T Y z, X z = h T f z Y z, f z X z where λ T is the metric on U obtained by projection of λ T on U The equality λ T f z Y z, f z X z = 0 shows us that f z X z for any z P U x,f But f z T z F T z F = ker θ f x = T f xf θ f z T z F T z F for every z P U x,f so f z X z T f xf θ for every z P U x,f We have P U x,f P U x,f Whence for any z, z P U x,f f z X z = f z X z = X f x The equality f = θ f shows us that X f x = f z X z = θ f x f z X z 11

13 and Thus X f x = f z X z = θ f x f z X z θ f x f z X z = θ f x f z X z The fact that θ f x : T f xf θ Tf xu is an isomorphism of vector spaces implies by using f z X z, f z X z T f xf θ Tf xf θ that f z X z = f z X z for any z, z Px,F U It follows from the foregoing that the F -transverse foliate vector field X is projeted following the plaques of F on the F -local quotients manifolds of distinguished open sets for F and F And, this shows that X is a F -foliate vector field As T F T F and X is tangent to T F then we have X which is tangent to T F Thus, X is a F -foliate vector field tangent to T F That shows that X is a F -transverse foliate vector field Ultimately, it can be said that lm, F lm, F Remark 42 We pose X = f X U where X U is the restriction to U of the F -transverse foliate field X Note that X is tangent to T F θ and is projeted by θ on X It results that X is F θ -transverse foliate It is noted that in general, thanks to the commutative diagram U f f U θ U, that f is a linear isomorphism between lu, F and lu, F θ By using propositions 32 and 41 and theorems 24 and 31 and the remark 25 we establish the following theorem which is the main result of this paper Theorem 43 Let be a Lie subalgebra of the structural Lie algebra G of a minimal Lie G foliation F on a compact manifold M, let ω be a 1-form of Fedida defining F and let F be the extension of F corresponding to Then: i lm, F = lm, F <h> h where F <h> is the extension of F corresponding to the Lie subalgebra < h > of G generated by h 12

14 ii In particular ω lm, F = {u / [u, h] = 0 for any h } ω lm, F <h> = {u < h > / [u, h] = 0} Proof i For every h F <h> F and F <h> is a Riemannian foliation Whence lm, F lm, F <h> for any h This implies that lm, F lm, F <h> h Let Y lm, F <h>, Z a vector field tangent to F h There exists X F X F, X and s A 1 M such that We have ] [Y, Z] = [Y, X F ] + [Y, sx Z = X F + sx ] = [Y, X F ] + Y s X + s [Y, X But Y is foliated for F <h> so Y is foliated for F because F F <h> and F is Riemannian This shows that [Y, X F ] is tangent to F F More ω X ] whence [Y, X is tangent to F <ωx > F Given that Y s X is tangent to F, it results from that precedes that [Y, Z] is tangent to F This means that Y is F -transverse foliate ence lm, F <h> lm, F h Ultimately, it can be said that lm, F = lm, F <h> h ii Let us show first that ω lm, F First of all, as F is an extension of Lie foliation F then according to the proposition 41 lm, F lm, F We note also that for every X lm, F and for evry x M we have ω x X x = ω X because the Lie foliation F is a foliation having dense leaves Let U an open of local trivialization of universal covering M of M distinguished both for F and F There is cfremark 25 a riemannian submersion f defining F on U and checking ω x X x = 1 L fx f x X x for every X x T x U That said, we have ω lm, F = ω x lm, F = L fx 1 f x lm, F But according to the remark 42 f x lm, F Kerθ fx = Lfx 13

15 then ω lm, F L fx 1 Lfx We also have 1 L fx Lfx = because L fx is an isometry It results from the foregoing that ω lm, F That said, for all X lm, F and Y X F = X F A 1 M where is the Lie algebra of F-transverse foliate vectors fields obtained from we have [X, Y ] X F which implies [ω X, ω Y ] because ω X F = and ω [X, Y ] = [ω X, ω Y ] On the other hand ω X and ω Y Whence according to proposition 32 [ω X, ω Y ] Thus,we obtain [ω X, ω Y ] Which means that [ω X, ω Y ] = 0 It follows from the foregoing that ω lm, F {u / [u, h] = 0 for any h } Consider now v {u / [u, h] = 0 for any h } et ṽ the F-transverse foliate vector field associated to v For all Z X F,there exists X F X F, X and s A 1 M such that Z = X F + sx We have ] [ṽ, Z] = [ṽ, X F ] + [ṽ, sx ] = [ṽ, X F ] + ṽ s X + s [ṽ, X We note that [ṽ, ] X F [] X F ence ] ω [ṽ, X F ] = 0 But ω [ṽ, X = v, ω X = 0 so ω [ṽ, Z] = ω ṽ s X = ṽ s ω X Which shows that [ṽ, Z] X F Which implies v ω lm, F Thus,it is found that {u / [u, h] = 0 for all h } ω lm, F It follows from the foregoing that In particular ω lm, F ={u / [u, h] = 0 forallh } ω lm, F <h> = {u < h > / [u, h] = 0} Corollary 44 Let G be the structural Lie algebra of a minimal Lie G foliation F on a compact manifold M We have the following assertions: i If is an ideal of G then the ortho-complémentary of in G is also an ideal, ii If F has a Lie extension F then F has also an other Lie extension F such thhat T F T F be orthogonal to T F T F 14

16 Proof Let ω be a 1-form of fédida defining the minimal Lie foliation F i Supposed that is an ideal of G From theorem 24,the extension F of F corresponding to is a Lie foliationwhence [3], [4] G dim ω lm, F = dim lm, F = dim = dim But ω lm, F so ω lm, F = This shows that is a Lie subalgebra of G The proposition 32 and the fact that is an ideal, implies that [h, u] = 0 for every h and all u Let v G There exists v and v such that Whence for every u, v = v + v [v, u] = [v, u] + [v, u] = [v, u] It results of fact that be a Lie subalgebra that [v, u] for every u and all v G ie is an ideal of G ii Supposed that F has a Lie extension F From the theorem 24 there is an ideal of G corresponding to F It results from i that is also an ideal of G Let F be the extension of F corresponding to we have ω T F T F = and ω T F T F = According to the remark 25, for all a M there exists an open neighborhood V a of a and a Riemannian submersion f : V a G such as for any X x T x V a, ω x X x = L fx 1 f x X x This implies that T F T F is orthogonal to T F T F References [1] RAlmeida et PMolino, 1986 Flot riemanniens sur les 4-variétés compactes Tôhoku Mathematical Journal, The Second Series, Vol 38, no 2, pp

17 [2] YCarrière, Flots Riemanniens In Structures transverses des feuilletages, Astérisque, , [3] CDadi, 2008 Sur les extensions des feuilletages Thèse unique, Université de Cocody, Abidjan [4] CDadi et Diallo, 2007 Extension d un feuilletage de Lie minimal d une variété compacte Afrika Matematika, Série 3, volume 18 pp [5] AEl Kacimi Alaoui, GGuasp and MNicolau, 1999 On deformation of trans-versely homogenous foliations Prépublication UAB, 4 [6] EFédida, 1974 Sur l existence des Feuilletages de Lie CRAS de Paris, 278, [7] CGodbillon, 1985 Feuilletage; Etude géométriques I Publ, IRMA, Strasbourg [8] PMolino, 1988 Riemannian foliations Birkhäuser [9] TMasson, 2001 Géométrie diff érentielle, groupes et algèbres de Lie, fibrés et connexions, laboratoire de Physique Théorique, Université Paris XI, Bâtiment 210, Orsay Cedex France 16