Lie algebras of vector fields and codimension one foliations

Publicacions Matemátiques, Vol 34 (1990), 311-321 . A bstract LIE ALGEBRAS OF VECTOR FIELDS AND CODIMENSION ONE FOLIATIONS ToMASz RYBICKI The main result is a Pursell-Shanks type theorem for codimension one foliations . This theorem can be viewed as a partial solution of a hypothetical general version of the theorem of Pursell-Shanks . Several propositions and lemmas on foliations are contained in the proof . 1 . Introduction The Lie algebra X(M) consisting of all vector fields onasmooth manifold M gives an important example of an infinite dimensional Lie algebra . A fundamental theorem proved by_ L .E . Pursell and M .E . Shanks in [10] states that the Lie algebra structure of X(M) completely determines the underlying topological and smooth structure of M . A whole series o£ papers followed, e .g . [1], [6], [9] . Our object of interest is the Lie algebra o£ a codimension one foliation . Let (Ml, F,) and (M2, F2) be smooth, second countable and smoothly foliated manifolds . By Y(Mi, F¡), i = 1, 2, we denote the Lie algebra o£ all lea£ preserving vector fields on Mi . I . Amemiya in [1] proved a theorem which can be formulated as follows . Theorem 1 .1 . If there exists a Lie algebra isomorphism oP of y(M 1 ,F 1 ) onto Y(M2,F2) then there exists a foliation preserving diffeomorphism cp of M l onto M2 such that So . = (P . This note is devoted mainly to a generalization o£ Theorem 1 .1 . For any foliated manifold (M, F) the symbol X(M, F) stands for the Lie algebra of all foliated vector fields i .e . vector fields with a flow transforming each leaf o£ F opto a lea£ of F . We recall that X is a foliated vector field if [X, Y] E Y(M, F) for any Y E y(M, F) . Thus y(M, F) is an ideal of X(M, F) . Now our result is the following 312  T . RYBICKI Theorem 1 .2 . Le¡ (Ml, FI) and (M2, F2) be one codirnensioal smooth nontrivial (i . e . dim Ml >_ 2, dim M2 >_ 2) foliations with Ml, M2 compact . If these is a Lie algebra isomorphism <D of X(M1,F1) onto X(M2,F2) then these is a foliation preserving diffeomorphism cp of Ml onto M2 such that cp * =,¿ . For simplicity sake we restrict our attention in the proof to transversely orientable foliations . Let us indicate analogous results to Theorems 1 .1 and 1 .2 . The case of compact foliations was solved in [6] as well as the case of transversely complete foliations in [111 . Two very special cases of codimension one foliations were also considered in [6, Theorems C and D] . In the last section of our note we formulate briefly a conjecture broadly generalizing these results . In this note all manifolds are second countable and all manifolds and foliations are of caass C°° . However, all considerations in the section 2 remain true in the case of C 2 foliations and C 2 functions . Finally I would like to thank Dr . Robert Wolak for fruitful conversations . I alsó express my deep gratitude to the referee whose critical remarks helped me to improve this note . 2 . Structure of open saturated sets Various authors have studied a theory of open saturated sets of one codimensional foliations . Referentes for the facts presented in this section are Cantwell and Cónlon [2], Hector [3] and Hector and Hirsch [4] . We shall formulate and prove here some preparatory results which will be useful in the sequel . Let (M, F) be a foliated compact manifold with codimension one . There is a increasing sequence Mk, k = 0, 1, 2, . . ., of closed saturated subsets of M such that 1) Mo is the union of all minimal sets of F ; 2) if Mk_1 is defined for k > 0 then the set Mk\Mk_1 is the union of all minimal sets of FIM\Mk_1 . Each leaf contained in Mk\Mk_1 is called a leaf at level k . In general, a leaf contained in some Mk is said to be at finite level . On the contrary, if these are any leaves in M\ U{Mk, k ='0,1, . . . } they are said to be at infinite level . Let a leaf L be at infinite level . The structure of L is the union of all leaves of L that are at finite level . Then Theorem 5 .0 in [2] states that the substructure of L is dense in L . Our object of interest are local minimal sets . A local minimal set is a minimal set of FlU, where U is some open saturated set . In particular, every proper leaf is alocal minimal set . Each leaf of a local minimal set lies at certain finite level . On the other hand, leaves that lie in no local minimal set are at infinite level . LIE ÁLGEBRA OF CODIMENSION ONE FOLIATION  31 3 We recall that minimal sets can be either closed leaves, or exceptional sets, or the whole manifold M . Denote by C(F), E(F) and Z(F) the union of all closed leaves, the union of all exceptional minimal sets and the union of all minimal sets of (M, F) respectively . Let U be any open saturated set . Then Z(F1U) 7É 0, C(F1U) is closed and E(F1U) is a finite union (cf . [3]) . In particular, the set Z(F1 U) is closed in U . From now on we fix a foliation T transversal to F ; it is one dimensional and orientable . Following [3] let us consider a situation near a proper leaf L . Having fixed T one can construct a pseudogroup P of global holonomy on an orbit C of T . Let x E Cf1L . It is essential that there is a compact neighborhood V of x contained in a T-plaque with a family fn, n = 1, 2, . . ., of functions generating P on V and defined on V . We may identify x with 0 and V with an interval, say [-1,1] . The action of P .,, the isotropy subpseudogroup at x, induces on V an equivalente relation . Shrinking V if necessary, we have the following three types of minimal subsets of [-1,1] : (i) a single point ; (ii) a cyclic orbit Le . a sequence (xj in (0,1] or [-1, 0) converging to 0 ; (iii) the intervals (0,1] or [-1, 0) . Denotation . Let U be a union of all leaves which are not locally dense and have an open saturated neighborhood U such that FI U is without holonomy . Then U is open and saturated . Of course, U need not contain all leaves without holonomy . Lemma 2 .1 . Le¡ L be a proper leaf at level k such that there is an open saturated neighborhood of L consásting of proper leaves at level k only . Then LcU . Proof .. Let V be a compact neighborhood of x E L contained in a T-plaque . We may assume that V meets only proper leaves at level k . Hence the types (ii) and (iii) above are not admissible on V . Hence FIV is without holonomy, where V is the saturation of V . As a consequence of Lemma 2 .1 we have the following Proposition 2 .2 .  Let L be a leaf at f pite level such that L ~ U . Then there is an open saturated neighborhood U of L with the following property : for any GC U, an open saturated neighborhood of L, there exisis W, an open saturated subset of G, such that L C W and any smooth function on G constant along ¡he leaves of FIG is constant on each componen¡ of W . ProoL The leaf L may be proper, locally dense or exceptional . The proof is trivial if L is locally dense . If L is exceptional then L is contained in an exceptional minimal set N . Then we apply Theorem 4 .1 .1 in [4] which estab_lishes the existente of an open saturated neighborhood U of N such that N C L' for any L' in U . Therefore any smooth function constant along the leaves which is defined on some W C U, a neighborhood of L, is a constand function . 314  T . RYBICICI Let us take up the case of L proper at level k . Let U be an open saturated set such that L is minimal of FIU . We set W = U\C(F1U) . Let W' be a connected component of W . Its boundary consists of a finite number of leaves Ll, . . . , L, . at level k . Each leaf in W' spirals onto one of the leaves Ll, ... , L, . or onto one of exceptional minimal sets N, . +1 , . . . , N ., contained in W' . Therefore a function on U constant along the leaves of FIU can assume only a finite number of values on W' and thus it is constant on W' . Finally, by Lemma 2 .1 and by the assumption, we get L C W . Notice that in the above argument U may be replaced by any open saturated G C U such that LC G . This completes the proof of the proposition . H . Imanishi and K . Yagi in [8] have studied a generalized Reeb component . We recall that a compact foliated manifold (N, F), where áN :~ 0, is called a generalized Reeb component if the following two conditions are satisfied : (1) all leaves in Int(N) are non-compact and proper ; (2) FlInt(N) has trivial holonomy . There was shown the following Proposition 2 .3 . (6, Proposition 2 .1 in [8]) . Leí (N,,F) be a generalized Reeb componen¡ . Then there is a transverse vector field X on Int(N) such that X has an orbií C and C fl L = {one point} for any leaf L in Int(N) . Such a vector field is said to be nice and C is a nice orbit . Let U bean open saturated set . By U` we denote the completion of U (cf . [2] or [4]) . Let i° : U`  + M be a canonical extension of the inclusion i : U --> M . Recall that U` is a manifold with boundary and i` maps diffeomorphically each component of U° onto a peripheral leaf of U . Now if U is a connected component of the set U we have the following two possibilities : (i) U` = L x [0, 1] is foliated as a product by the foliations F and T ; (ii) U` is a manifold with boundary, there is a finite number of leaves which are the connected components of aU` and the restriction FlInt(U`) is a fibraton over S' . Thenwe have the following corollary . Corollary 2 .4 .  There is a nice vector field on U . 3 . Maximal ideals of X'(M,F) By X'(M, F)we denote the derived ideal of X(M, F) Le . the ideal generated by all brackets [X, Y], where X, Y E X(M, F) . We begin with the following proposition which is true for an arbitrary foliation (N, F) . LIE ALGEBRA OF CODIMENSION ONE FOLIATION  31 5 Proposition 3 .1 . The derived ideal Y(N, .F) coincides with y(N, .F) . The proof is a modification of that of Lemma 14 in [11] and is omitted . Corollary 3 .2 . y(N, .F) is an ideal of X'(N, .F) . Corollary 3 .3 . For any manifold N the derived ideal X'(N) coincides with X(N) . Let X E X(M, F) . We have the unique decomposition X = X l + X2, where X I is tangent to F and X2 is tangent to the transversal foliation T . Before giving a description of maximal ideals we shall show that any X E X'(M, .F) is tangent to F on M\U, that is X2 = 0 on M\U . Let LC M\U be an arbitrary leaf at finite level . If there does not exist any X E X(M, F) such that X is not tangent to L then we are done . Suppose that X E X(M, F) such that X2 q¿ 0 on L . Hence X2 :~ 0 on an open saturated neighborhood G of L .  Let us consider any bracket [Y, Z] E X'(M, F) such that Y = Y1 + Y2 and Z = Z 1 +Z 2 belong to X(M, F) .  Then there exist smooth functions f and g on G, constant along leaves, such that Y 2 = f X2 and Z 2 = gX2 . By Proposition 2 .2, there is an open saturated subset W of G such that LC W and f, g are constant on each connected component of W . Hence [Y, Z] = [Y1 + Y 2 , Z 1 + Z2] = [Y1, Zl] + [Y 1 , Z 2 ] + [Y2, ZI] + [Y2, Z2] = [Y1, Z I ] + g[Y1, X2] + f [X2, ZI] is tangent to F on W and, by continuity, also on L . Next, let L be at infinite level . The substructure of L is obviously contained in M\U and by the above argument X'(M, F) is tangent to F on the substructure . Hence, by continuity, any element of X'(M, F) is tangent to L . Thus we have the proof of the following Theorem 3 .4 . Le¡ (M, F) be an arbitrary one codimensional foliation on a compaci manifold M . There is an open saturated subset U of M such that FIU is without holonomy andany X E X'(M, F) is tangent to F on M\U . Remark . It is known that any one codimensional foliation without holonomy admits an extension of any vector at a point to a foliated vector field . Our theorem completes this property : the transversal part of X'(M, F) simply vanishes outside an open set without holonomy . Let C be a nice orbit on U of T (now we consider T on U only) . 31 6  T . RYBICKI Corollary 3 .5 . The transversal part X2 of X E X'(M, F) is completely determined by its restriction X 2 ¡C . In fact, the flow of X2 P uniquely determines the flow of X2 . Deflnition . From now on, by Trans(M) we denote the union of all leaves L such that there is X E X'(M, F) with X2 :~ 0 on L . Trans(M) is open, saturated and contained in U by Theorem 3 .4 . The following examples justify the definition . Examples : (3 .6) Let M = T 2 and F be the well known foliation by a rotation with rational slope . Here the transversal part of X(M, F) is isomorphic to X(S' ) . Since, by Corollary 3 .3, X'(S') = X(S 1 ) we have X'(M, F) _ X(M, F) . In particular, Trans(M) =M . (3 .7) The situation is quite different in the case of the Reeb foliation of S 3 . Obviously Ll = S3 \T', where T 2 denotes the unique compact leaf . A nice orbit consists of two disjoint circles . However, the transversal part of X(M, F) is finite dimensional as it was shown by K . Fukui in [5] . Consequently Trans(M) = 0 . Now we shall try to describe maximal ideals . Let N be a manifold and p E N . By m P we denote the ideal of X(N) formed by all vector fields vanishing at p with all its derivatives . By abuse of notation, m p denotes also such an ideal in any subalgebra of X(N) . The following fact is fundamental . Proposition 3 .8 . Let (N, .F) be any compact foliated manifold and dim .F > 0 . The ideals m p , p E N, are ¡he unique maximal ideals in y(N, .F) . In particular, m P are ¡he unique maximal ideals in X(N) . The second part was proved in [10] . The proof of the first part is essentially the same . Denotation . For any algebra A we denote by A* the set of all maximal ideals of A . For simplicity sake we shall write A instead of X'(M, F) . One can easily see that m p E A* for p E M\Trans(M) . Let L be a leaf in Trans(M) . Then PL denotes the ideal in A of all vector fields with their transversal parts infinitely flat on L . A standard argument shows that PL E A* . Lemma 3 .9 . For any p E Trans(M) the ideal m p is contained in PL, where pEL . In fact, if the flow of a foliated vector field fixes p then it fixes, of course, the leaf L . Now let us consider the Lie algebra A restricted to Trans(M) . By the above arguments, the transversal part of A may be identified with a subalgebra, say A, of X(C), where C is a topological sum of a number of S l or (0,1) . In particular, LIE ALGEBRA OF CODIMENSION ONE FOLIATION  31 7 the ideals PL, where L C Trans(M), correspond to m,,, x - E C . On the other hand, there are some maximal ideals in A different from m z . Let {A ;} be the totallity of such ideals . For any ideal A ¡ there is the unique corresponding ideal A ; E A* . Such ideals exist, even for very simple foliations, as the following example shows . Example 3 .10 .  Let a sequence (xk) C (0, l), k = 1, 2, . . ., converge to 0 . By A(xk) we denote the ideal of all X E X((0,1)) such that X is ak-flat at xk for some sequence of positive integers % tending to oo . A maximal ideal containing A(xk) gives an example of such A ; . Proposition 3 .11 . The unique elements of A* are m p , p E M\Trans (M), PL, L C Trans(M), and ~i . Proof . . Let I be a maximal ideal of A such that I is neither contained in m p , nor in pL, nor in a i . Let P = I fl y(M, F) . Then, by Proposition 3 .8 and Lemma 3 .9, we get P = Y(M, f) and y(M, F) C I . Therefore there is a unique J, an ideal of A, corresponding to I . Then J being proper must be contained in m x or A¡ . Hence I is contained in PL or Ai . This contradiction proves the proposition . Definition . Maximal ideals pL and al are called "horizontal" ; m p are called "vertical" . We shall denote E= the totallity of /UL, a= the totallity of m p , A= the totallity of Ai . 4 . Stone topology ofA* and the proof of Theorem 1 .2 In the situation described above we have to find an algebraic property which characterizes the horizontal ideals . This property being preserved by isomorphisms ensures that p,L cannot be mapped onto m p and vice versa . Our characterization is based on topological properties of A* . We introduce the Stone topology on the set A* in the following way . Let 52 be a subset of A* . Then the closure of 52 is defined by 52= {mEA* :n{wE52}Cp} . In particular, ~ = 0 . Remark . Let (N, F) be any foliated manifold with dim .F > 0 . It is an easy consequence from Proposition 3 .8 that the mapping N Dp -> m pE y(N, F)* is a homeomorphism . Specifically, N and X(N)* are homeomorphic . Now we consider the partition of A* by connected components in the Stone topology . We want to show that such a component must be "vertical" Le . formed by vertical ideals, or "horizontal" Le . formed by horizontal ideals . First we strengthen Theorem 3 .4 . 31 8  T . RYBICKI Theorem 4 .1 . The transversal part X of a vector faeld belonging to A vanishes with all its derivatives on M\Trans(M) . Proof . . Let pEL and L 9~- Trans(M) . First case : L C U . There is a foliated vector field Y on M tangent to T, Y(p) :~ 0 . Let (U, xl, .. . , x n ) be a bidistinguished chart at p (with respect to F and T) such that Y= a1ax n near p . Suppose that there is k > 0 such that the k-th derivative a k XIaxñ ~ 0 at p . We have [Y, X] E X(M, F) and [Y, X] = OXIax n on a neighborhood of p . Moreover, X E A implies [Y, X] E A . Continuing this procedure k times we get Xo E A such that X o (p) :~ 0 . This contradicts the definition of Trans(M) . Second case : L 9~- U . Since Trans(M) C U is an open saturated set without holonomy one can construct a non-vanishing foliated vector field Y on Trans(M) tangent to T ITrans(M) . Now, if X = f Y on Trans(M) then f is a smooth function on Trans(M) constant along leaves . First we assume L to be at finite level . If L is locally dense or exceptional then X = 0 on a neighborhood of p (see the proof of Prop . 2 .2) . Next, let L be proper and U be an open saturated neighborhood of L such tha t L is a minima l set of U . Then W = U\C(F1 U) satisfies LC LV . Let L 9~- W n Trans(M) . There is G, an open neighborhood of L, such that G fl W f1 Trans(M) = 0 and Gf1W :~OsinceL~U . ThenX=OonGnWáx=OonGnWandso on . Thus X is infinitely flat on G fl W and at p . Next, let L C W fl Trans(M) . As in the proof of 2 .2, the function f is constant on each connected component of W fl Trans(M) . Therefore f' = af = 0 on W f1 Trans(M) and aX = 0 on W f1 Trans(M) and so on . In fact, locally Y = a1ax,, in some distinguished chart . Hence, by continuity, X is infinitely flat at p . Finally, let us consider L at infinite level . It is visible that for any L a of the substructure of L there is an open set W a such that X is infinitely flat on W 1 , and L a C W a . Hence X is infinitely flat on the union W = U W a and L C U L a C W . Thus X is infinitely flat at p . This completes the proof . Corollary 4 .2 . We have E fl ° _  and A n  _ ~ . In particular, the components of A* are purely "vertical" or purely "horizontal" . Proof .. E U A = EUA follows from the evident property that the vanishing with derivatives of transversal parts of foliated vector fie1ds cannot yield the vanishing of therr tangent parts . On the other hand, the above theorem implies " = E . In fact, let A be any horizontal ideal . If X is a transversal of an element of A which does not belong to Athen, by the theorem, X E n E . Then the definition of the topology implies that A 1 " . Deflnition . pE A* is said to be distinguished if p belongs to an element of a unique minimal family R of connected components of A* chaxacterized by the equality where R = U R . LIE ALGEBRA OF CODIMENSION ONE FOLIATION  31 9 n{,,E(nñ)*}=6, Lemma 4 .3 . There does not exist any connected componen¡ of A* consisting of elements of A only . In fact, it is immediate from the definition that E = E U A . Proposition 4 .4 . The horizontal ideals are uniquely characterized by the property "to be distinguished" . Proof :: In fact, the intersection of all horizontal ideals gives y(M, F) . It is clear that y(M, F)* satisfies the definition by Proposition 3 .8 . Suppose now that R does not contain some horizontal componen¡, say E' . By Lemma 4 .3, there are a connected componen¡ U of Trans(M) and a leaf LCU such that JIL 0 R = U R . Now, it follows from the definition of the topology that p V E E' for every L' C U . Consequently, n {ti E (n R)* } contains all vector fields tangent to FIU with supports in U and thus it is non-zero . This and the minimality of R implies the proposition . Corollary 4 .5 . Y(M, F) is characterized in A as ¡he intersection of all distinguished ideals . Now we have to reformulate slightly Theorem 1 .1 . Corollary 4 .6 . If 4 is a Lie algebra isomorphism of X(M I ,F I ) onto X(M2, F2) such that $(Y(M1, Fl)) = Y(M2, F2), then there is a foliation preserving diffeomorphism cp of Ml onto M2 such that cp * = ~¿ on X(MI,FI) . The proof is in [11] . The proof of Theorem 1 .2 is now a consequence of Corollaries 4 .5 and 4 .6 and of the fact that an isomorphism of Lie algebras induces an isomorphism of their derived ideals . Remark . Theorem 1 .2 can be formulate in slightly more general version . Namely, X(MI, FI) and X(M2, F2) can be replaced by any their subalgebras A1 and A2 respectively such that Y(MI, F I ) C A1 and Y(M2, F 2 ) C A2 . The proof is the same . 5 . Conjecture A . Lichnerowicz introduced a notion of Jacobi manifold (see e .g . [7]) which correponds to a notion of local Lie algebra over R . In particular, this notion