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UDC 514.763.23
Complete Foliations with Transverse Rigid Geometries and Their Basic Automorphisms
N.I. Zhukova
Department of Mathematics and Mechanics Nizhny Novgorod State University Gagarina ave., 23, korp. 6, Nizhny Novgorod, Russia, 603950
The notion of rigid geometry is introduced. Rigid geometries include Cartan geometries as well as rigid geometric structures in the sense of Gromov. Foliations (M, F) with transverse rigid geometries are investigated. An invariant go of a foliation (M, F) with transverse rigid geometry, being a Lie algebra, is introduced. We prove that if, for some foliation (M, F) with transverse rigid geometry, g0 is zero, then there exists a unique Lie group structure on its full basic automorphism group. Some estimates of the dimensions of this group depending on the transverse geometry are obtained. Examples, illustrating the main results, are constructed.
Key words and phrases: rigid geometry, foliation, basic automorphism, holonomy group.
1. Introduction
One of the basic objects associated with a geometric structure on a smooth manifold is its automorphism group. Among the central problems, there is the question whether the automorphism group can be endowed with a (finite-dimensional) Lie group structure [1].
In the theory of foliations with transverse geometries, automorphisms are understood as diffeomorphisms mapping leaves onto leaves and preserving transverse geometries. The group of all automorphisms of a foliation (M, F) with transverse geometry is denoted by A(M, F). Let Al(m, F) be the normal subgroup of A(M, F) formed by automorphisms mapping each leaf onto itself. The quotient group A(M, F)/Al(m, F) is called the full basic automorphism group and is denoted by Ab (M, F).
In the investigation of foliations (M, F) with transverse geometry it is naturally to ask the above problem about the existence of a Lie group structure for the full group Ab (M,F) of basic automorphisms of (M,F).
Leslie [2] was first who solved a similar problem for smooth foliations on compact manifolds. For foliations with complete transversal projectable affine connection this problem was studied by Belko [3].
The leaf space M/F of the foliation is a diffeological space, and the group Ab (M,F) can be considered as a subgroup of the diffeological Lie group Diff(M/F). For Lie foliations with dense leaves on a compact manifold, the diffeological Lie groups Diff(M/F) are computed by Hector and Macias-Virgos [4].
In this work we introduce a notion of a rigid structure. Cartan geometries [1] and rigid geometric structures in the sense of Gromov [5, 6] are rigid structures in our sense. At the same time almost complex and symplectic structures don't belong to rigid structures. A manifold equipped with a rigid structure is called a rigid geometry.
We investigate foliations admitting rigid geometries as transverse structures and call them by foliations with transverse rigid geometries (TRG). Cartan foliations [7,8] and G-foliations, where G is a Lie group of finite type, are foliations with TRG. In particular, Riemannian, pseudo-Riemannian, Lorenz, projective and conformal foliations belong to the class of foliations under investigation. The category of foliations with TRG is denoted by Ftrg. The group Ab (M, F) is an invariant of (M, F) in the category Ftrg.
Received 31st October, 2008.
This work was supported by the Russian Foundation for Basic Research, project no. 06-01-00331-a.
We assume that all the foliations under consideration are modelled on effective rigid geometries. We construct the foliated bundle for a foliation (M, F) with TRG and reduce problems on the automorphism groups and the basic automorphism groups of (M,F) to the analogous problems for e-foliations (Theorems 3 and Proposition 9). Emphasize that these statements are proved without assumption of completeness of (M,F ).
For any complete foliation (M, F) with TRG we define the structure Lie algebra Qo(M, F) and show that Qo(M, F) is an invariant of this foliation in the category Ftrg (Proposition 5). One of the main results of this work is the proof of the theorem asserting that if q0(M,F) is zero, then there exists a unique Lie group structure on Ab (M,F). We also obtain some estimates of the dimensions of these Lie groups depending on the transverse geometry (Theorem 5).
We give different interpretations of holonomy groups of complete foliations with TRG (Theorem 4) and find some other sufficient conditions for the existence of a Lie group structure on Ab (M,F) (Theorem 6). In particular, it is shown that the structure Lie algebra of any complete proper foliation with TRG is zero, and Ab (M, F) is a Lie group (Corollary 2).
We demonstrate that, for a foliation with TRG covered by a fibration, the condition go(M,F) = 0 is equivalent to the discreteness of its global holonomy group (Theorem 7).
Examples of computations of the full basic automorphism group of a foliation with TRG are constructed. Examples 1 and 2 also show that the group Ab (M, F) depends on the transverse rigid geometry of the foliation (M, F).
2. Rigid geometries
Parallelizable manifolds. Recall that a manifold admitted an e-structure is called parallelizable. In other words, a parallelizable manifold is a pair (P,u), where P is a smooth manifold and u is a smooth non-degenerate Rm-valued 1-form u on P, i. e., oju : TUP ^ is an isomorphism of the vector spaces for each u E P. Here m = dim P.
Rigid structures. We will use notations from [9]. Denote by P(N, H) a principal fl-bundle with the projection p : P ^ N. Suppose that the action of H on P is a right action and Ra is the diffeomorphism of P, corresponding to an element a E H.
Two principal bundles P (N, H) and P(N, H) are called isomorphic if H = H and there exists a diffeomorphism r: P ^ P such that r o Ra = Ra o r, Va E H, where Ra is the transformation of P, corresponding to an element a.
Def 1. Let P(N,H) be a principal fl-bundle and (P,u) be a parallelizable manifold satisfying the following condition:
(S) there is an inclusion h C of the vector space of the Lie algebra h of the Lie group H into vector space such that u(A*) = A, VA E h, where A* is the fundamental vector field on P corresponding to A.
Then £ = ( P( N, H),u) is called a rigid structure on the manifold N. A pair (N, £) is called a rigid geometry.
Def 2. Let £ = (P(N,H),u) and |= (P(TV,H),Ôj) be two rigid structures. An isomorphism r: P ^ P of the principal bundles P (N, H ) and P(TV, H) satisfying the equality r*cv = u is called an isomorphism of the rigid structures £ and V.
Any isomorphism r of rigid structures £ and V defines a map 7 : N ^ N such that p o r = 7 o p, and 7 is a diffeomorphism from N to N. The projection 7 is called an isomorphism of the rigid geometries ( N, £) and ( N, V).
Induced rigid geometries. Let £ = (P(N,H),u) be a rigid structure on a manifold N with the projection p: P ^ N. Let V be an arbitrary open subset of the manifold N, let Pv := P-1(V) and uy := (¿Ipv . Then := (Py (V, H),uy) is also a rigid structure.
Def 3. The pair ('V,£v) defined above is called an induced rigid geometry on the open subset V of N.
Gauge transformations. Let be the group of all automorphisms of a rigid structure £ = (P(N, H),u). It is a Lie group as a closed subgroup of the group A(P, u) of all automorphism of a parallelizable manifold (P, u). Denote by A(N, £) the group of all automorphisms of the geometry (N,£), i. e., A(N,£) := G Diff(N) 13 r G A(Q : p o r = 7 o p}. Consider the natural group epimorphism % : ^ A(N, £) : r ^ ■j, where 7 is the projection of r with respect to p : P ^ N.
Def 4. Let £ = (P(N,H),u) be a rigid structure on a manifold N with the projection p : P ^ N. The group Gauge(£) := {r G | p o r = p} is called a group of gauge transformations of the rigid structure
Remark that Gauge(£) is a closed normal Lie subgroup of the group because it is the kernel of the natural group epimorphism % : ^ A(N,£).
Effectiveness of rigid geometries.
Def 5. A rigid structure £ = (P(N,H),u) is called effective if for an arbitrary open subset V in N the induced rigid structure = (Pv(V,H),i^v) has the trivial group of gauge transformations, i. e., Gauge(£y ) = {idpv }. A rigid geometry (N,£) is said to be effective if £ is an effective structure.
Pseudogroup of local automorphisms. Let (N,£) be a rigid geometry. For arbitrary open subsets V, V' C N an isomorphism V ^ V' of the induced rigid geometries ((V,^v) and (V',£v') is called a local automorphism of (N,£). The family H of all local automorphisms of a rigid geometry (N, £) forms a pseudogroup of local automorphisms. Denote it by H = H(N,£). Recall that a pseudogroup H of local diffeomorphisms of manifold N is called quasi-analytic if the existence of an open subset V C N and an element 7 G H such that 7lv = idy implies that 7^(7) = id£>(7) in the entire (connected) domain D(j) on which 7 is defined.
Proposition 1. The pseudogroup H = H(N,£) of all local automorphisms of an effective rigid geometry (N,£) is quasi-analytic.
Proof. Let 7 be an element of H = H(N,£) such that 7lv = idy for some open subset V in N. The effectiveness of the rigid geometry (N,£) implies r = idpv, where r is a local automorphism of £ having the projection 7lv with respect to p : P ^ N. Let the domain D = D(ry) of 7 be an open connected subset of N such that D \ V = $. Consider an automorphism r of the induced rigid structure with the projection j. Since r*ud = wd , r is an isomorphism of the parallelizable manifold (Pd ). It is known that two automorphisms of a connected parallelizable manifold, which coincide at one point, coincide at any point. Therefore it follows from the equality rIpv = r = idpv that t~lcpD = idcp0 for each connected component CPd of Pd. Thus, r = idpc, hence 7 = idp>. □
3. Foliations with transverse rigid geometries.
Foliated bundles
Foliations with transverse rigid geometries (TRG). A foliation (M,F) of codimension q on an n-manifold M has a transverse rigid geometry (N, £), where N is a ^-manifold, if (M,F) is defined by a cocycle iq = {Ui,fi, {jij}} modeled on (N,£), i. e.,
1) Wi} is an open covering of M ;
2) fi : Ui ^ N are submersions with connected fibres;
3) hj o fj = fi on Uî n Uj,
with 7ij is a local automorphism of (N, £). The topological space N is not assumed to be connected. Without loss of generality, we will suppose that N = Ui€jfi(Ui) and the family {( Ui, fi)} is maximal as it is generally used in manifold theory.
Let £ be the set of fibres of the submersions fi belonging to the cocycle rj. One can easily check that £ is a base of a certain topology t in M. The connected components of the topological space (M, t) form a partition F = {Lal a E A}.
Def 6. We call (M, F), where F is the partition mentioned above, a foliation with transverse rigid geometry ( N, £), and La are called its leaves. The cocycle r] modelled on (N, £) is said to be an ( N, £)-cocycle.
Let ( M,F) be a foliation defined by an (N, £)-cocycle ^ = {Ui, fi, {7ij}}, where ( N, £) is an effective rigid geometry. Effectiveness of £ guarantees the existence of a unique isomorphism rij- of the induced rigid structures (u.nu) and u.nu), whose projection coincides with 7^. Hence, in the case Ui n Uj nUk = the equality 7-ij o 7jk = 7ik implies the equality (ri) rij o rjk = rik.
The following two equalities are direct corollaries of the effectiveness of ^ and (r1) : (r2 )Ti = id Pi and (r3)rij = (rji)-1.
Assumptions. In this work we will assume that each rigid geometry is effective and all the foliations under consideration are modeled on effective rigid geometries.
Notations. We denote by X(N) the Lie algebra of smooth vector fields on a manifold N. If Q is a smooth distribution on M, then Xq(M) := {X E X(M) | Xu E Qu, Vu E M}. If Q is an integrable distribution and defines a foliation F, where Q = TF, we also use notation Xp(M) for Xq(M).
Foliated bundles. Now we construct the foliated bundle for a foliation with TRG.
Theorem 1. Let ( M,F) be a foliation with a transverse rigid geometry (N, £), where £ = ( P ( N,H ),w). Then there exist a principal H-bundle n : ' ^ M, an H -invariant foliation (', T) whose leaves are projected by n onto the leaves of (M, F) and an Rm-valued 1-form Cj on where m = dim P, that satisfy the following conditions:
(i) the map Cju : TU(P) ^ Rm, Vu E is surjective; moreover, kercvu = TUT;
(ii) there is an inclusion h C of the vector space of the Lie algebra h of the Lie group H into such that ¡V(A*) = A, VA E h, where A* is the fundamental vector field on ' corresponding to A;
(iii) the foliation (', T) is an e-foliation;
(iv) the restriction nc on an arbitrary leaf C of the foliation (', T) is a regular covering map onto a leaf of (M, F), and the subgroup H(C) := {a E H | Ra(C) = C} of the Lie group H is the group of deck transformations.
Proof. Supppose that the foliation ( M, F) with transverse rigid geometry is defined by a (N, ^-cocycle {Ui, fi, {7ij}}, where £ = (P(N,H),u), and let p: P ^ N be the projection of the principal fl-bundle P (N, H ). Denote Vi := fi(Ui), Pi := _p-1(Vi) and Pi := pIpî. Without loss of generality, we can assume that Ui and Vi are contractible open sets. Let ' := f*Pi := {(x, z) E Ui x Pi | fi(x) = Pi(z)}, fi : ' ^ Pi : (x, z) ^ z and ni : ' ^ Ui : (x, z) ^ x, V(x, z) E '. We have pi o fi = fi o ni. The formula (x, z) - a := (x,z ■ a), V(x, z) E ', Va E H, defines a right action of the group H on '. Thus we have the principal H -bundle ni : ' ^ Ui with the simple H -invariant foliation T i = {fi~ 1(z) | z E Pi}. Let := i^Ipî. We have the Rm-valued 1-form ûi := f*i^i defined on '. Moreover, Qi(X) = 0 for X E ) if and only if X E XTi (P ).
Let Y := Lk j ' be the disjunct union of the manifolds ' . Let us introduce an equivalence relation p in Y. Denote a point u eY n' by the pair (i ,u). Two points (i, u) and (j, w) in Y are said to be p-equivalent if
(1) ir_i(u) =nj(w) E Ui nUj;
(2) fi(u) = (r^- o fd)( w),
where r^- is the isomorphism of the rigid structures (UinUj) and ^fi(uinuj) whose projection 7ij belongs to the (N, £)-cocycle {Ui, fi, {jij}}.
The above equality (r2) implies that p is reflexive. The relation (r3) guarantees the symmetry of p, while the relation (Ti) implies the transitivity of p. Thus p is indeed an equivalence relation in Y. Hence we have the quotient space K := Y/p, the quotient mapping p : Y ^ K and the surjective projection n : K ^ M, where n maps the equivalence class [(z, u)] G K of a point (i,u) G Y to the point ni(u) G M. The restriction pi := p\ni : K ^ K is injective. Therefore pi is a bijective map onto image Ui := p(K); it will be denoted by pi : K ^ Ui. A smooth structure in K is well defined by assuming that each bijection pi is a diffeomorphism of K and [/¿.
Let x be any point in K and Ui 3 x. Set x ■a := p-^x) ■a, y a G H. This definition does not depend on the choice of Ui containing x because all Tij are isomorphisms of the corresponding principal fl-bundles. Thus K becomes the total space of the principal fl-bundle. The quotient manifold K/H can be identified with the manifold M, while the projection onto the quotient can be identified with n : K ^ M.
Define an 1-form Co on K by the formula Co\j. := (p-1)*^. If Ui n Uj = fy, then Ui nUj = 0 and r*^ = Uj because T^ is an isomorphism of the respective rigid structures, which lies over the local automorphism 7^ of (N, £). Since ooi = f*Ui, we have the equality ( i-1 )*ûi = (ip-1)*ûj on Ui n Uj. Thus the 1-form Co is well defined.
The foliations Ti on Ki and hence the foliations (tpi)*Ti on Ui are glued together by p into a foliation T on the manifold K such that T\jj. = (pî)*Ti. It follows from the definition of Co that Co(X) = 0 for X G X(K) if and only if X G (K).
The invariance of the foliations T, i G J, with respect to the action of the group H implies the fl-invariance of the foliation T on K.
The equality Cj(A*) = A, y A G h, is a consequence of the equality (S) for u and the definitions of the principal fl-bundle n : K ^ M with the 1-form Co.
We emphasize that the ( P, w)-cocycle {Ui, fi, {r^}} defines the foliation (K, T). Thus (K, T) is an e-foliation.
From the construction of the foliation (K, T) it follows that the restriction n\c onto an arbitrary leaf C of (K, T) is a covering mapping onto some leaf L of the foliation ( M, F ). Fix a point x G L and a point u G C n n-1(x). For any point u' G Cn n-1(x) there exists a unique element b G H such that u' = u ■ b. Invariance of the lifted foliation (K, T) with respect to the action of the Lie group H implies that Rb(C) = C, hence b G H (C) := {a G H \ Ra(C) = C}. Thus the subgroup H (C) of the group H acts transitively on the set C n n-1(x), with L = C/H(C). Therefore the covering mapping n\c : C ^ L is regular, and H(C) is its deck transformation group. □
Def 7. The principal fl-bundle K(M, H) with the fl-invariant foliation (K, T) constructed in the proof of Theorem 1 is called the foliated bundle for the foliation ( M, F) with transverse rigid geometry (N, £) and (K, T) is called the lifted foliation.
Remark 1. The lifted e-foliation (K, T) is defined by (P, w)-cocycle {Ui, fi, {r^}}.
Remark 2. If H is disconnected, K may be also disconnected. In this case all the connected components of K are mutually diffeomorphic, and we will consider one of them. Thus, we assume that the space of the foliated bundle K is connected.
4. Completeness and a structure Lie algebra of a
foliation with TRG
Completeness of foliations with TRG. Let ( M,F) be an arbitrary smooth foliation on a manifold M and TF be the distribution on M formed by the vector spaces tangent to the leaves of the foliation F. The vector quotient bundle TM/TF is called the transverse vector bundle of the foliation (M, F). Let us identify TM/TF
with an arbitrary smooth distribution M on M that is transverse to the foliation ( M,F), i. e., TM = TF ® M.
Let ( M, F) be a foliation with TRG and (P, T) be the lifted foliation. It is natural to identify the transverse vector bundle TP/TT with a distribution M := n*M on P, i. e., with a distribution defined by the equality Mu := {Xu E TUP | n*Xu E Mx}, where x = n(u) and u E P.
Def 8. A foliation (M, F) with transverse rigid geometry is said to be M-complete if any transverse vector field X E X^O such that cv(X) = const is complete. A foliation ( M, F) with TRG of arbitrary codimension q is said to be complete if there exists a smooth ^-dimensional transverse distribution M on M such that (M, F) is M-complete.
Remark 3. In other words, (M, F) is an M-complete foliation iff the lifted e-foliation (P, T) is complete with respect to the distribution M in the sense of Con-lon [10]. Remark that a complete e-foliation in the sense of Conlon is also complete in the sense of Molino [11].
Ehresmann connections for foliations. Let ( M, F) be a foliation of codimension q and M be a smooth ^-dimensional distribution on M that is transverse to the foliation F. The piecewise smooth integral curves of the distribution M are said to be horizontal, and the piecewise smooth curves in the leaves are said to be vertical. A piecewise smooth mapping H of the square p x I2 to M is called a vertical-horizontal homotopy if the curve Hlsxi2 is vertical for any s E h and the curve H|IlXt is horizontal for any t E I2. In this case, the pair of paths ( HlIlX{0},H|{o}xl2) is called the base of H. It is well known that there exists at most one vertical-horizontal homotopy with a given base. A distribution M is called an Ehresmann connection for a foliation ( M, F) (in the sense of Blumenthal and Hebda [12]) if, for any pair of paths (a, h) in M with a common starting point a(0) = h(0), where a is a horizontal curve and h is a vertical curve, there exists a vertical-horizontal homotopy H with the base (a,h). If the distribution M is integrable, then the connection is said to be integrable. For a simple foliation F, i. e., such that it is formed by the fibers of a submersion r: M ^ B, a distribution M is an Ehresmann connection for F if and only if M is an Ehresmann connection for the submersion r, i. e., if and only if any smooth curve in B possesses horizontal lifts.
Proposition 2. If ( M,F) is an M-complete foliation with TRG, then M is an Ehresmann connection for this foliation.
Proof. The distribution M := n*M is an Ehresmann connection for the lifted foliation (P, T), because ( M,F) is M-complete. So in view of F = n*T and M = n*M, we see that M is an Ehresmann connection for ( M,F). □
Structure Lie algebra. Applying of the relevant results of Molino [11] on complete -foliations, we obtain the following theorem.
Theorem 2. Let ( M,F) be a complete foliation with TRG and (P, T) be its lifted -foliation. Then:
(i) the closure of the leaves of the foliation T are fibers of a certain locally trivial fibration : P ^ W;
(ii) the foliation (C, T|^) induced on the closure C is a Lie foliation with dense leaves with the structure Lie algebra g0, that is the same for any C E T.
Def 9. The structure Lie algebra g0 of the Lie foliation (C, T^) is called the structure Lie algebra of the complete foliation ( M, F) and is denoted by g0 = Q0(M, F).
Remark 4. If (M, F) is a Riemannian foliation on a compact manifold, this notion coincides with the notion of a structure Lie algebra in the sense of Molino [11].
Def 10. The fibration : P^W satisfying Theorem 2 is called a basic fibration for ( M,F).
5. Category of foliations with TRG
Category of foliations. Denote by Fol the category of foliations, objects of which are foliations, morphisms of two arbitrary foliations ( M,F) and (M',F') are smooth maps M — M' mapping leaves of the foliation (M, F) into leaves of the foliation ( M' ,F'); a composition of morphisms coincides with the composition of maps.
Category of foliations with TRG. Let ( M,F) and (M',F') are foliations with transverse rigid geometries ( N, £) and ( N', £') defined by an ( N, £)-cocycle V = {Ui, fi, {^iij}} and an ( N', £')-cocycle r]' = {U'r, f'r, {7^}}, respectively. Let f: M — M' be a morphism which is a local isomorphism in the category Fol.
Hence for any x G M and y := f(x) there exist neighborhoods Uk 3 x and U'k 3 y from r] and rf respectively and a diffeomorphism A : Vk — V^, where Vk := fk (Uk) and Vg := f's(U's), satisfying the relations f(Uk) = U's and A o fk = f's o f\Uk. We will say that preserves transverse rigid structure if the diffeomorphism A : Vk —Vg is an isomorphism of the induced rigid geometries ( Vk, £vk) and ( Vg, ').
This notion is well defined, i. e., it does not depend of the choice of neighborhoods Uk and U'k from the cocycles r] and r(.
By a TRG-morphism of two foliations ( M,F) and (M',F') with transverse rigid geometries we mean a morphism f: M — M ' in the category Fol which preserves transverse rigid structure. The category Ftrg objects of which are foliations with TRG, morphisms are TRG-morphisms, is called the category of foliations with transverse rigid geometries.
Isomorphisms in the category Ftrg. Remark that for any e-foliation (K, T) the lifted foliation coincides with (K, T). Using this we easily get the following lemma.
Lemma 1. Let (K, T) and (K', T') be two e-foliations with transverse rigid geometries ( P,u) and (P',u/) respectively, where (P,^) and (P',u>') are parallelizable manifolds. Let Cj and Cj' be the basic 1-forms on K and K' defined according to Theorem 1. Then a diffeomorphism f : K — K' is an isomorphism in the category Ftrg if and only if f is an isomorphism in the category Fol and f*ùj' = Cj.
Proposition 3. Let (M,F) and (M',F') be two foliations with TRG, let (K, T) and (K', T') be the corresponding lifted foliations. Then a diffeomorphism f : M — M' is an isomorphism in the category Ftrg if and only if there exists an isomorphism f : K — K' of the lifted foliations in the category Ftrg such that R'a o f = f o Ra, y a G H, where Ra, R'a are the right translations by an element a G H on K and K' accordingly.
Proof. We will use the notations introduced in the proof of Theorem 1. Let the foliation with TRG ( M,F) is defined by an (N, £)-cocycle r] = {Ui, fi, {^iij}}. Recall
that the lifted foliation (K, T) is defined by a ( P, w)-cocycle fj = {Ui, fi, {r^}}, where Ui := n-1(Ui), and the local isomorphisms r^ of the rigid structure £ lie over the local isomorphisms 7^ of the rigid geometry (N, £). For the objects, concerning to the foliation ( M',F'), we will use primes.
At first, suppose that : K — K is an isomorphism of the -foliations (K, T) and (K', T') satisfying the condition R'a of = f o Ra, ya G H. Then the projection f : M — M' of f is well defined by the equality n' of = f on, where n : K — M and n' : K' — M' are the projections of the foliated bundles.
Consider an arbitrary point x G M and y := f(x) G M'. There are neighborhoods Uk 3 x and U's 3 y from the (N, £)-cocycle and the (N',£')-cocycle defining the foliations ( M,F) and (M',F') respectively, with f(Uk) = U's. Then f(Ûk) = Û's. The lifted e-foliations ( Uk, T\jk ) and ( U's, T'\j' ) are defined by the submersions fk : Uk — Pk and f's : U's — P's accordingly. Besides, R'a of = f o Ra, ya G H. Hence, according to Lemma 1, a diffeomorphism r: Pk — P's defined by the relation r o fk = f's o f\u' is a local isomorphism of the rigid structures £ and . Put Vk = fk (Uk) and V's = f's(U's).
Let 7 :Vk ^ V's be the projection of r, then 7 is an isomorphism of the rigid geometries induced on Vk and VS. Thus, f is an isomorphism of the foliations (M, F) and (M', F') in the category Ftrg.
Converse, suppose that f : M ^ M' is an isomorphism of the foliations (M, F) and ( M ' ,F ') in the category Ftrg. Construct f: P ^ P' in the following way. Let x be any point in M and y := f(x) E M'. Let Uk 3 x and U's 3 y be neighborhoods from the cocycles r] and rf respectively, with f(Uk) = U's. Consider Pk := f**(Pk),
where Pk = fk(Uk). Then Pk = {(x, z) E Uk x Pk | fk(x) = Pk(z)}, P's is defined similarly. Since f is an isomorphism in the category Ftrg, by definition, there exists a diffeomorphism 7 : Vk ^ VS which is an isomorphism of the induced rigid geometries ( Vk, ivk) and ( VS, ), and 7 o fk = fS o fluk.
Since the rigid geometries ( N, £) and (N', £') are effective, there is a unique isomorphism r : Pk ^ PS of the induced rigid structures £vk and ' with the projection 7. Then r*w' = u and r o Ra = R'a o r, Va E H. Define a map h : Pk ^ P's by the equality
h(x, z):=( f(x), r(z)), V(x, z) E Pk.
According to the definition of the foliated bundle for (M,F), the bijections : P-i ^ Ui, are isomorphisms of the simple foliations with TRG defined by the submersions fi : Pi ^ Pi and fi : Ui ^ Pi respectively. An analogous assertion holds for the foliation ( M',F'). Hence h : Pk ^ P's is an isomorphism of the foliations mentioned above in the category Ftrg.
Put, by definition, /U := o h op-1 for any neighborhood Uk from the cocycle r]. It is not difficult to check that this equality defines the map f: P ^ P!, where f satisfies the following conditions: (i ) f*Cv' = cj and (ii) R'a of = f o Ra, Va E H. Therefore, by Lemma 1, f is an isomorphism of the lifted e-foliations satisfying (ii). □
Proposition 4. Let ( M,F) and (M', F') be two foliations with transverse rigid geometries ( N, £) and ( N'£') accordingly. Let f1 and f2 : P ^ P' be two isomorphisms of (P, T) and (P', T') satisfying the equalities R'a o fi = fi o Ra, i = 1,2, Va E H. If their projections hi : M ^ M' coincide: h1 = h2, then f1 = f2.
Proof. The map f := f-1 o f1 : P ^ P is an isomorphism of (P, T) satisfying the relation R'a o f = f o Ra, Va E H, where the projection of f is f = h-1 o h1 = id^. For any x E M we have y = f(x) = x. Therefore, we can take Us = Uk 3 x in the definition of morphisms of the category Ftrg. Then we have Uk = Us. As above, let fk : Uk ^ Vk be a submersion from the cocycle defining (M, F) and pk := plpk. Then 7 = idyfc and r opk = pk, where r: Pk ^ Pk is an automorphism of the induced rigid structure £vk = ( Pk(H,Vk),uk). Therefore r E Gauge(£yfc). Due to the effectiveness of the transverse rigid geometry ( N, £), we necessarily have r = id pk. The equality
r o fk = fk o flok implies fk = fk o flok, i. e., f)ûk E ( Uk, Tl ùk). For any x em,
the neighborhoods {Uk | x E Uk} from the (N, ¿;)-cocycle r] = {Ui, fi, {7ij}} form a base of the topology of the manifold M at x. Hence f(u) = u, Vu E n-1(x). Since x is an arbitrary point in M, we have f = id^.. Thus, f1 = f2. □
A foliated natural functor. By analogy to Proposition 3 and 4 it is not difficult to prove that for any morphism f : M ^ M' of foliations (M, F) and (M, F') in the category Ftrg there exists a unique morphism f: P ^ P' of the lifted foliations (P, T) and (P', T') satisfying the equality R'a of = f o Ra Va E H. Set $(M, F) := (P, T) and $( f) := f, then we get a covariant functor $ from the category Ftrg to the category of foliated bundles. This functor is a foliated natural bundle in sense of Wolak [13], [14, Chapter II].
Automorphism groups of foliations with TRG. Let ( M,F) be a foliation with a fixed transverse rigid structure ( N, £). Denote by A(M,F) the group of all automorphisms of ( M, F) in the category Ftrg . We say also that A(M, F) is the full group of automorphisms.
Theorem 3. Let (M,F) be a foliation with TRG. Let (K, T) be the lifted foliation and AH(K, T) = {f G A(K, T) \ f o Ra = Ra o f, ya G H}. Then the map p : AH (K, T) — A(M,F): f — f, where f is the projection of f G AH (K, T) with respect to n : K — M, is a natural group isomorphism.
Proof. By Proposition 3, the map p is well defined and surjective. It is clear that V is a group homomorphism. According to Proposition 4, p is injective. Thus, p is a group isomorphism. □
Remark 5. Due to Theorem 3, problems concerning to automorphism groups of foliations with TRG are reduced to the analogous problems for automorphism groups of the lifted -foliations.
Invariance of the structure Lie algebra. The following statement shows that the structure Lie algebra g0(M,F) of a foliation (M,F) with TRG is an invariant in the category Ftrg.
Proposition 5. Let ( M,F) and (M',F') be two foliations with TRG isomorphic in the category Ftrg. Then their structure Lie algebras q0(M,F) and g0(M',F') are isomorphic.
Proof. Let (K, T) and (K', T') be the lifted foliations for ( M,F) and (M',F') respectively. Suppose that there exists an isomorphism f : M — M' of the foliations ( M,F) and (M',F') in Ftrg. Then by Proposition 3 there exists a map f : K — K' which is an isomorphism of (K, T) and (K', T'). Let C be an arbitrary leaf of (K, T), then C = /(C) is a leaf of (K', T'). Since f is a homeomorphism, f maps the closure C of C onto the closure C' of C, i. e., /(C) = C'. Thus, : C — C' is an isomorphism of the induced Lie foliations (C, T\^) and (C', T'\-j-j) with dense leaves. It is known [11] that the structure Lie algebra of a Lie foliation with dense leaves is an invariant in the category of foliations Fol. Therefore the Lie algebras g0(C, T\^) and g0(C', T'\^r) are isomorphic. By definition q0(M,F) = g0(C, T\^) and q0(M',F') = g0(C',T'\jr), hence the Lie algebras g0( M,F) and Q0(M', F') are isomorphic. □
6. Different interpretations of holonomy groups
Holonomy groups of foliations with Ehresmann connections. Let ( M, F)
be a foliation with an Ehresmann connection M (see Section 3). Let Qx be the set of horizontal curves with an initial point x. It is not difficult to prove that the map : Q x x n\(L, x) ^ Qx : (a, [ft,]) ^ a, where [h] G k\(L, x), H is a vertical-horizontal homotopy with the base ( a,h), and <r(s) := H(s, 1), s G h, defines a right action of the fundamental group ^i(L,x) of the leaf L = L(x) on the set Qx.
Def 11. Since KM(L,x) := {[h] G v1(l,x) | §x(a, [h]) = a, Va G = ker^ is a normal subgroup in n1(L,x), the quotient group hm(l,x) := K1(L,x)/ker$x is well defined [12]. The group Hm(L, x) is called the M-holonomy group of the leaf L of the foliation ( M, F) with the Ehresmann connection M.
It is known that there is a natural group epimorphism 5: Hm(L, x) ^ r(L, x) onto the germ holonomy group r( L, x) of the leaf L = L(x) such that
P = 5 o a, (*)
where a: n1(L, x) ^ Hm(L,x) and P: n1(L, x) ^ r(L, x) are the natural projections onto the corresponding quotient groups.
The following assertion is a consequence of Theorem 7 proved by the author in [15].
Proposition 6. Let ( M,F) be a foliation with an Ehresmann connection M. The natural group epimorphism 5: Hm(L,x) ^ Y(L,x) satisfying the relation (*) is an isomorphism if and only if the holonomy pseudogroup of the foliation ( M, F) is quasi-analytic.
Equivalent approaches to the notion of holonomy groups.
Theorem 4. Let ( M,F) be an M-complete foliation with TRG defined by an (N, £)-cocycle {Ui, fi, {^ij}}. Let L = L(x), x G M, be an arbitrary leaf of this foliation and C = C(u), u G n-1(x), be the corresponding leaf of the lifted foliation (R, T). Then the germ holonomy group r(L, x) of the leaf L is isomorphic to each of the following five groups:
(i) the M-holonomy group Hm(L, x);
(ii) the group Hv formed by germs of local diffeomorphisms belonging to the isotropy subpseudogroup of the holonomy pseudogroup H of local automorphisms of the transverse rigid geometry ( N, £) at point v = fi(x), where x G Ui;
(iii) the group of deck transformations of the regular covering map nle: C ^ L;
(iv) the subgroup H(C) = {a G H | Ra(C) = C} of the Lie group H;
(v) the holonomy group $( u) of the integrable connection T(T^-i^)) in the principal H-bundle with the projection n^-i^): n-1(L) ^ L.
Proof. According to Proposition 2, an M-complete foliation ( M, F) with TRG has an Ehresmann connection M. Recall that the holonomy pseudogroup H is a subpseudogroup of the pseudogroup H( N, £) of all local automorphisms of the transverse rigid geometry ( N, £). According to Proposition 1 H is a quasi-analytic pseudogroup. Therefore applying Proposition 6 we see that v: Hm(L,x) ^ r(L, x) is a natural group isomorphism.
Recall that according to Theorem 1 the restriction nle : C ^ L is a regular covering with the deck transformations group H(C). Then there is a normal subgroup p*(u) of the fundamental group n1 (L, x) and a group isomorphism pu: n1 (L, x) /p* (u) ^ H(C).
Denote by a: n1(L,x) ^ r(L, x) and 0: n1(L,x) ^ $(u) the natural group epi-morphisms. It is enough to show that kera = ker0 = p*(u). Let [h] G ker a, then h is a loop at x. Consider a chain U1,..., Uk, Ui fl Ui+1 = Vi G {1,... ,k — 1}, of neighborhoods from the ( N, £)-cocycle r] that covers the set h([0,1]). Let fi: Ui ^ Vi be submersions and jjs be the corresponding local automorphisms of the rigid geometry ( N, £) from rj. According to Proposition 4 for each jjs there is a unique local automorphism rjs of £ lying over 7js.
The composition of projections 7 := 71k o 7kk-1 o ... o 721 is defined in a neighborhood of the point v := f1(x) of the manifold N. The triviality of the germ of 7 at v is a consequence of the choice of [h] G kera. Therefore there exists a neighborhood V 3 v in N such that 7lv = idy. Due to the effectiveness of £, the automorphism r := r1k o r kk-1 o ... o r21 satisfies the equality T|pv = idpv.
Denote by h the path in the leaf C with the origin u = h(0) covering the loop h. In the sequel, we will use notations of the proof of Theorem 1. From the definition of the lifted foliation (R, T) it is follows that the chain U1,...,Uk, where Ui = n-1(Ui), covers the set h([0,1]). As f1(u) G Pv, the equality T|pv = idpv implies r( fi(u)) = fi(u). Hence /i(h(1)) = fi(h(0)). Therefore h(1) = h(0) = u and h G ker0. Thus, kera C ker0.
The equality ker0 = p*(u) follows directly from the definition of $(u) [7]. To complete the proof, we have to show the implication p*(u) C kera. Take any [h] G p*(u). Let h be the loop in C covering h with the origin at u = h(0). Then h(1) = h(0) = u. Consider an arbitrary chain U1,..., Ur, Ui f Ui+1 = Vi G {1,... ,r — 1}, of neighborhoods belonging to the ( N, £)-cocycle r] that covers the set h([0,1]). Let 7js be the corresponding local automorphisms of ( N, £) from r¡. Let rjs be the unique local automorphism of £ with the projection 7js. It is well known that any e-foliation has no holonomy. Then the holonomy diffeomorphism r := r1r o rrr-1 o ... o r21 has the
trivial germ at the point f1(u). Therefore its projection 7 := 71r o 7rr-1 o ... o 721 has the trivial germ at point v = /1 (x). Since 7 is a local holonomy diffeomorphism along the loop h, we have [h] G ker a. □
7. Foliations with the zero structure Lie algebra
Proposition 7. Let ( M, F) be a complete foliation with TRG. Suppose that g0( M, F) = 0. Let : R^W be the basic fibration. Then:
(i) the formula
$w : W xH ^ W: (w, a) ^ Kb(Ra(u)) V(w,a) GW x H, Vu Gk-1(w)
defines a smooth locally free action of the Lie group H on the basic manifold W;
(ii) there is a homeomorphism s: M/F ^ W/H between the leaf space M/F and the orbit space W/H satisfying the equality k = s o qo^, where k: W ^ W/H is the quotient map onto W/H, q: M ^ M/F is the quotient map onto M/F;
(iii) the equality k*lu = tO defines an Rm-valued non-degenerate 1-form to onW such that oj(AW ) = A, where AW is the fundamental vector field on W defined by an element A G h C Rm.
Proof. (i) Since q0(M,F) = 0, by Theorem 2 the lifted foliation (R, T) is formed by the fibres of the basic fibration : R^W. The action $w is well defined, because the lifted foliation (R, T) is H-invariant. Smoothness of the action of H on R and smoothness of imply smoothness of $w. Take any point w G W and u G n-1(w). Let C = C(u) and L := n(C), then x = k(u) G L. Recall that H(£) = {a G H | Ra(C) = C}. Let Hw be the isotropy subgroup of H at w. From the definition of the action $w of the Lie group H on W it follows that H(C) = Hw. The condition g0( M,F) = 0 implies that the lifted foliation (R, T) is proper, hence the induced foliation (k-1(L), T|,r-i(£)) is also proper. Therefore the orbit u H (C) = CflK-1(x) is a discrete subset of the orbit u-H. So H(C) is a discrete subgroup of the Lie group H.
Thus, each isotropy group of the action $w is discrete, i. e., $w is a locally free action.
(ii) Consider an arbitrary point x G M, u G K-1(x) and w = Kb(u). From (i) it is follows that nb(n-1 (L(x)) = w ■ H. Hence the map s: M/F ^ W/H: [L] ^ [w ■ H], where [ L] is the leaf L considered as a point of M/F and [w ■ H] is the orbit of H considered as a point of W/H, is well defined and satisfies the equality k= soqow stated in (ii). Since k and are open and continuous maps, this relation implies that the bijection is a homeomorphism.
(iii) This statement is a consequence of the assertion (ii) of Theorem 1 and of the definition of the 1-form to. □
Corollary 1. If g0( M,F) = 0, then the holonomy group r(L,x) is isomorphic to the isotropy group Hw, where w G Kb(ir-1(x)), of the induced action $w of the Lie group H on the basic manifold W.
Proof. As shown in the proof of Proposition 7, H(C) = Hw. Therefore, according to Theorem 4, the holonomy group r(L, x) of a leaf L of this foliation is isomorphic to the isotropy group Hw. □
8. The groups of basic automorphisms of foliations
with TRG
Let A(M, F) be the full automorphism group of a foliation (M, F) with TRG. We denote by p: AH (R, T) ^ A(M,F) the group isomorphism defined in Theorem 3.
Leaf automorphisms. The group
Al(M, F) := [f e A(M, F) | f(La) = La, VLa e F}
is a normal subgroup of A(M, F) which is called the leaf automorphism group of (M, F).
Proposition 8. Consider the subgroup of leaf automorphisms AH (R, T) := [f e AH(R, T) I f(Ca) = Ca, VCa e T} of the group AH(R, T). Then the restriction
»l := : AH(R, T) ^Al(M,F)
is a group isomorphism.
Proof. Let f e AIf (R, T) and f := »(f). Consider an arbitrary leaf L e F. There exists a leaf C e T such that nIc: C ^ L is a covering map. As /(C) = C and n of = f o n then f(L) = L, hence f e Al(M,F ). Thus we have an inclusion »(AH(R,T)) CAl(M,F).
Let us show that the map »l is surjective. Take an arbitrary element g e Al(M,F). According to Theorem 3 there is a unique element g e AH (R, T) lying over g. Let u be an arbitrary point in R, u' := g(u), x = n(u), L = L(x). There are neighborhoods Uj 3 x and Ui 3 y = g(x) from the (N, ¿;)-cocycle r] = [Ui, fi, [^ij}} defining the foliation ( M,F). Remark that the points v := fj(x) and v' := fi(y) belong to the same orbit of the holonomy pseudogroup H(M, F) of the foliation (M, F). Recall that each element of H(M, F) is a local automorphism of the transverse rigid geometry (N, £). Therefore there exists a local automorphism e H(M, F) such that li.j(v) = v'. Effectiveness of the transverse rigid geometry (N, £) implies the existence of a unique local automorphism Tj of the rigid structure £ from the holonomy pseudogroup H(R, T) with the projection 7^. In the notations of the proof of Theorem 1 V = [Ui, f-i, [rj}} is a ( P, w)-cocycle defining the e-foliation (R, T). Let w = fj(u), w' = fi(u') then w' = rj(w), i. e., the points w and w' belong to the same orbit of the holonomy pseudogroup H(R, T). Therefore the points u and u' belong to the same leaf C of (R, T), i. e., g(C) = C. Hence g e Aj^(R, T).
Thus »l is an isomorphism of the groups Aj^(R, T) and Al(M,F). □
Basic automorphisms of foliations with TRG. Remark that the quotient group AH(R, T) := AH(R, T)/AH(R, T) is well defined.
Def 12. The quotient group
AB( M, F) := ^(M, F)/Al(M, F)
is called the basic automorphism group of the foliation ( M, F) with TRG.
Let ( M,F) be a foliation with TRG. Let M/F be the leaf space of (M,F), and q: M ^ M/F be the natural projection onto the leaf space which maps any x e M to the leaf L(x) considered as a point [L(x)] in M/F. Each f e A(M,F) maps an arbitrary leaf L of F onto some leaf of this foliation. Hence the equality f([L]) = [f(L)\ defines a mapping of the leaf space M/ F onto itself such that the following diagram
M
M/F f
M
M/F.
(1)
f
is commutative. Since q is an open and continuous mapping, (1) implies that f is a homeomorphism of the leaf space M/F. Denote by A(M/F) the set of all such homeomorphisms of M/F. Then
q: A(M, F) ^ A(M/F) :f»f
is a group epimorphism with the kernel ker q = Al(M,F). Therefore the basic automorphism group Ab ( M,F) is canonically isomorphic to the group A(M/F). Thus the basic automorphism group Ab ( M,F) can be considered as a group, A(M/F), of homeomorphisms of the leaf space M/ F of this foliation.
Let us emphasize that the basic automorphism group Ab ( M,F) of a foliation ( M,F) with TRG is an invariant of this foliation in the category Ftrg .
Proposition 9. Let ( M, F) be a foliation with TRG and (R, T) be the lifted foliation. Denote by AH(R, T) the quotient group AH(R, T)/AjH (R, T). There exists a natural group isomorphism x: AH(R, T) ^ Ab (M,F) satisfying the commutative diagram
AH (R, T) —— A( M,F)
(2)
AH (R, T) -— Ab ( M,F), where r and s are the associated group epimorphisms onto the quotient groups.
Proof. By Theorem 3, the map x: ЛH(R, T) ^ Ab (M,F): h ■ AH(R, T) ^ h ■ Al(M,F), where h is the projection of h G AH(R, T) with respect to i: R ^ M, is well defined. According to Proposition 8, ^(kerr) = ker s, where kerr and ker s are the kernels of the epimorphisms r and s, respectively. Hence there exists an isomorphisms of the quotient groups AH (R, T) and Ab ( M,F) satisfying the diagram (2). □
9. Conditions guarantee that Ab(M,F) is a Lie
group
Uniqueness of a Lie group structure. The next proposition follows from Proposition 1 proved by Bagaev and the author in [16].
Proposition 10. Let A(P,o) be the Lie group of all automorphisms of a paral-lelizable manifold ( P,o). If a group G is realized as a closed subgroup of A(P,u), then G admits a unique topology and a unique smooth structure that make it into a Lie group. This topology coincides with the compact-open topology.
The case g0( M,F) = 0. A leaf L of a foliation (M,F) is called closed if L is a closed subset in the topology of the manifold M. Further we use the term "a closed leaf" only in this sense.
Let (M, F) be a complete foliation with TRG and Kb: M ^ W be the basic fibration. Suppose that g0( M,F) = 0, then according to Theorem 2 the leaves of the lifted foliation (R, T) coincide with the fibres of the basic fibration Kb: R ^ W. Hence the basic manifold W can be identified with the leaf space R/T of the foliation (R, T), and Kb can be identified with the projection q: R ^ R/T.
Applying the commutative diagram (1) to the foliation (R, T) we see that each automorphism h G AH(R, T) induces a diffeomorphism h of the manifold W such that Kb o h = h o Kb. Since h*u = o, then, from the definition of the non-degenerate Rm-valued 1-form o on W satisfying Proposition 7, it is follows that h*ui = tO. From the definition of the action H on W it is follows that h o Ra = Ra o h, Va G H.
As above, we denote by A(W, o) the group of all automorphisms of the paralleliz-able manifold ( W,o), i. e., A(W,u) := {f G DiffW | f*o = O}. It is well known
that A(W, oo) admits a unique Lie group structure. There is a natural bijection between the identity component Ae(W,oo) of A(W,co) and the orbit Ae(W,oj) ■ v of a point v G W, being a closed submanifold of W. This bijection induces a smooth structure on Ae(W,oo) [9]. According to Proposition 10, the topology of A(W,oj) is the compact-open topology.
Let AH ( W) := {f G A( W,o) | f o Ra = Ra of} and let AH (W) be the identity component of AH( W). Then AH( W) and AH( W) are closed Lie subgroups of A(W, oo).
Proposition 11. Let (M, F) be a complete foliation with TRG and g0(M, F) = 0. Then the map
v\ AH (R, T) ^Ah ( W): ft ■AH (R, T) ^ ft
where ft G AH (R, T) and ft is the projection of ft with respect to nb: R ^ W is a group isomorphism onto an open-closed Lie subgroup of the Lie group AH ( W).
Proof. At first, consider the map a: AH(R, T) ^ AH(W): h^ ft, where ft is the projection of ft with respect to Kb: M ^ W. As shown above, ft G AH(W). It is clear that a is a group homomorphism with the kernel kera, being equal to the normal subgroup AjH(R, T). Therefore, there exists a group isomorphism u: AH(R, T) ^ AH ( W), satisfying the equality a = v o r, where r: AH (R, T) ^ AH (R, T) is the natural projection onto the quotient group AH(R, T) = AH(R, T)/AH(R, T).
It is enough to prove that im a is an open-closed subgroup in AH ( W).
If AH( W) is a discrete Lie group, then im v = im a is also a discrete Lie group.
Now suppose that dim AH ( W) > 1. Let a be the Lie algebra of the Lie group AH( W) and B be any element of a. Denote by B* the fundamental vector field defined by B. Then X := B* is a complete vector field and it defines a 1-parameter group pf-, t G (-x, x), of diffeomorphisms of W. The condition G AH(W), Vt G (-x, x) is equivalent to the following relations: 1) LxAw = 0, VA G h; 2) Lxoj = 0.
Since Kb: R^W is a submersion with an Ehresmann connection M, there exists a unique vector field Y G X^(R) such that Kb*Y = X. Remark that completeness of the vector field X implies completeness of the vector field Y. Hence Y defines a 1-parameter group ^Y, t G (-x, x), of diffeomorphisms of the manifold R. Let us show that ^Y G AH (R, T), Vt G ( -x, x ), i. e., we have to check the validity of the following facts: 1) the map ^Y, t G (-x, x), is an isomorphism of (R, T) in the category Fol; 2) LyO = 0; 3) LyA* = 0, VA G h.
1) The equality Kb*Y = X implies the relation Kb o ^Y = o Kb for any fixed t G (-x, x), hence (k-1(w)) = n-1((px(v)), Vv G W, and is an isomorphism of the lifted foliation (R, T) in the category Foi.
2) Take arbitrary u gR and Zo G Mu. There is a unique vector field Z G X^(R) such that Z^ = Zo and oo(Z) = Oo(Zo) = const. Put Zw := Kb*Z and apply the following formula [9]:
(LxO)(Zw) = X(J(Zw)) - j([x, zw]). (3)
The relation o = Oo o Kb* implies that oj(Zw) = to(Z0) = const, so X(oo(Zw)) = 0. By the choice of X we have Lxoj = 0. Hence the equality (3) gives
to([X,Zw ]) = 0. (4)
In the formula
( LyO)(Z) = Y(O(Z)) -O([Y,Z]) (5)
the first term Y(oo(Z)) = 0, because to(Z) = const. The relations to = O o Kb* and (4) imply the following chain of equalities:
Ù([Y,Z]) = û(nb*[Y,Z]) = co([TTb*Y,nb*Z]) = û([X,Zw ]) = 0.
Therefore (5) implies that ( Lyuj)(Z) = 0 and (Lylo)(Z0) = 0. Thus, Lylo = 0.
3) Denote by ( W, FH) the foliation formed by the connected components of orbits of the action $w of H on W. Let (R, TH) be the foliation formed by the connected components of orbits of the Lie group H on R.
At any point u G R there is an neighborhood W foliated with respect to both foliations (R, T) and (R, TH) which meets each leaf of these foliations in at most one connected subset. We can suppose that the basic fibration -b: R ^ W is trivial in the neighborhood --1(V), where V := nb(W). Put U = n(W). Let r: U ^ U/(Flu) and s: V ^ V/(TH|v) be the quotient maps. We can identify U/(Flu) and V/(TH|v) with the manifold V such that the diagram
W —^ V
U —— V,
(6)
where the restrictions of - and -b onto W are denoted by the same letters, is commutative. Without loss of generality, we can assume that M| u is an Ehresmann connection for the submersion r and M|w is an Ehresmann connection for the submersion -b.
By the choice of X, for any A G h we have the equality LxAW = 0, i. e., [^4W, X] =0. Since the fundamental vector fields span the tangent spaces to the leaves of the foliation ( W,FH), it is not difficult to check that X is a foliated vector field for this foliation. Hence the vector field Xy := s*X|y is well defined. There is a unique vector field Yu G Xm(U) such that r*Yu = Xy. In other words, Yu is the M-horizontal lift of Xy. The commutative diagram (6) implies the relation -b*Yw = Yu, hence Y is a foliated vector field with respect to the foliation (R, TH). Therefore,
[ A*,Y] G X^h(R). (7)
According to Theorem 1, we have the equalities uj(A*) = oj(AW) = A, hence A* is a vector field foliated with respect to (R, T). So we have the following chain of equalities
-b*[A*,Y ] = [-b*A*,-b*Y ] = [AW ,X ] = 0,
hence,
[ A*,Y] G X^(R). (8)
The relations (7) and (8) imply the equality [ A* ,Y] = 0, VA G h.
Thus, we proved the inclusion Af?( W) C ima = imv. Therefore imv is an open-closed Lie subgroup of the Lie group AH( W). □
Theorem 5. Let (M,F) be a complete foliation with a transverse rigid geometry ( N, £), where £ = ( P( N,H),w). Suppose that the structure Lie algebra (M,F) is zero. Then:
(i) the full basic automorphism group Ab ( M,F) is realized as an open-closed subgroup of the Lie group AH( W) and admits a Lie group structure with the following estimate of its dimension:
dim Ab (M,F) < dimP; (9)
(ii) if either there exists an isolated closed leaf L or the set of closed leaves of the foliation ( M, F) is countable, then
dim Ab (M,F) < dimH; (10)
(iii) there exists a unique topology and a unique smooth structure on the full group Ab ( M,F) of basic automorphisms of the foliation (M,F), making Ab (M,F) into a Lie group. This topology coincides with the compact-open topology, when Ab ( M,F) is realized as a subgroup of the group AH ( W).
s
Proof. (i) Applying Propositions 9 and 11, we get that the map / := v o X-1: Ab (M,F) ^ AH (W) is a group isomorphism of the full group of basic automorphisms Ab( M, F) onto an open-closed subgroup im/ of the Lie group AH(W). We identity Ab ( M,F) with im/ and consider Ab (M,F) as an open-closed subgroup of the Lie group AH( W). Hence Ab( M,F) admits a Lie group structure, and the following estimates of dimensions hold:
dim Ab (M,F) < dim AH(W) < dim A(W,u) < dim W = dimP = m.
(ii) Suppose that there exists a closed leaf L of the foliation (M,F). Then -b(--1 (L)) is a closed orbit of the action $w of the Lie group H on W. Let us to fix an arbitrary point v in this orbit. Let Lw = Lw (v) be a leaf of the foliation (W, Fw). Then the leaf Lw is a closed subset in W. It is known [1] that the smooth structure of the Lie group A'H( W) coincides with the smooth structure induced by the bijection of the identity component AH ( W) of the Lie group AH ( W) onto the closed submanifold A,H ( W )-v of W, where AH (W )-v is the orbit of the point v. Any g G AH (W) maps each closed orbit of the Lie group H onto some closed orbit of H. Since AH(W) ■ v is connected and either the orbit v-H is isolated or if the set of closed orbits of H is countable, so AH( W) ■ v C Lw. Therefore dim Ab (M,F) < dim AH (W) -v < dim FH = dim H.
(iii) Applying the statement (i) proved above and Proposition 10 to the group Ab( M,F) we get the statement (iii). □
Remark 6. Theorem 5 does not exclude the triviality of the full group Ab (M, F).
Remark 7. The main result of the work [3] by Belko is the theorem asserting that if there exists a closed leaf of a foliation ( M, F) with complete transversally projectable affine connection, then the group Ab (M, F) is a Lie group. This statement is not correct. It's proof essentially uses the fact that existence of a closed leaf of this foliation implies that the lifted foliation is simple. It is not true, in general. Let us consider a foliation ( M,F) from Example 3 (in Section 10), when r = 1/-, as affine foliation. It has a compact leaf, but g0(M, F) = R1 = 0, hence the lifted foliation is not simple. Thus the foliation ( M, F) is a Lie foliation with non-zero structure Lie algebra g0( M,F), and the group Ab(M,F) is not a Lie group.
Discrete holonomy groups of leaves. Let ( M, F) be a complete foliation with TRG. Let -: R ^ M be the projection of the foliated bundle over (M, F).
Def 13. We say that the holonomy group of a leaf L 3 x of the foliation (M,F) is discrete if there exists a point u G --1(x) such that the group H(C) := {a G H | Ra(C) = C, C = C(u) G T} is a discrete subgroup of the Lie group H.
Let u' G --1(x) and u' =G C = C(u). In this case the subgroup H(C') is conjugate to the subgroup H(C) in the Lie group H. Hence H(C) is a discrete subgroup of H iff H(C') is a discrete subgroup of H. Thus, by Theorem 4 the notion of discrete holonomy group of leaf L is well defined.
Recall that a leaf L of a foliation (M, F) is said to be proper if L is an embedded submanifold in M. A foliation (M, F) is called proper if each its leaf is proper.
Proposition 12. Let ( M,F) be a complete foliation with TRG. If there exists proper leaf L with discrete holonomy group then the structure Lie algebra g0(M,F) is zero.
Proof. Let L be a proper leaf with discrete holonomy group. Let x G L, u G —-1(x), and C = C( u). Since L is proper, there exists a foliated neighborhood U of the point x such that L meets U in a connected subset L n U. Then there is a neighborhood U of u foliated with respect to (R, T) such that an embedded submanifold (L) of R meets U in a connected subset. Since the subgroup H(C) of H is discrete, Cn--1(x) = u H(C) is a discrete subset of the fiber --1(x). Therefore, there exists a neighborhood V CU of u foliated with respect to the foliation (R, T) such that CnV
is connected. By [17, Theorem 4.11], it follows that the leaf C is proper. Thus the complete e-foliation (R, T) has a proper leaf C. It is known [10] that such a foliation is formed by the fibers of a locally trivial fibration. Hence all leaves of the lifted foliation (R, T) are closed and the structure Lie algebra g0( M,F) is zero. □
Theorem 6. Let ( M, F) be a complete foliation with transverse rigid geometry ( N, £), where £ = ( P( N, H),o). If at least one of the following conditions holds:
(i) there exists a proper leaf L with discrete holonomy group;
(ii) there is a closed leaf L with discrete holonomy group;
(iii) there exists a proper leaf L with finite holonomy group;
(iv) there is a closed leaf L with finite holonomy group,
then the basic automorphism group Ab ( M,F) admits a Lie group structure of dimension at most dim P, and this structure is unique.
Proof. Remark that any closed leaf of a foliation is proper and each finite holonomy group is a discrete one. Hence we have implications (iv) ^ (iii) ^ (ii) ^ (i). According to Proposition 12 the existence of a proper leaf L with discrete holonomy group guarantees the equality g0( M,F) = 0. Thus, applying Theorem 5 we get the required assertion. □
It is well known that any foliation has leaves without holonomy. Therefore, the following statement is a consequence of the assertion (iii) of Theorem 6.
Corollary 2. For any proper complete foliation ( M, F) with TRG the basic automorphism group Ab ( M,F) admits a unique Lie group structure.
10. Foliations covered by fibrations
( G, X)-foliations. It is said that a group of diffeomorphisms of a manifold X acts quasi-analytically, if existence of an element g G G and an open subset U in X such that gh; = id u implies g = id x.
Let G be a Lie group of diffeomorphisms of a manifold X, which acts quasi-analytically on X. Recall that (M,F) is a (G,X)-foliation if (M,F) is defined by an ry-cocycle {Ui, fi, {7ij}}, where ji: Ui ^ Vi is a submersion onto an open subset of X, for each 7^ there is g G G such that 7^ = g^.(Uinu.) and {Vi} is a covering of X. Uniqueness of such a g G G is a consequence of quasi-analyticity of the action of G.
Def 14. Let (X, £) be a rigid geometry, where £ = ( P( H,X),o), let G be an automorphism group of (X,£). A ( G,X)-foliation (M,F) is called a (G,X)-foliation with transverse rigid structure.
Foliations covered by fibrations. Let f: M ^ M be the universal covering map.
Def 15. We say that a foliation ( M,F) is covered by a fibration if the induced foliation F := f*F on M is formed by the leaves of a submersion r: M ^ B onto a ^-dimensional manifold B, where q is codimension of the foliation (M,F).
Proposition 13. Let ( M,F) be a foliation with TRG, which admits an Ehresmann
connection M and is covered by a fibration r: M ^ B, where f:M ^ M is the universal covering map. Then:
(i) B is simply connected, the submersion r: M ^ B is a locally trivial fibration;
(ii) the manifold B admits a rigid geometry Q locally isomorphic to (N, £);
(iii) there is a group epimorphism a: k1 (M) ^ ^ onto some subgroup ^ of automorphisms of the rigid geometry ( B, Q: the group ^ is called the global holonomy group of the foliation covered by a fibration; the foliation ( M,F) is a (^, B)-foliation with TRG;
(iv) for any x G M, the holonomy group Y(L,x) of the leaf L = L(x) is isomorphic to the isotropy subgroup ^ y, where y G r(f-1(x)).
Proof. (i) By Proposition 2, the distribution M is an Ehresmann connection for foliation ( M,F). It is not difficult to check that the distribution M = f *M is an Ehresmann connection for the foliation (MM, F), hence M is an Ehresmann connection for the submersion r: M ^ B. It is well known that a submersion admitting an Ehresmann connection is a locally trivial fibration. Thus, the foliation ( M, F) is formed by the fibres of the locally trivial fibration r: M ^ B. Hence r: M ^ B is a fibration with the covering homotopy property. Applying the exact homotopy sequence and the fact that the leaves of r are arcwise connected and the manifold Mi is simply connected, we see that the basic manifold B is also simply connected.
(ii) Let fi : Ui ^ Vi be a submersion from the (N, £)-cocycle г/, defining (M,F). Without loss of generality, we can assume that Ui is a regularly covered neighborhood, i. e., f-1( Ui) = is a disjunct sum of neighborhoods Wa such that flwa : ^ Ui is a diffeomorphism. Therefore there exists a diffeomorphism 7a: Va = r(Wa) ^ Vi satisfying the equality 7a or = fi of on Wa. The diffeomorphism 7a induces a rigid geometry = ( Pa(Va, H),ua), where Pa := 7**Pa on Va, such that 7a : Va ^ Vi is an isomorphism of (Va, Qa) and ( Vi, ^). By a straightforward verification, one can show that there exists a unique rigid structure Q = ( P>(H, B),fi) on B such that = (a.
(iii) Let us consider the fundamental group k1(M,x), x G M, as the group G of deck transformations of the universal covering map /: M ^ M. Since each g G G is an isomorphism of the induced foliation ( M, F) and the basic manifold B of the fibration r: M ^ B can be considered as the leaf space MI/F, g defines a map ф: B ^ B satisfying the relation год = ф o r. Hence ф is a diffeomorphism of B. Moreover, from the definition of the rigid geometry ( on B it follows that ф G A(B, Q. Denote by Ф the group of all such ф. Then there is a group epimorphism %: ■k1(M,x) ^ Ф: g ^ ф, where o = ф o .
(iv) As f: M ^ M is the covering map, we can consider (M,F) as a (^,B)-foliation. Then the holonomy pseudogroup H of ( M, F) is determined by the group Ф. Since Ф acts quasi-analytically on B, for each у G B the group Hy, which consists of germs at of transformations from the isotropy subpseudogroup of the holonomy pseudogroup H, is isomorphic to the isotropy subgroup Ф y. According to Theorem 4, the holonomy group r(L, x) is isomorphic to Hy, where у G r(f-1 (x)), and hence the group r(L, x) is isomorphic to Фу. □
According to Proposition 2 a complete foliation ( M, F) with TRG admits an Ehresmann connection, hence the following assertion is true.
Corollary 3. If (M,F) is a complete foliation with TRG, then the statements of Proposition 13 are valid for (M,F).
Basic automorphism groups of foliations with TRG covered by fibra-tions. In the following theorem we give and apply another interpretation of the structure Lie algebra of a foliation ( M, F) with TRG covered by a fibration.
Theorem 7. Let ( M,F) be a complete foliation with TRG covered by a fibration r: M ^ B, where f: M ^ M is the universal covering map. Let Ф be the global holonomy group of ( M,F) considered as a subgroup of the Lie group A(B, Q of all automorphisms of the rigid geometry ( B, Q, which was introduced in Proposition 13. Then:
(i) the structure Lie algebra go(M, F) is isomorphic to the Lie algebra of the Lie group Ф, where Ф is the closure of Ф in the full Lie group of automorphisms A(B, Q which is a Lie group;
(ii) the equality g0( M,F) = 0 is equivalent to the condition that Ф is a discrete subgroup of the Lie group A(B, Q;
(iii) if Ф is a discrete subgroup of the Lie group A(B, Q, then the full group of basic automorphisms Ав ( M, F) admits a Lie group structure, and this structure is unique.
Proof. As above, let ■к: P ^ M be the projection of the foliated bundle over ( M,F) and f: M ^ M be the universal covering map. Put TP := f *P = {(y,u) G
M xK\ f(y) = i(u)}, 7f: K ^ M: (y, u) ^ y, pp: K ^ K: (y ,u) ^ u. A right action of H on K is defined by the equality (y,u) ■a = (y,u ■ a), Va G H. It is easy to see that a principal H-bundle -ff: K ^ M equipped with a foliation (K, T), where T = p*T, is the foliated bundle for the foliation (M, F). Since (MI, F) is a simple foliation defined by submersion r: M ^ B, so (K, T) is also simple foliation defined by the projection of the basic fibration I: K ^ W. In general, when the Lie group H is not connected, the manifold K is not simply connected. Remark that the lifted e-foliation (K, T) is covered by fibration — : K ^ WW. The fundamental group G = i1(M,x) acts on K by the formula g(y,u) := (g(y),u), V(y,u) G K, Vg G G. Hence g o Ra = Ra o g, Va G H, Vg G G. Moreover, each g is an automorphism of the foliation (K, T) in the category Fol. Therefore G induces a group 11 C Dif f(W). Let s: W ^ B be a map defined by the equality s o Ib = I o r. Analogously to proof of Proposition 13, a rigid structure £ = ( W(H,B), 9) with the projection s: W ^ B is defined, and ( B, Q is a rigid geometry, 1 c ^(C). Furthermore the group isomorphism A(Q ^ A(B, 0 maps l onto 1.
Consider any leaf C = C(u), u G K, of (K, T). Let z G p-1 (u) and d = ifb(z).
Since 11 C A(0 C A(W, 0), so l ■ d = (c№) ■ d, where 11 ■ d is the closure of the orbit 1 ■ d in W, and c№ is a closure of 1 in the Lie group ^.(C). Hence the closure C of C in K satisfies the equality C = p(I-1((cZl1) ■ d)). Denote by (cZlI)e the identity component of the Lie group c 11, then ( ell) e ■d and L := I-1 ((cl1)e ■d) are connected smooth manifolds, with : L ^ C is a regular covering map. The induced foliation ( p\L)*(T\-¡=) is simple and is defined by a submersion Ib\L: L ^ (cl1)e ■ d = (cZ1I)e. It is known [11] that this implies that the structure Lie algebra of the Lie foliation (C, T\-£) with dense leaves is isomorphic to the Lie algebra of the Lie group ( cl 1)e. Since 1 is the projection of 1 with respect to s: WW ^ B, so effectiveness of £ implies that the Lie groups 1 and cl1 are isomorphic.
The statement (ii) is a direct consequence of the statement (i). Therefore the assertion (iii) follows from Theorem 5. □
Basic automorphisms of foliations with integrable Ehresmann connections.
Proposition 14. Let ( M,F) be an M-complete foliation with TRG. Suppose that the distribution M is integrable and, therefore, defines a foliation (M,Fl), where TFl = M. Then:
(i) the universal covering manifold M can be identified with the product L xB of some manifolds L and B, and (M, F) is covered by the trivial fibration r:L x B ^ B, where r is the canonical projection onto the second factor;
(ii) if the global holonomy group 1 is a discrete subgroup of the Lie group A(B, Q of all automorphisms of the induced rigid geometry ( B, Q, then the full group of basic automorphisms Ab ( M,F) admits a unique Lie group structure.
Proof. By assumption, M is endowed with two transverse foliations (F, Fl) of dimensions p and q, respectively, where p + q = dim M. According to Proposition 2, the distribution M = TFl is an Ehresmann connection for the foliation (M, F).
Let f: M ^ M be the universal covering map. Let F := f*F, 1 := f*Fl be the induced foliations on M and MM := Tft. Remark that MM = f*M is an integrable Ehresmann connection for the foliation ( M,I). In the terminology of Section 3, the simply connected manifold M is endowed with two transverse foliations (F, I11) such that for any pair of curves ( a, h) with a common initial point a(0) = h(0), where a is a horizontal curve and h is a vertical curve, there exists a vertical-horizontal homotopy H with the base ( a,h). In other words, the conditions of the famous Kashiwabara's theorem about the decomposition of manifolds [18] (rediscovered by Blumenthal and
Hebda [12]) are satisfied. According to this theorem there exists a diffeomorphism $ of M onto a product of manifolds L xB which is an isomorphism in the category Fol of two pairs of foliations: first, of ( M, F) and (L x B, F1), where F1 = {L x {z} | z G B}, second, of ( M, Fl) and (L x B,F2), where F2 = {{y} x B ^ G L}. We identify M with L x B by means of while the foliation (M, F) is identified with the trivial foliation ( L x B,F1). Therefore the foliation (M,F) is covered by the trivial fibration r: L xB ^ B. Thus, (M, F) satisfies Theorem 7. □
11. Examples
Foliations obtained by suspension of a homomorphism. Let p: -1 (B, bo) ^
Diff(T) be a homomorphism of the fundamental group of a manifold B 3 b0 into the group of diffeomorphisms of a ^-dimensional manifold T, and let p: B ^ B be the universal covering mapping. Then we have a right action of the group n := -1(B, b0) on B by deck transformations. The equality
(x, t) ■g := (x ■ g,p(g-1)(t)), V(x, t) G B xT, Vg G n,
defines a free right properly discontinuous smooth action of the group n on the product of manifolds B x T; therefore the quotient manifold M := B xnT is defined. Let k: B xT ^ M be the natural projection. Then F := {k(B x {t}) | t G T} is a foliation of codimension on M; in this case, it is said that the foliation ( M, F) is obtained by suspension of the homomorphism p. For this foliation we will use the notation ( M,F) := Sus(T, B,p) suggested in [19]. The image ^ := imp is the global holonomy group of ( M, F).
Transversally similar and transversally homothetic foliations. Let G be the similarity group of the Euclidean space E9, q ^ 1, and R+ be the multiplicative group of positive real numbers. Then G = CO(q) XR9 is the semidirect product of the conformal group CO(q) = R+ ■ O(q) and the group R9. Let H = CO(q) and p: G ^ G/H = E9 be the canonical principal H-bundle. Let g be the Lie algebra of the Lie group G, and u be the Maurer-Cartan g-valued 1-form on G. Then £ = (G(E9,H),u) is an effective rigid geometry. Foliations with this transverse geometry (E 9, £) are called transversally similarity foliations [7].
Denote by E the neutral element of the group O(q). If G = (R+-E)XRq, H = R+-E, and u is the Maurer-Cartan g-valued 1-form on the Lie group G, then foliations with the transverse effective rigid geometry (E9, £), where £ = ( G(E 9, R+ ■ E),u), are called transversally homothetic foliations [7].
Example 1. Let B be a smooth p-dimensional manifold whose fundamental group -1 (B, b) contains an element a of infinite order. For an arbitrary natural number q ^ 1, denote by E9 a ^-dimensional Euclidean space. Define a homomorphism p: n := -1(B, b) ^ Diff(E9) by setting p(a) = where ^ is the homothetic transformation of the Euclidean space E9 with the coefficient A = 1, i. e. ^(x) = Xx, Vx G E9, and p(/) = idg« for any element / G -1 (B, b) such that / = ak with some integer k. Then ( M, F) = Sus(E 9, B, p) is a proper transversally similar foliation with a unique closed leaf diffeomorphic to B.
According to Corollary 2, the full basic automorphism group Ab (M,F) of this foliation ( M,F) admits a Lie group structure. Let us compute the group Ab (M,F) and show that this fact is indeed true.
The group n0 := ker p acts on B x E9 properly discontinuously, hence the quotient manifold B xno E9 = B0 x E9, where B0 := B/n0, is defined. The quotient group
:= n/n0 = Z acts from the right on the product of manifolds B0 x E9 such that M = B0 x^0 E9 and the quotient map k : M0 := B0 x E9 ^ M is a regular covering map with the deck transformation group The foliation (M0,F0), where F0 := k*F,
is formed by the fibres of the projection pr2: M0 = B0 x E9 ^ E9 onto the second factor.
The group Д(£) is equal to the group of left translations of the Lie group G = CO(q) X R, hence we can identify A(Eq, £) = A(0 with G. For any h e G the transformation h' = (id в0,h) of B0 x E9 belongs to A(M0,F0). Therefore, the map a: A(M0,F0) ^ G: h' ^ h, where h о pr2 = pr2 о h', is a group epimorphism with kera = Al(M0, F0). Let us emphasize that f e A(M0,F0) lies over an automorphism f e A(M, F) if and only if it satisfies the relation f о Ф0 = Ф0 о f. Remark that а(Ф0) = Ф С Л(Е9, £) = G is the global holonomy group of the foliation (M,F). Let N(Ф) be the normalizer of Ф in the Lie group G. It is not difficult to check that the map
P: Ав (M,F) ^ N(Ф)/Ф: f • Al (M, F) ^ a(f) • Ф,
where f e A(M0,F0) lies over f with respect to the map к, is a group isomorphism, hence Ав (M, F) ^ N(Ф)/Ф.
In our case Ф = (ф) and N(Ф) = R+ • O(q), therefore Ав(M,F) ^ U(1) x O(q), where U(1) = (R+ • E)/Ф is the compact 1-dimensional abelian group. If q = 1, then O(q) = Z2 and Ав (M, F) ^ U(1) x Z2.
Example 2. Consider the foliation ( M, F) constructed in Example 1 as a transver-sally homothetic foliation, i. e., with a different transverse rigid geometry. In this case the Lie group Ав(M,F) is isomorphic to the quotient Lie group N(Ф)/Ф, where N(Ф) is the normalizer of Ф in the Lie group (R+ • E) X R9. Since N(Ф) = R+ • E, so Ав( M, F) = U(1).
Remark 8. In both examples 1 and 2 the foliation ( M, F) has a closed leaf and, in Theorem 3, the equality is achieved in the estimate (ii) of the dimension of Ав ( M, F).
Example 3. Let ф be the rotation of the plane E2 about the point 0 e E2 through the angle 5 = 2ттг. Consider an Euclidean metric g on E2. Denote by Iso(E2, g) the full isometry group of (E2, g). Let p: ki(S 1, b) = Z ^ Iso(E2, g) be defined by the equality p(1) := ф, 1 e Z. Then we have a suspended Riemannian foliation (M,F) := Sus(E2,S1,p). This foliation has a unique closed (compact) leaf.
There exists a group isomorphism between Ав ( M,F) and the quotient group N(Ф)/Ф, where Ф = (ф) and N(Ф) is the normalizer of Ф in the Lie group Iso(E2,g) identified with O(2) X R2. Since N(Ф) = O(2), so Ав(M,F) = 0(2)/Ф. Hence Ав ( M,F) admits a Lie group structure if and only if Ф is a closed subgroup of O(2) or, equivalent, when 5 = 2ттг for some rational number r.
If 5 = 2ттг, where r is a non-zero rational number, then Ав ( M,F) = O(2).
References
1. Кобаяси Ш. Группы преобразований в дифференциальной геометрии. — М.: Наука, 1986. — 224 с.
2. Leslie J. A Remark on the Group of Automorphisms of a Foliation Having a Dense Leaf // J. Diff. Geom. — 1972. — Vol. 7. — Pp. 597-601.
3. Белько И. В. Аффинные преобразования трансверсальной проектируемой связности на многообразии со слоением // Мат. сборник. — 1982. — Т. 117, № 2. — С. 181-195.
4. Hector G., Macias-Virgos E. Diffeological Groups // Reseach and Exposition in Math. — 2002. — Vol. 25. — Pp. 247-260.
5. D'Ambra G., Gromov M. Lectures on Transformation Groups: Geometry and Dynamics, Surveys in Differential Geometry (Cambridge, Mass., 1990). — Bethlehem, Penn.: Lehigh University, 1991. — Pp. 19-111.
6. Gromov M. Rigid transformations groups // Geometrie Differentielle (Paris, 1986). Travaux en Cours. — 1988. — Vol. 33. — Pp. 65-139.
7. Жукова Н. И. Минимальные множества картановых слоений // Труды матем. института им. В.А. Стеклова. — 2007. — Т. 256. — С. 115-147.
8. Blumenthal R. A. Cartan Connections in Folated Bundles // Michigan Math. J. — 1984. — Vol. 31. — Pp. 55-63.
9. Кобаяси Ш., Номидзу К. Основы дифференциальной геометрии. — М.: Наука, 1981. — Т. 1, 344 с.
10. Conlon L. Transversally Parallelizable Foliations of Codimension 2 // Trans. Amer. Math. Soc. — 1974. — Vol. 194. — Pp. 79-102.
11. Molino P. Riemannian Foliations. Progress in Math. — Birkhauser Boston, 1988. — 339 p.
12. Blumenthal R. A., Hebda J. J. Ehresmann Connections for Foliations // Indiana Univ. Math. J. — 1984. — Vol. 33, No 4. — Pp. 597-611.
13. Wolak R. A. Foliated and Associated Geometric Structures on Foliated Manifolds // Ann. Fac. Sci. Toulouse Math. — 1989. — Vol. 10, No 3. — Pp. 337-360.
14. Wolak R. A. Geometric Structures on Foliated Manifolds // Publ. del Dep. de Geometria y Topologia, Universidad de Santiago de Compostela. — 1989. — Vol. 76.
15. Жукова Н. И. Свойства графиков эресмановых слоений // Вестник ННГУ. Сер. Математика. — 2004. — Вып. 1. — С. 73-87.
16. Багаев А. В., Жукова H. И. Группы изометрий римановых орбифолдов // Сиб. Мат. Журнал. — 2007. — Т. 48, № 4. — С. 723-741.
17. Тамура И. Топология слоений. — М.: Мир, 1979. — 317 с.
18. Kashiwabara S. The Decomposition of Differential Manifolds and its Applications // Tohoku Math. J. — 1959. — Vol. 11. — Pp. 43-53.
19. Chubarov G. V., Zhukova N. I. Aspects of the Qualitative Theory of Suspended Foliations // J. of Difference Equations and Applications. — 2003. — Vol. 9. — Pp. 393-405.
20. Kamber F., Tondeur P. Foliated Bundles and Characteristic Classes // Lecture Notes in Math. — Springer, 1975. — Vol. 494.
УДК 514.763.23
Полные слоения с трансверсальными жесткими геометриями и их базовые автоморфизмы
Н. И. Жукова
Кафедра математики и механики Нижегородский государственный университет им. Н.И. Лобачевского пр. Гагарина, д. 23, корп. 6, г. Нижний Новгород, Россия, 603950
Введено понятие жестких геометрий. Жесткие геометрии включают картановы геометрии, а также жесткие геометрические структуры в смысле Громова. Исследуются слоения (М, Р) с трансверсальными жесткими геометриями. Найден инвариант до (М, Р) слоения (М,Р), представляющий собой алгебру Ли. Доказано, что при д0(М,Р) = 0 группа базовых автоморфизмов слоения (М, Р) допускает структуру группы Ли, причем эта структура единственна. Получены оценки размерностей этих групп в зависимости от трансверсальных геометрий. Построены примеры вычисления групп базовых автоморфизмов слоений.
