The Erwin Schrodinger International Boltzmanngasse 9. Institute for Mathematical Physics A-1090 Wien, Austria

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1 ESI The Erwin Schrodinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria A Note on Generalized Flag Structures Tomasz Rybicki Vienna, Preprint ESI 545 (1998) April 10, 1998 Supported by Federal Ministry of Science and Transport, Austria Available via

2 A NOTE ON GENERALIZED FLAG STRUCTURES Tomasz Rybicki Abstract. Generalized ag structures occur naturally in the modern geometry. By extending a well-known Stefan's statement on generalized foliations we show that such structures admit distinguished charts. Several examples are included. 1. Introduction It has been rst established by P.Stefan in [13,14] that the orbits of any 'isotopically connected' set of local C r -dieomorphisms, 1 r!, t together to form a generalized foliation. The career of this statement in the geometry is justied by the fact that a nontransitive geometric structure usually induces a foliation with singularities. Further facts concerning generalized foliations and the integrability of distributions can be found in [1,2,15,17]. However, it seems that there is surprisingly little known on this subject when comparing with the theory of regular foliations. In this note we give some introductory remarks on generalized ag structures. This notion describes a bit more complicated situation which arises for instance in the multisymplectic geometry, Riemannian foliations, Jacobi structures, Hamiltonian actions on Poisson manifolds (cf. Section 3). Given generalized foliations F i (i = 1; 2) on a manifold M we write F 1 F 2 and say that F 1 is a subfoliation of F 2 if every leaf of F 1 is contained in a leaf of F 2. By a generalized ag structure on M we shall mean any nite sequence F 1 F k of foliations of M. Throughout we shall drop the term "generalized" in the above notions. Our purpose is to formulate the notion of a distinguished chart for a ag structure. We show that any ag structure admits distinguished charts. This fact is well-known and trivial for regular ag structures (see, e.g., [18]) contrary to the case of (regular) almost-product structures where one needs more than merely the integrability of each distribution to obtain the integrability of the whole structure. One of the consequences of our result is that the pseudo-n-transitivity can be formulated in more general context (Section 4) Mathematics Subject Classication. 57R30. Key words and phrases. generalized foliation, subfoliation, ag structure, distinguished chart. Partially supported by KBN 2 P03A Typeset by AMS-T E X 1

3 2. The existence of distinguished charts Let us recall some concepts from [13] and [14]. A foliation of class C r, 1 r!, is a partition F of M into weakly imbedded submanifolds (see below), called leaves, such that the following condition holds. If x belongs to a k-dimensional leaf, there exists an (inverse) chart (U; ) with (0) = x and U = V W, where V is an open ball in R k and W is an open ball in R n?k, such that if L 2 F then (U) \ L = (V l); for l = fw 2 W : (0; w) 2 Lg: A subset L of a C r -manifold M endowed with a C r -dierentiable structure which makes it an immersed submanifold is weakly imbedded if for any locally connected topological space N and a continuous map f : N! M satisfying f(n) L, the map f : N! L is continuous as well. It follows that such a dierentiable structure is necessarily unique. A smooth mapping of an open subset of R M into M is said to be a C r -arrow if (1) (t; :) = t is a local C r -dieomorphism for each t, possibly with empty domain, (2) 0 = id on its domain, and (3) dom( t ) dom( s ) whenever 0 s < t. Given an arbitrary set of arrows A let A be the totality of local dieomorphisms such that = (t; :) for some 2 A; t 2 R. Next, A stands for the set consisting of all local dieomorphisms being nite compositions of elements from A or (A )?1 = f?1 : 2 A g, and of the identity. Then the orbits of A are called accessible sets of A. For x 2 M we let A 0 (x); A(x), and A(x) be the vector subspaces of Tx M spanned by f _ (0; x) : 2 Ag; f _ (t; y) : 2 A; t (y) = xg; and fd y (v) : 2 A ; (y) = x; v 2 A(y)g; respectively. Clearly A 0 (x) A(x) A(x). Theorem 1 [13]. Let A be an arbitrary set of C r -arrows on M. Then: (i) Every accessible set of A admits a (unique) C r -dierentiable structure of a connected weakly imbedded submanifold of M. (ii) The collection of accessible sets denes a foliation F = F(A). (iii) f A(x)gx2M is the tangent distribution of F(A). Any dieomorphism group G(M) Diff r (M) denes uniquely a set of arrows. Namely, by a C r -smooth path (or isotopy) in G(M) we mean any family ff t g t2r with f t 2 G(M) such that the map (t; x) 7! f t (x) is smooth. Next, G(M) 0 denotes the subgroup of all f 2 G(M) such that there is a smooth path ff t g t2r with f t = id for t 0 and f t = f for t 1. The totality of f t as above constitutes a set of arrows. This set determines uniquely a foliation, F(G), which coincides with the orbits of G(M) 0. Likewise, any set of local vector elds denes a foliation as any ow is an arrow. 2

4 A set of arrows A is said to be homogeneous if A(x) = A(x) 8x 2 M. Next, A is symmetric if 2 A implies that?1 is a composition of elements of A. Lemma. To each set of arrows A one can assign a homogeneous set of arrows A such that F(A) = F(A) and A 0 (x) = A(x) 8x 2 M. Moreover, if A 1 A 2 then A 1 A 2 where A i is assigned to A i. We reproduce the proof from [14] for the sake of completeness. First we enlarge A by setting ~A = f t+s ( s )?1 ; s?t ( s )?1 j 2 A; s 2 Rg: It is visible that ~ A = A and F( ~ A) = F(A). Furthermore, ~ A is symmetric, ~A 0 (x) = ~ A(x) = A(x), and ~ A is homogeneous whenever so is A. Next we put ^A = f = t?1 j dom = (?; ) V; V open; 2 ~ A; 2 A g: Then by a straightforward inspection ^A(x) = A(x). Since F( ^A) = F(A) it follows by Theorem 1 that ^A is homogeneous. Therefore A = e^a satises the claim. Let F 1 F k be a ag structure on M and let x 2 M. If x 2 L i 2 F i we write p i (x) = dim L i, p i (x) = p i (x)? p i?1 (x) (i = 2; : : : ; k) and q i (x) = m? p i (x). Denition. A chart (U; ) of M with (0) = x is called a distinguished chart at x with respect to F 1 F k if U = V 1 V k W with V 1 R p 1(x), V i R p i(x) (i 2) and W R q k(x) open balls such that for any L i 2 F i we have () (U) \ L i = (V 1 V i l i ); where for i = 1; : : : ; k. l i = fw 2 V i+1 V k W : (0; w) 2 L i g Observe that actually the above is an inverse chart; following [13] we call it a chart for simplicity. Notice as well that in the above denition one need not assume that F i is a foliation but only that it is a partition by weakly imbedded submanifolds; that F i is a foliation follows then by denition. Theorem 2. Let A 1 A k be an increasing sequence of sets of arrows of M. Then F(A 1 ) F(A k ) admits a distinguished chart at any x 2 M. Specically, so does any ag structure on M. Proof. Fix x 2 M and let p i = p i (x); p i = p i (x); q i = q i (x). We have A i (x) = T x F i, i = 1; : : : ; k. In view of Lemma each A i can be replaced by A i such that A 0 i (x) = Ai (x), F(A i ) = F(A i ), and A 1 A k. 3

5 First we choose j 2 A 1; j = 1; : : : ; p 1, such that _ j (0; x) form a basis of T x F 1. Next we extend this basis to a basis of T x F 2 by means of _ j (0; x) with j 2 A for 2 j = p 1 + 1; : : : ; p 2, and so on. Thus we obtain a basis of T x F k of the form _ 1 (0; x); : : : ; _ p k (0; x) where 1 ; : : : ; p i 2 A i for i = 1; : : : ; k. Now let Q be a q k -dimensional submanifold of M with x 2 Q and T x M = T x F k T x Q. Shrinking Q if necessary we may and do assume that _ pk?1+1 (0; z); : : : ; _ p k (0; z) are linearly independent for any z 2 Q. By choosing k > 0 suciently small we get that the mapping k : (t pk?1+1; : : : ; t pk ; z) 2 V k Q 7! p k?1+1 t pk?1 +1 p k t pk (z) 2 M is an imbedding, where V k = ft = (t pk?1+1; : : : ; t pk ) 2 R p k : jtj < kg, j:j being the usual norm. Then T x M = T x F k?1 T x (im k ). Again, possibly shrinking Q and V k we have that _ p k?2+1 (0; z 0 ); : : : ; _ p k?1 (0; z 0 ) are linerly independent for any z 0 2 im k. Then with some small k?1 > 0 the mapping k?1 : (t pk?2+1; : : : ; t pk ; z) 2 V k?1 V k Q 7! p k?2+1 t pk?2 +1 p k t pk (z) 2 M is an imbedding, where V k?1 = ft = (t pk?2+1; : : : ; t pk?1) 2 R p k?1 : jtj < k?1 g. We have T x M = T x F k?2 T x (im( k?1 ). By continuing this procedure we obtain a chart of the form 1 = : (t 1 ; : : : ; t pk ; y) 2 V 1 V k W 7! 1 t 1 p k t pk ((y)) 2 M; where W R q k is an open ball and : W! Q with (0) = x is an (inverse) chart. Let us check that is distinguished, i.e. that () is fullled. Fix 1 i k. Let z 2 U and (z) = 1 t 1 p k t pk ((y)) = 1 t 1 p i t pi (v) for some y 2 W and v 2 im( k?i+1 ). But im( k?i+1 ) = (0 0 V i+1 V k W ). Hence (z) 2 L i if and only if v 2 L i \ im( k?i+1 ), i.e. v = (0; w) where w 2 l i. In fact, 1 t ; : : : ; p i t preserve all the leaves of F i. For the second assertion let F 1 F k be a ag structure. If A(F i ) is the set of ows of all (local) vector elds tangent to F i then obviously F(A(F i )) = F i. The proof follows by the rst part. In view of the above proof we get Corollary. Let F 1 F k on M and let (L; ) be a leaf of F k. Then F 1 jl F k?1 jl and a distinguished chart at x for L is the restriction to L of a distiguished chart at x for M. 4

6 3. Examples We give some geometric examples of subfoliations or ag structures which motivate our interest, mainly in disciplines being nowadays intensively developed Let F i ; i = 1; 2, be a foliation of class C 1 and D i = ft x F i g its tangent distribution. Then F 1 \ F 2 F i if and only if D = D 1 \ D 2 is of class C 1 (i.e. for any x 2 M there are vector elds X 1 ; : : : ; X k(x) such that D x is spanned by X 1 (x); : : : ; X k(x) (x)), cf.[1] Let G i (M) Diff r (M) be a locally arcwise connected group of diffeomorphisms. If G 1 (M) G k (M) we get a ag structure F(G 1 ) F(G k ) Recall that any set of local vector elds V(M) on M denes uniquely a foliation F(V). So that if V 1 V k is an increasing sequence of sets of local vector elds one has F(V 1 ) F(V k ). In particular, if X is any vector eld tangent to a foliation F then the orbits of its ow constitute a subfoliation of F On M = R 3 we dene two foliations: F 1 = f all circles parallel to the x 2 x 3 -plane with center on the x 1 -axisg; and F 2 = f all spheres centered at 0g: Then F 1 F 2. We have three types of points: (i) if x = 0 then p 1 (x) = p 2 (x) = 0; (ii) if x lies on the x 1 -axis, x 6= 0 then p 1 (x) = 0 and p 2 (x) = 2; (iii) p 1 (x) = 1 and p 2 (x) = 2 for x o the x 1 -axis. This example can be obviously extended to R n+1 with F 1 F n Let (M; F) be a regular Riemannian foliation and let F be the set of closures of the leaves of F. Then F is a singular Riemannian foliation (cf.[9]) and obviously F F. Also orbit-like foliations introduced by P.Molino in [10] have the property that the foliation F by the closures of leaves of F is again Riemannian, and the relation F F holds Let i be a Poisson structure on M, i = 1; : : : ; k, and let F( i ) be the corresponding symplectic foliation (cf.[16]). If F( 1 ) F( k ) we shall say that ( 1 ; : : : ; k ) constitutes a Poisson ag structure. Observe that contrary to Theorem 2 and the existence of the canonical charts for Poisson manifolds (a splitting theorem of Weinstein [16]), a common canonical chart for i would exist only in very special cases. Likewise, a Jacobi ag structure arises when one has a sequence of Jacobi structures ( i ; E i ) such that F( 1 ; E 1 ) F( k ; E k ), where F( i ; E i ) is the characteristic foliation of ( i ; E i ) (see (3.10) below). 5

7 3.7. (Reduction of Poisson manifolds, cf.[16].) Let N be a submanifold of a Poisson manifold (M; ) such that D N = (] Ann(T N)) \ T N is a distribution of constant dimension along N. Here ] : 1 (M)! X (M) is dened by ]()() = (; ), and Ann denotes the annihilator. Then D N is dierentiable and integrates to a foliation F N which is called a subcharacteristic foliation of N. If N is transversal to F() then F N F() \ N Consider a Hamiltonian action of a compact Lie group G on a Poisson manifold (M; ), cf.[11]. Then the orbits of G form a subfoliation F G of F(). The same is true for canonical manifolds, cf.[6] A homogeneous Poisson structure (; Z) on a manifold M is a Poisson structure and a vector eld Z such that L Z =?, where L is the Lie derivative. These structures play a central role in the theory of Jacobi manifolds, see [3]. Let F(; Z) be the foliation generated by F() and Z (at some points Z is tangent to F() and at some is not). Then F() F(; Z). This is still the case of locally homogeneous structures (cf.[3, Prop. 2.16]) Recall that a pair (; E) is a Jacobi structure on M if is an antisymmetric (2,0)-tensor, E is a vector eld, and the equalities [; ] = 2E ^ ; [E; ] = 0 are satised. Here [:; :] is the Schouten-Nijenhuis bracket. For u 2 C 1 (M) one denes a Hamiltonian vector eld by X u = [; u] + ue. The orbits of the set of all Hamiltonian vector elds form a characteristic foliation of (; E), denoted by F(; E). Next, let F (; E) be a foliation determined by all X u with u 2 C 1 (M) satisfying L E u = 0. Then F(E) F (; E) F(; E); where F(E) is the foliation given by the orbits of E, cf.[3, p.119] Interesting examples arise on the ground of multisymplectic geometry (see, e.g., [4,5,7]). One of them is produced by Nambu-Poisson manifolds. Let us recall that a skew-symmetric n-linear mapping f; : : : ; g : C 1 (M) C 1 (M)! C 1 (M) is called a generalized almost Poisson (g.a.p.) bracket of order n if it veries the Leibniz rule: fu 1 v 1 ; : : : ; u n g = u 1 fv 1 ; : : : ; u n g + v 1 fu 1 ; : : : ; u n g for all u 1 ; : : : ; u n ; v 1 2 C 1 (M). Equivalently, a g.a.p. manifold of order n is given as the pair (M; ), where is a skew-symmetric (n; 0)-tensor on M. The relation between and the n-bracket f; : : : ; g is expressed by the equality (du 1 ; : : : ; du n ) = fu 1 ; : : : ; u n g. Then we dene a linear mapping ] : n?1 (M)! X (M) by setting < ]( 1 ^ : : : ^ n?1 ); >= ( 1 ; : : : ; n?1 ; ) for any 1 ; : : : ; n?1 ; 2 1 (M). X (M) 1 (M). Here <; > is the natural pairing on 6

8 For any u 1 ; : : : ; u n?1 2 C 1 (M) we dene a Hamiltonian vector eld X u1 ;::: ;u n?1 = ](du 1 ^ : : : ^ du n?1 ): A vector eld X is called an innitesimal automorphism of (M; ) if L X = 0. It is visible that this condition amounts to claiming that X is a derivation of the bracket f; : : : ; g. Now a g.a.p. manifold of order n is called a Nambu-Poisson manifold if any Hamiltonian vector eld is a derivation of the bracket. Notice that for n = 2 the above condition is equivalent to [; ] = 0 and, consequently, the Nambu-Poisson manifolds of order 2 coincide with the Poisson manifolds. Given a Nambu-Poisson tensor let us dene a smooth distribution D = fd x g x2m where D x = ](^n?1 T x M). This distribution is called characteristic. We have the following structural theorem. Theorem 3 [4]. Let (M; ) be an m-dimensional Nambu-Poisson manifold of order n 3. Then: (1) The characteristic distribution D is completely integrable and, consequently, it denes a foliation, denoted by F(). There are two kind of leaves of F(): (a) if x = 0 then then leaf passing through x reduces to x itself, and (b) if x 6= 0 then the leaf meeting x has dimension n and restricted to it induces a Nambu-Poisson structure which comes from a volume form. (2) In case (b) there exists a distinguished chart (x 1 ; : : : ; x n ; y 1 ; : : : ; y q ) at x (q = m? n) such that 1 ^ : : : n i i. Now for any u 2 C 1 (M) we put u = du. It is easily seen [4] that u is a g.a.p. (resp. Nambu-Poisson) structure of order n? 1 if is a g.a.p.(resp. Nambu-Poisson) structure of order n. This can be iterated by setting uv = ( u ) v and so on. Therefore for any choice u 1 ; : : : ; u n?1 2 C 1 (M) we obtain a ag structure F( u1 ;::: ;u n?1) F( u1 ;::: ;u n?2) F( u1 ) F(): As in (3.6) there are no common canonical charts for this ag Only recently the authors of [5] have introduced the concept of Nambu- Jacobi structure as a counterpart of the Jacobi structures in the multisymplectic geometry. Specically, if ( 1 ; 2 ) is a Nambu-Jacobi structure of order n, n > 2, then 1 (resp. 2 ) is a Nambu-Poisson structure of order n (resp. n? 1). Consequently further examples of ag structures occur. 4. Locality and pseudo-n-transitivity In [12] we have proven that the locality of a dieomorphism group yields its pseudo-n-transitivity. This can be extended to the ag case. Recall that G(M) Diff r (M), where r 1 (resp. r =!) satises the locality condition if for any open relatively compact U; V M with U V, and a C r -dieotopy ff t g in G(M) with f 0 = id, there exist > 0 and a smooth dieotopy fg t g in G(M) such that g t = f t on U and supp(g t ) V 7

9 for jtj < (resp. g t is suciently C 1 near f t on U and g t is suciently C 1 near the identity outside V for jtj < ). By an orbit of a sequence of dieomorphism groups G 1 (M) G k (M) we mean any orbit of each G i (M) 0. Next, for x; y 2 M belonging to a common orbit, a minimal orbit containing x; y is a unique orbit the least dimension passing through x; y. The sequence G 1 (M) G k (M) is said to be pseudo-n-transitive if for any two n-tuples of pairwise distinct points (x 1 ; : : : ; x n ) and (y 1 ; : : : ; y n ) of M such that x j ; y j belong to the same orbit and each orbit of dimension 1 contains at most one x j there exists f 2 G k (M) 0 satisfying f(x j ) = y j and preserving all the minimal orbits containing the pairs x j ; y j. Observe that the concept of pseudo-n-transitivity is an extension of the well-known notion of n-transitivity (see, e.g.,[8]) to arbitrary groups of diffeomorphisms. Theorem 4. Suppose that each group of an increasing sequence G 1 (M) G k (M) Diff r (M) (1 r!) satises the locality condition. Then this sequence is pseudo-n-transitive for every n 1. The proof makes use of Lemma and repeats an argument from [12]. References [1] M.Bauer, Feuilletage singulier deni par une distribution presque reguliere, These, Univ. Louis Pasteur (Strasbourg), Publ. I.R.M.A [2] P.Dazord, Feuilletages a singularites, Indag. Math. 47 (1985), [3] P.Dazord, A.Lichnerowicz, C.M.Marle, Structure locale des varietes de Jacobi, J.Math. pures et appl. 70(1991), [4] R.Iba~nez, M.de Leon, J.C.Marrero, D. Martin de Diego, Dynamics of generalized Poisson and Nambu-Poisson brackets, J. Math. Phys. 38(1997), [5] R.Iba~nez, M.de Leon, J.C.Marrero, E.Padron, Nambu-Jacobi and generalized Jacobi manifolds, preprint (September 1997). [6] C.M.Marle, Lie group actions on a canonical manifold, Symplectic Geometry, ed. by A.Crumeyrolle and J.Grifone, , Pitman, Boston, [7] P.W.Michor, A.M.Vinogradov, n-ary Lie and associative algebras, preprint ESI 402, Vienna (1996). [8] P.W.Michor, C.Vizman, n-transitivity of certain dieomorphism groups, Acta Math. Univ. Comenianae 63(1994), [9] P.Molino, Riemannian foliations, Progress in Math. 73, Birkhauser, [10] P.Molino, Orbit-like foliations, Proc. of "Geometric Study of Foliations" held in Tokyo, November 1993, World Scientic, Singapore (1994), [11] R.Ouzilou, Hamiltonian actions on Poisson manifolds, Symplectic Geometry, ed. by A.Crumeyrolle and J.Grifone, , Pitman, Boston,

10 [12] T.Rybicki, Pseudo-n-transitivity of the automorphism group of a geometric structure, Geom. Dedicata 67(1997), [13] P.Stefan, Accessibility and foliations with singularities, Bull. Amer. Math. Soc. 80(1974), [14] P.Stefan, Accessible sets, orbits and foliations with singularities, Proc. London Math. Soc. 29(1974), [15] H.J.Sussmann, Orbits of families of vector elds and integrability of distributions, Trans. Amer. Math. Soc. 180(1973), [16] I.Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Math. 118, Birkhauser, Basel, [17] V.P.Vilyantsev, Frobenius theorem for dierential systems with singularities, Vestnik Moskow. Univ. 3(1980), [18] R.A.Wolak, Characteristic classes of almost-ag structures, Geom. Dedicata 24(1987), Institute of Mathematics Pedagogical University ul.rejtana 16 A Rzeszow, POLAND rybicki@im.uj.edu.pl 9