Pierrot's theorem for singular Riemanian foliations

Publicacions Matemátiques, Vol 38 (1994), 433-439 . PIERROT'S THEORE M FOR SINGULAR RIEMANIAN FOLIATION S ROBERT A . WOLA K A bst r act  Let .T be a singular Riemannian foliation on a compact connecte d Riemannian manifold M . We demonstrate that global foliate d vector fields generate a distribution tangent to the strata define d by the closures of leaves of .~ and which, in each stratum, is transverse to these closures of leaves . The aim of this short note is to prove M . Pierrot's theorem for singula r Riemanian foliations, cf . [5], namely . Theorem 1 . Let F be an SRF on a compact manifold M . Then th e vector space of global foliated vector fields is transitive to the closures o f leaves in each closure stratum . 1 . Preliminarie s First we recall some and prove other results about SRF -s (singula r Riemannian foliations), cf . [3] and [4] . Assume that the manifold M is compact and connected (or the metri c is complete) . Then the closure of any leaf is a submanifold . Let k be any number between o and n . Defin e E k ={xEM :xEdimL, = k} . The leaves of ,1 ' is Ek are of the same dimension, however they ca n have holonomy . P . Molino demonstrated that the sets E k or rather thei r connected components are submanifolds ofM and C U i<k Ei . Not e that for some i the sets Ei can be empty . Moreover, let ko be th e maximum dimension of leaves of F . Then the set E h is open and dense 434  R . A . WOLA K in M . It is the principal stratum . In fact, the partition {E k }ó is a n abstract stratification . Let W be a compact submanifold of M . The geodesics define th e exponential mapping exp : N(W) ~ M . Denote by S r (W) = {v E N(W) : il v il = r} (resp . D r (W) _ {v E N(W) : Ç r}) and b y S (W, r ) (resp . D (W, r )} its image by exp . If W is a closed leaf o r the closure of a stratum then it is not difficult to notice that leaves o f the foliation F live on S(W, r), cf . [3], [4] . Moreover, the homotethie s (along the geodesics) ha : D(W, r) --~ D(W, I a i r ), ha(exp(v)} = exp(ñv) preserve the foliation . The leaf passing through exp(v) has the sam e dimension and holonomy as the leaf passing through exp(av) . Connected components of E i are submanifolds of M . They can be o f different codimension and it can happen that some connected componen t of E i is a compact submanifold . Since the foliation is Riemannian th e closure of a leaf from a stratum Ei remains in it . In fact, let OEi = V I U • • • U 14 where Us—, V $ _ E i — E i, each V S being a connected submanifol d of M . I n 1 tubular neighbourhood of V s leaves of live on the spher e bundies 5(V5, r ) . Thus if L C S ( V s , r ) , so does .its closure 1 . Therefor e for all our purposes the foliation F IEi behaves like a RF on a compac t manifold . Therefore we can define the subspace s Ei i = {xEE i :xELE .~, dimL= j} . Each Ei .7 is a submanifold of ~i and OE i3o C U5< z Es U<0 E . Th e closures of leaves of F induce a regular RF  of compact leaves o n Ei3 . The leaves of have finite holonomy . Using the exponentia l mapping restricted to the normal bundle of a leaf one easily learns tha t the holonomy of a leaf is conjugated to the linear holonomy of this leaf . The linear holonomy is a finite subgroup of the linear orthogonal group . The linearholonomy groups h(L, x) at different points x of a given leaf L are conjugated ; let us denote this conjugacy class by h (L ) . If a denote s a conjugacy class of a subgroup of the linear orthogonal group then le t E i_jrx = { x E E ii :xE L Eh(L) =a} . In [5] M . Pierrot uses aslightly rougher stratification for regular RFs , namely E Pik ={xELEF :dimL= j,#h(L,x) =k } where p = dim .~, and the holonomy is considered in the stratum E j . However, in a tubular neighbourhood of a compact leaf L, the foliatio n by the closures of leaves, is conjugated to the natural foliation of the fia t bundle L x G R s where G is the lipear holonomy group of the leaf L an d s = codim Ei L . It is not difficult to notice that in these tubular neighbourhoods leaves of ,J ' have their linear holonomy groups conjugated to PIERROT ' S THEOREM FOR SINGULAR FOLIATIONS  43 5 a subgroup of G . It means t hat for any a, G E a, #G = k E pio , c E p i k and the submanifolds Ep i k are separated . If Ep i a and Epo are two suc h sets then the lemma concerning the homotethies, cf . [3], [4], ensure s that E p ia n E p o = O . Therefore connected components of E p ic , are als o connected components of E p ik . Thus connectedcomponents of these set s define the same stratification {E .} . The stratification {E} possesses a natural partial arder E y Ç  iff E l , c ~ .~~ . The strata defined aboye we call the closure strata of the foliation .F t o distinguish them from the strata defined by the dimension of leaves . In [3], [4] P . Molino describes a way of desingularization of SRFs . Let E be a minimal stratum . E is a closed submanifold . Let N(E ) be the normal bundle of E . Leaves of .~ also live on sphere bundle s 5 (E, r) over E . Take M° = (M — E) x {0}, M 1 = (M — E) x {1} and S = S (E, r ) x (—1, 1) for some r ~ 0 . Then M° , mi and S glue togethe r to become a compact manifold M 1 , i .e . 5 (E, r) x {t} is identified wit h 5 (E,i t i r ) x {0} CM° ift C OandwithS(E,itlr) x {1} CM 1 ift~ 0 . M 1 pro j ects onto M, p : M 1 --} M . Over M — E p is a double coverin g and p' (E) = 5 (E, r) . P . Molino proves that on M 1 there exists an SRF  which does no t have leaves of the type encountered in E, and including the old foliatio n on M° and M 1 . After a finite number of steps we get a regula r Riemannian foliation on a compact manifold 1% . Using the exponential mapping it is quite easy to prove a followin g lemma . Lemma 1 . For any 0 < S 1 < 6 2 <E there exists a basic smoot h function a(5 1 , s 2 ) : D(E, E) -> [ 0 = 1 ] such that supp )i(6 1 , S 2 ) c D (E, 62) and a(6 1 , S2) 1 D (E, S1} = 1 . In our future considerations we shall need the following relations between basic functions on the foliated manifolds (M, ,F) an d Lemma 2 . Let f be a basic function on (M 1 , .F1 ) . Then for any poin t x E M° there exists a foliated neighbourhood U of x in M° and a basi c function fu on (M, ,F) such that fuplU = f ¡U . Proof : The set D (E, E) — E = D° (E, e) can be considered as (via p ) an open subset of m° . Therefore we have to consider two cases : (a ) x D°(E, E) and (b) x E D°(E, E) . 436  R . A . WOLA K In the case (a) as U we can take M - -- D(E, 62), 0 < 62 C E and as f u the function f (z)  z D ° (E, E ) (1 - -- (61,62))f(z) z E D° (E , E), o C S 1 < 6 2 1 f(z) = o  z E E . In the case (b) let x E S (E, r), o < r Ç E . Then we take U = M — D(E, r/2) and define the function as in the case (a) taking o < S1 < 52 C r/2 . ■ Lemma 3 . Let f be a basic function on the foliated manifold (M , Then for any point x of M---E there exists an open foliated neighbourhoo d U of x in M — E and a basic function fu on (M 1 , .F 1 ) such that f i Up = fuI M ° rlp -1 (U) . Proof : It is analogous to that of Lemma 2 . Using this constructio n we obtain a basic function fu with compact support on (M °, Y' i ) ; w e extend it to M 1 putting 0 on E and W . ■ Let us recail the definition of the `musical ' isomorphism, for exampl e cf . [1} . b :TM —1 T * M is given by : for X E TM x X I ' is the only 1 -form such tha t g(X, Y) = X b (Y) for any Y E TM~ . # :T * M ---+ T M for any w E T * M s w ~ is the only vector for such tha t g(w # , Y) = w(Y) far any Y E TM~ . Therefore to any function f on M we associate a vector field x f by th e formula g(X f , Y) = df(Y) for any Y E TM or X(x) _ (df) . Now we shall study the properties of vector fields associated to basi c functions . First let us notice that for any basic function f the vecto r field X f is orthogonal to the leaves of the foliation . Moreover if th e function f is global the vector field X f is orthogonal to the closures o f leaves . Lemma 4 . If f is a basic function then the vector fLeld X f is a n infinitesimal automorphism of the foliation . The proof is a straightforward calculation . PIERROT ' s THEOREM FOR SINGULAR FOLIATIONS  43 7 2. Regular cas e Let .F be an RF . We shall look at the existence of global basic functions . Denote X O (M, .F) the vector space of global vector fields of th e form x f for some global basic function f on (M, .F} . The closures of leaves form an SRF and we can consider strata fo r this foliation, cf . [5] . These strata are just our closure strata for .F as . F being regular we have just the principal stratum for this foliation . It i s obvious that global infinitesimal automorphisms must be tangent to th e closure strata . Let E be one of these strata . Lemma 5 . For any vector X E TE S orthogonal to the closure S o f the leaf L in E passing through x, there exists a global basic function f such that df (X) 0 . Proof : There exists e > o such that the mapping exps : B E (X) –~ M is an embedding . Then there is a leaf L ' , with the closure S ' , of the sam e stratum E on the geodesic with the initial condition X at the distanc e less than E such that the mapping exps, : B(8') ~ M is an embedding , cf . [2] . Then the function f s► (y) -= d (y, S')' is a smooth basic functio n on exps► ( B E (S' }} for which df S ► (X ) O . f s, can be easily extended t o a global basic function . ■ Combining Lemmas 4 and 5 we get the following proposition which , in fact, is a variant of the theorem due to M . Pierrot, cf . [5] . Proposition 1 . Let (M, .F) be a compact foliated manifold with F being a regular RF . Then the vector space X # (M, .F) is transitive to th e closures of leaves in each closure stratum . 3. Singular cas e Now let ,F be an SRF on M . First we prove the singular version o f Lemma 5 . Lemma 6 . Let (M, .F) be a compact foliated manifold with ,F bein g an SRF . Let E be a closure stratum of .F . For any vector X E TE x orthogonal to the closure S of the leaf L passing through x there exists a basic function f such that df (X) O . Proof : Using the blowing up procedure and Lemma 2 we can reduc e our considerations to the case where the point x belongs to the singula r stratum Eo of the foliation F . Thus E is a submanifold of Eo and a 438  R . A . WOLA K closure stratum of (Eo, .~} which is compact RM . Therefore according t o Lemma 5 there exists a basic function fe on Eo such that dfo (X) O . According to the next lemma this basic function can be easily extende d to a global basic function on (M, F) . ■ Lemma 7 . Any basic function on a stratum E can be extended to a global basic function on M . Proof : Since the pro j ection p : B (E, E) -4 E maps leaves onto leaves , for any basic function f on E, the function f p is basic on B (E, 6) . The n using a function A(6 1 ,62) we can extend f p to a global basic function o n (M,F) . m Far vectors which are not tangent tostrata we have the followin g lemma . Lemma S . Let .~ be an SRF on a compact manifoldM . If a vecto r field X is not tangent to the closure of a leaf L at a point x, then ther e exists a global basic function f such that the germ at x of the functio n df (X ) is not O . Proa . Let S be the closure of the leaf L . It is a compact submanifol d of M . Let N(S) be its normal bundle . Far some E > o the exponentia l mapping defined by the geodesics starting from vectors of N(S) is a diffeomorphisms of B,(S) {v E N(S) : C E} onto the imag e B (S, E), cf . [4] . Using a similar method as in Lemma 2 we can exten d any basic function on B(S, E) to a global one . Therefore we have reduce d our problem to a local one . Then t he functio n M y } = d(L, y ) 2 satisfies the conditions of the lemma . ■ 4 . Proof of Theorem 1 Let x be any point of a closure stratum E . Let V be the subspac e of T x E orthogonal to T s S, S = L s . We know that for any global basi c function f Xl E V . Lemma 6 ensures that there does not exista vector i n V which is orthogonal to all X l e . It means precisely that SPAN{XD = V . Therefore we have proved the following theorem : Theorem 2 . Let M be a compact connected manifold and .~ be a n SRF on M . Then the vector space X O (M, . .1 '} is transitive to the closure s of leaves in each closure stratum of (1V1,  . Of course Theorem 2 is just a more detailed version of Theorem 1 . PIERROT ' S THEOREM FOR SINGULAR FOLIATIONS  43 9 Reference s 1. GALLOT S ., HULIN D . AND LAFONTAINE J ., "Riemannian Geometry," Springer, 1987 . 2. HIRSCH M ., " Differential Topology," Springer, 1976 . 3. M .CLINQ P ., " Riemannian Foliations," Progress in Math . 73 , Birkháuser, 1988 . 4. MOLINO P ., " Feuilletages riernanniens reguliers et singuliers, Géometrie et Physique," Paris, 1986 . 5. PIERROT M ., Orbites des champs feuilletés pour un feuilletage s riemanniens sur une variété compacte, C . R . Acad . Sc . Paris 30 1 (1985), 443-445 . Instytut Matematyk i Uniwersytet Jagiellonsk i W1 . Reymonta 4 30-059 Krako w POLAN D Rebut el 29 d'Abril de 1994