Local and Global Stability of Compact Leaves and Foliations
The equivalence of the local stability of a compact foliation to the completeness and the quasi analyticity of its pseudogroup is proved. It is also proved that a compact foliation is locally stable if and only if it has the Ehresmann connection and the quasianalytic holonomy pseudogroup. Applicatio...
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irk-123456789-1067622016-10-05T03:02:16Z Local and Global Stability of Compact Leaves and Foliations Zhukova, N.I. The equivalence of the local stability of a compact foliation to the completeness and the quasi analyticity of its pseudogroup is proved. It is also proved that a compact foliation is locally stable if and only if it has the Ehresmann connection and the quasianalytic holonomy pseudogroup. Applications of these criterions are considered. In particular, the local stability of the complete foliations with transverse rigid geometric structures including the Cartan foliations is shown. Without assumption of the existence of an Ehresmann connection, the theorems on the stability of the compact leaves of conformal foliations are proved. Our results agree with the results of other authors. Доказана эквивалентность локальной устойчивости произвольного компактного слоения полноте и квазианалитичности его псевдогруппы голономии. Мы доказали, что компактное слоение локально устойчиво тогда и только тогда, когда оно допускает связность Эресмана и имеет квазианалитическую псевдогруппу голономии. В качестве приложения показана локальная устойчивость полных компактных слоений с жесткой трансверсальной структурой, включающих в себя полные картановы слоения. Без предположения о существовании связности Эресмана доказаны теоремы о стабильности компактных слоев конформных слоений. Установлена связь с результатами других авторов. 2013 Article Local and Global Stability of Compact Leaves and Foliations / N.I. Zhukova // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 3. — С. 400-420. — Бібліогр.: 37 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106762 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The equivalence of the local stability of a compact foliation to the completeness and the quasi analyticity of its pseudogroup is proved. It is also proved that a compact foliation is locally stable if and only if it has the Ehresmann connection and the quasianalytic holonomy pseudogroup. Applications of these criterions are considered. In particular, the local stability of the complete foliations with transverse rigid geometric structures including the Cartan foliations is shown. Without assumption of the existence of an Ehresmann connection, the theorems on the stability of the compact leaves of conformal foliations are proved. Our results agree with the results of other authors. |
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Article |
| author |
Zhukova, N.I. |
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Zhukova, N.I. Local and Global Stability of Compact Leaves and Foliations Журнал математической физики, анализа, геометрии |
| author_facet |
Zhukova, N.I. |
| author_sort |
Zhukova, N.I. |
| title |
Local and Global Stability of Compact Leaves and Foliations |
| title_short |
Local and Global Stability of Compact Leaves and Foliations |
| title_full |
Local and Global Stability of Compact Leaves and Foliations |
| title_fullStr |
Local and Global Stability of Compact Leaves and Foliations |
| title_full_unstemmed |
Local and Global Stability of Compact Leaves and Foliations |
| title_sort |
local and global stability of compact leaves and foliations |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/106762 |
| citation_txt |
Local and Global Stability of Compact Leaves and Foliations / N.I. Zhukova // Журнал математической физики, анализа, геометрии. — 2013. — Т. 9, № 3. — С. 400-420. — Бібліогр.: 37 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT zhukovani localandglobalstabilityofcompactleavesandfoliations |
| first_indexed |
2025-07-07T18:57:37Z |
| last_indexed |
2025-07-07T18:57:37Z |
| _version_ |
1837015664605265920 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2013, vol. 9, No. 3, pp. 400–420
Local and Global Stability of Compact Leaves and
Foliations
N.I. Zhukova
Department of Mechanics and Mathematics Nizhny Novgorod State University
23 Gagarin Ave., korp. 6, Nizhny Novgorod, 603095, Russia
E-mail: n.i.zhukova@rambler.ru
Received January 17, 2012, revised March 18, 2013
The equivalence of the local stability of a compact foliation to the com-
pleteness and the quasi analyticity of its pseudogroup is proved. It is also
proved that a compact foliation is locally stable if and only if it has the
Ehresmann connection and the quasianalytic holonomy pseudogroup. Ap-
plications of these criterions are considered. In particular, the local stability
of the complete foliations with transverse rigid geometric structures includ-
ing the Cartan foliations is shown. Without assumption of the existence
of an Ehresmann connection, the theorems on the stability of the compact
leaves of conformal foliations are proved. Our results agree with the results
of other authors.
Key words: foliation, compact foliation, Ehresmann connection for a
foliation, holonomy pseudogroup, local stability of leaves.
Mathematics Subject Classification 2010: 57R30, 53D22.
Introduction
The notion of the stability of leaves of foliations was introduced by Ehresmann
and Reeb, the founders of the theory of foliations.
Remind that a subset of the foliated manifold is called saturated if it can be
represented as a union of some leaves of the foliation.
Definition 1. A leaf L of a foliation (M, F) of codimension q is said to be
locally stable in the sense of Ehresmann and Reeb if there exists a family of its
saturated neighbourhoods {Wk|k ∈ N} satisfying the following conditions:
This work was supported by the Federal Target Program ”‘Scientific and Scientific-
Pedagogical Personnel”’, Project No. 14.B37.21.0361, and the Russian Federation Ministry of
Education and Science, Project No. 1.1907.2011.
c© N.I. Zhukova, 2013
Local and Global Stability of Compact Leaves and Foliations
1) there exists a submersion f1 : W1 → L such that for every k ∈ N the triplet
(Wk, fk, L), where fk = f1|Wk
is a locally trivial fibration with a q-dimensional
disk Dq as the standard fiber such that its fibers are transversal to the leaves of
the foliation (Wk,F|Wk
);
2) for any point x ∈ L, the set {Wk
⋂
f−1
1 (x) | k ∈ N} forms a base of the
topology of the fiber f−1
1 (x) at x.
According to well-known Reeb’s theorem [23]), any compact leaf of the folia-
tion with finite holonomy group is locally stable.
Remind that a foliation is said to be compact if every its leaf is compact.
For a compact foliation (M, F), the local stability of a leaf L by Definition 1 is
equivalent to the existence for L a basis of saturated neighborhoods of L in M .
G. Reeb [20] proved that the leaf space of every smooth compact foliation
of codimension one is Hausdorff. In [11], D. Epstein proved that any leaf of a
compact foliation (M, F) has finite holonomy group iff the leaf space M/F is
Hausdorff.
In [16], K. Millett made the following conjecture:
Every leaf of a compact foliation on a compact manifold has finite holonomy
group.
Due to the Reeb theorem mentioned above, the Millett conjecture is called
the problem on the local stability of compact foliations.
R. Edwards, K. Millett and D. Sullivan [8] and independently E. Vogt [24]
proved that in the case of codimension q = 2 the Millett conjecture is valid. For
the one-dimensional compact foliations on the closed 3-manifolds it was proved
earlier by D. Epstein [10].
If the foliated manifold M is not compact, then the analog of the Millett
conjecture is not true in general for the compact foliations (M, F) of codimension
q = 2. Now it is known that, generally speaking, for q = 3 the Millett conjecture
is not valid. The first counterexample was constructed by Sullivan. He found a
smooth unstable flow on a closed 5-manifold [22], in which each orbit is periodic.
This example shows that some additional hypothesis of a global character on M
is required in general.
The examples of the unstable compact foliations were also constructed by D.
Epstein and E. Vogt [12], Thurston [22] and others.
After the Sullivan counterexample, there appeared a number of works con-
taining criterions and sufficient conditions for the Millett conjecture to be true.
According to Epstein’s assertion mentioned above, the validity of the Millett
conjecture is equivalent to the property of the leaf space of the foliation to be
Hausdorff. Decesaro and Nagano [7] stated that the Hausdorff separation prop-
erty for the topology of the leaf space is equivalent to the boundedness of the
volume of leaves function near every given leaf with respect to any Riemannian
metric on M .
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 401
N.I. Zhukova
H. Rummer [21] proved that the local stability of a compact foliation (M, F)
on a compact manifold M is equivalent to the existence of a Riemannian metric
on M such that, with this metric, each leaf is a minimal Riemannian submanifold.
A survey of the results on the Millett conjecture can be found in [8] and [14].
Remark that according to the papers [11], [30] and [17], a compact foliation
(M, F) is locally stable iff there exists a complete bundle-like metric on M with
respect to (M, F), that is equivalent to the existence of a natural structure of a
smooth q-dimensional orbifold on the leaf space M/F.
The compact foliations (M, F) with a Hausdorff separation property for the
leaf space M/F are referred to the Hausdorff foliations and are studied in [6]. As
indicated above, the Hausdorff foliations are locally stable.
A. Gogolev [15] and P.D. Carrasco [5] studied the partially hyperbolic diffeo-
morphisms with compact center foliation, where the local stability of this foliation
plays an important role.
Definition 2. A pseudogroup H of local diffeomorphisms of a manifold N is
said to be quasianalytic if the existence of an open connected subset V in N such
that h|V = idV for an element h ∈ H implies h = idD(h), where D(h) is the
connected domain of definition of h that contains V .
For instance, holonomy pseudogroups of G-foliations are quasianalytic. Let
us emphasize that the holonomy pseudogroup of every foliation with transverse
rigid geometry in the sense of [36] is quasianalytic.
Definition 3. A pseudogroup H of local diffeomorphisms of a manifold N is
called complete if for every pair of the points x and x′ on N there exist the open
neighbourhoods U and U ′ such that:
if y ∈ U and y′ = γ(y) ∈ U ′ for some γ ∈ H there exists a prolongation h ∈ H
of the local diffeomorphism γ to the entire neighbourhood U .
Here we prove the following new criterion of the stability of compact foliations.
Theorem 1. A compact foliation (M, F) of arbitrary codimension q is locally
stable if and only if the holonomy pseudogroup of this foliation is complete and
quasianalytic.
In Subsection 1.4. we remind the definition of (G,X)-foliations.
Corollary 1. Any compact (G,X)-foliation is locally stable.
The notion of the Ehresmann connection for the foliations was introduced
by Blumenthal and Hebda [1] as a natural generalization of the notion of the
Ehresmann connection for the submersions (the definition is contained in Sec. 2).
402 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Local and Global Stability of Compact Leaves and Foliations
In this work, we give a detailed proof and apply the following criterion of the
local stability of compact foliations formulated by us (without proving) in [32].
Theorem 2. For the compact foliation (M, F) of arbitrary codimension q to
be locally stable, it is necessary and sufficient that the following two conditions
should hold:
1) the holonomy pseudogroup of the foliation (M, F) is quasianalytic;
2) there exists an Ehresmann connection for (M, F).
Accentuate that Theorems 1 and 2 were proved by us without assumption of
the compactness of foliated manifolds.
The effectiveness of the second criterion is confirmed by the following state-
ment obtained as an application of Theorem 2.
Theorem 3. Each compact foliation on an n-manifold of an arbitrary codi-
mension q, 0 < q < n, belonging to at least one of the following classes:
1) complete foliations with transverse rigid geometry in the sense of [36];
2) transversally holomorphic foliations with Ehresmann connections;
3) transversally real analytic foliations admitting an Ehresmann connection;
4) G-foliations with Ehresmann connection,
is locally stable.
R e m a r k 1. The class of foliations with transverse rigid geometry introduced
in [36] contains the Cartan foliations with effective transverse Cartan geometries
as well as the foliations admitting transversely complete, transversely transitive
foliated system of differential equations with the unique solution property in the
sense of Wolak [26].
Observe that each holonomy group of a compact foliation (M, F) can be lin-
earized iff (M, F) is a locally stable foliation. Therefore, applying Theorem 2, we
obtain the following statement.
Corollary 2. Let (M, F) be a compact foliation with quasi analytic holonomy
pseudogroup. Then each holonomy group can be linearized iff (M, F) admits an
Ehresmann connection.
In particular, each holonomy group of every foliation satisfying Theorem 3
can be linearized.
The holonomy group Γ(L, x) of a leaf L of the foliation (M, F) at the point
x ∈ L, usually used in the foliation theory [23], consists of the germs of local
holonomy diffeomorphisms along the loops based at x of a transversal at the
point x ∈ L. We will call it the germinal holonomy group to distinguish from the
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 403
N.I. Zhukova
M-holonomy group HM(L, x) for the foliation (M, F) with Ehresmann connection
M (see Sec. 2).
As an application of Theorem 2, we obtain the following theorem ([31], The-
orem 7.4) on the global stability of a compact leaf with finite holonomy group
(and also with finite fundamental group).
Theorem 4. Let (M, F) be a foliation with quasianalytic holonomy pseu-
dogroup and Ehresmann connection. If there exists a compact leaf with finite
germinal holonomy group (or finite fundamental group), then each leaf of this fo-
liation is compact and has finite germinal holonomy group (or finite fundamental
group, correspondingly), and (M, F) is a locally stable foliation.
Theorem 4 is an analog of the famous Reeb theorem [23] on the global stability
of a compact leaf with finite fundamental group for the Cr foliations, r ≥ 2, of
codimension one on the compact manifolds.
Corollary 3. Let (M, F) be a foliation belonging to at least one of the classes
1)–4) in Theorem 3. If there exists a compact leaf with finite germinal holonomy
group, then every leaf has finite germinal holonomy group, and (M, F) is a locally
stable compact foliation.
In particular, if (M, F) is a complete conformal foliation of codimension q > 2,
then the main result of the Blumenthal paper [2] follows from Corollary 3. In the
case when M is compact, the statement analogous to Corollary 3 was proved by
Wolak for the complete G-foliations of finite type [27].
It is known (see, for example, [1]) that for a totally geodesic foliation (M, F) of
codimension q on a Riemannian manifold (M, g) with complete induced metric on
leaves the orthogonal q-dimensional distribution M is an Ehresmann connection.
Therefore the following assertion follows from Theorem 4.
Corollary 4. Let (M, F) be a totally geodesic foliation on the Riemannian
manifold (M, g). If the induced metric on the leaves is complete and the holon-
omy pseudogroup is quasianalytic, then the existence of a compact leaf with finite
germinal holonomy group implies the compactness of every leaf and the finiteness
of its germinal holonomy group, i.e., the compactness and local stability of this
foliation.
Let (M, F) be an arbitrary smooth foliation, where F = {Lα |α ∈ A}. Recall
the construction of the graph of the foliation given by Winkelnkemper in [25].
Take any two points x and y from a leaf Lα. Denote by A(x, y) the set of piecewise
smooth paths in the leaf Lα connecting x with y. Two paths h and g from A(x, y)
are said to be equivalent h ∼ g if the loop, which is equal to the product h ·g−1 of
404 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Local and Global Stability of Compact Leaves and Foliations
the paths h and g−1, defines the trivial element of the germinal holonomy group
Γ(Lα, x). The equivalent class containing the path h is denoted by < h >. The
set G(F) of triplets of the form (x,< h >, y), where x ∈ M , y ∈ L(x), h ∈ A(x, y),
is said to be the graph of the foliation (M, F), and the maps
p1 : G(F) → M : (x,< h >, y) 7→ x, p2 : G(F) → M : (x,< h >, y) 7→ y
are called the canonical projections. One can define a structure of the smooth
(n + p)-manifold on the graph G(F), where n is a dimension of M and p is a
dimension of the foliation (M, F). The topological space of G(F) is not Hausdorff
in general. The family
F = {L = p−1
1 (Lα) |α ∈ A}
forms the induced foliation on the graph G(F). Winkelnkemper [25] proved a
criterion of the property of the graph G(F) to be Hausdorff (see Sec. 4, Prop. 2).
Suppose that a foliation (M, F) admits an Ehresmann connection M. Repla-
cing the germinal holonomy group Γ(L, x) of each leaf L with the M-holonomy
group HM(L, x) in the definition of the graph of a foliation, we obtain the defi-
nition of the graph GM(F) = {(x, {h}, y)} of the foliation with Ehresmann con-
nection. We proved that GM(F) is equipped in a natural way with the structure
of the smooth manifold [31] (see also [33, 34]). We showed that the topological
space of GM(F) is always Hausdorff unlike the one of the graph G(F). The map
β : GM(F) → G(F) : (x, {h}, y) 7→ (x,< h >, y)
is a local diffeomorphism. Both graphs G(F) and GM(F) are equipped with a
groupoid structure, and β is a groupoid epimorphism.
A holonomy vanishing cycle for a foliation (M, F) [29] is a mapping c : S1 ×
[0, 1] → M such that for any t ∈ [0, 1] the loop ct = c|S1×{t} belongs to a leaf
of the foliation, and for every t > 0 the loop ct induces the trivial element of
the corresponding germinal holonomy group unlike c0, which induces a nontrivial
element of the germinal holonomy group of the leaf containing it.
The following theorem sums up our results on the local stability of compact
foliations along with the known results of other authors.
Theorem 5. For any compact foliation (M, F) of an arbitrary codimension q
on an n-dimensional manifold M , where 0 < q < n, the following conditions are
equivalent:
1) the foliation (M, F) is locally stable;
2) the holonomy pseudogroup of the foliation (M, F) is complete and quasi-
analytic;
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 405
N.I. Zhukova
3) the foliation (M, F) has an Ehresmann connection and a quasianalytic
holonomy pseudogroup;
4) the foliation (M, F) has an Ehresmann connection, and its graph G(F) is
Hausdorff;
5) the foliation (M, F), where F = {Lα |α ∈ A}, admits an Ehresmann con-
nection M such that the holonomy groups HM(Lα) and Γ(Lα), ∀α ∈ A, are
isomorphic in a natural way;
6) there exists an Ehresmann connection M for (M, F) such that the map
defined above, β : GM(F) → G(F), is a groupoid isomorphism;
7) all the fibres of the canonical projections pi : G(F) → M , i = 1, 2, are
compact;
8) the induced foliation (G(F),F) on the graph G(F) is compact;
9) the foliation (M, F) is Riemannian;
10) the leaf space M/F is Hausdorff;
11) the leaf space M/F of the foliation (M, F) carries a structure of a smooth
orbifold such that the quotient map is a morphism in the category of orbifolds;
12) there exists a complete bundle-like metric with respect to the foliation
(M, F);
13) for every Riemannian metric g on M the function for the volume of the
leaves is locally bounded;
14) on M there exists the Riemannian metric with respect to which every leaf
is a minimal submanifold;
15) all the germinal holonomy groups of (M, F) are finite;
16) the foliation (M, F) has no holonomy vanishing cycles.
Corollary 5. A compact foliation (M, F) on a compact manifold is locally
stable iff its graph G(F) is compact.
Without assumption of the compactness of the foliation (M, F), the equiva-
lence of conditions 3) and 5), 6) was proved in [33] while the equivalence of 4)
and 16) was proved in ([29], Theorem 1).
Let (M, F) be a transversally complete G-foliation of finite type on a compact
manifold M . In this case Theorems 3 and 5 imply Theorem 1 from [28], according
to which the foliation (M, F) is compact iff its orbit space is a smooth orbifold.
Basing on the results from [37], we obtain the following two theorems on the
stability of leaves of conformal foliations without assumption of the existence of
the Ehresmann connection.
Theorem 6. Any compact conformal foliation of codimension q > 2 is locally
stable.
Theorem 7. Let (M, F) be a conformal foliation of codimension q > 2 on a
compact manifold M . If there exists a compact leaf with finite germinal holonomy
406 Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3
Local and Global Stability of Compact Leaves and Foliations
group, then every leaf of this foliation is compact with finite germinal holonomy
group, i.e., (M, F) is a locally stable foliation.
R e m a r k 2. The analogous theorem for the holomorphic foliations on
compact complex Kaehler manifolds was proved by Pereira in [19].
1. The Holonomy Pseudogroups of Foliations
1.1. The Definition of a Foliation by an N-cocycle
Suppose that there are given:
1) an n-dimensional manifold M and a possibly disconnected q-dimensional
manifold N , where 0 < q < n;
2) an open locally finite covering {Ui|i ∈ J} of the manifold M ;
3) submersions with connected fibres fi : Ui → Vi on Vi ⊂ N ;
4) diffeomorphism γij : fj(Ui ∩ Uj) → fi(Ui ∩ Uj) between the open subsets
in the manifold N satisfying the equality fi = γij ◦ fj on Ui ∩ Uj for any i, j ∈ J
such that Ui ∩ Uj 6= ∅.
Condition 4) implies the equality γik = γij ◦ γjk for Ui ∩ Uj ∩ Uk 6= ∅ and
γii = id|U i.
The maximal (by inclusion) N -cocycle {Ui, fi, {γij}}i,j∈J satisfying conditions
1)–4) defines a new topology τ on M called the leaf topology, which has as a base
the set of all fibres of the submersions fi. The connected components of the
topological space (M, τ) form a division of M denoted by F = {Lα|α ∈ A}, and
the pair (M, F) is called a foliation given by an N -cocycle {Ui, fi, {γij}}i,j∈J with
leaves Lα, α ∈ A. The manifold N is said to be a transversal manifold. As
every N -cocycle belongs to the unique maximal N -cocycle, to define the foliation
(M, F) it is sufficient to take any N -cocycle satisfying conditions 1)–4).
1.2. Transverse Geometric Structures
Let (M, F) be a foliation given by an N -cocycle {Ui, fi, {γij}}i,j∈J . A holonomy
invariant, i.e., invariant under all local diffeomorphisms γij , i, j ∈ J , geometric
structure on the manifold N is called the transverse geometric structure of this
foliation.
A foliation admitting a G-structure as a transverse geometric structure, where
G is a subgroup of the Lie group GL(R, q), is said to be the G-foliation. If there
exists a natural number k such that the kth prolongation of the G-structure on
N is the e-structure, then it is said that (M, F) is a G-foliation of finite type. The
G-foliation is called the ∇ − G-foliation if the transverse manifold N admits a
holonomy invariant G-connection [27]. These foliations were studied by Molino,
who called them the foliations with transversally projectable G-connection.
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 407
N.I. Zhukova
A foliation admitting a Cartan geometry, as a transverse geometric structure,
is called a Cartan foliation ([3, 35]).
In particular, a foliation (M, F) of codimension q is said to be Riemannian if
it is given by an N -cocycle {Ui, fi, {γij}}i,j∈J , and the manifold N admits a Rie-
mannian metric gN such that all {γij} are local isometries. By now Riemannian
foliations have been most deeply studied [17].
1.3. Holonomy Pseudogroup of the Foliation
Remind the main notions ([4, 17], Appendix D).
Definition 4. Let N be a smooth q-dimensional manifold which can be dis-
connected. A smooth pseudogroup of the local transformations H on N is a set
of the diffeomorphisms h : D(h) → R(h) between the open subsets of N satisfying
the following axioms:
1. Let g, h ∈ H and R(h) ⊂ D(g), then g ◦ h ∈ H.
2. If h ∈ H, then h−1 ∈ H.
3. idN ∈ H.
4. Let h ∈ H and let W ⊂ D(h) be an open subset, then h|W ∈ H.
5. If h : D(h) → R(h) is a diffeomorphism between the open subsets of N ,
and for any w ∈ D(h) there is a neighbourhood W in D(h) such that h|W ∈ H,
then h ∈ H.
Definition 5. Let A be a family of the local diffeomorphisms of N containing
idN . The pseudogroup obtained by adding h−1 for each h from A and restrict-
ing local diffeomorphisms on the open subsets, compositions and unions of the
elements from A, is called the pseudogroup generated by A.
Definition 6. Let (M, F) be a foliation given by an N -cocycle {Ui, fi, {γij}}ij∈J .
The pseudogroup generated by the local diffeomorphisms γij of the manifold N is
called the holonomy pseudogroup of this foliation and is denoted by H = H(M, F).
1.4. (G,X)-foliations
Let X be a connected manifold and G be some group of the diffeomorphisms
of X. The group G acts on X quasianalytically if no element of G, except the
identity, fixes a nonempty open set in X.
Definition 7. Assume that the group G of diffeomorphisms of a connected
manifold X acts on X quasianalytically. A foliation (M, F) given by an X-cocycle
ξ = {Ui, fi, {γij}}i,j∈J is called the (G,X)-foliation if for any Ui∩Uj 6= ∅, i, j ∈ J ,
there exists an element g ∈ G such that γij = g|fj(Ui∩Uj).
Remark that a holonomy pseudogroup of any (G,X)-foliation is complete and
quasianalytic.
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2. Ehresmann Connection for Foliations
2.1. A Vertical-Horizontal Homotopy
Remind the notion of the Ehresmann connection introduced by R.A. Blu-
menthal and J.J. Hebda [1]. We use the term a vertical-horizontal homotopy
introduced previously by Hermann. All mappings are supposed to be piecewise
smooth.
Let (M, F) be a foliation of arbitrary codimension q ≥ 1. A distribution
M on the manifold M is called transversal to the foliation F if for any x ∈ M
the equality TxM = TxF ⊕Mx holds, where ⊕ stands for the direct sum of the
vector spaces. The vectors from Mx, x ∈ M , are called horizontal. A piecewise
smooth curve σ is horizontal (or M-horizontal) if each of its smooth segments is
an integral curve of the distribution M. The distribution TF tangent to the leaves
of the foliation (M, F) is called vertical. In other words, a curve h is vertical if h
is contained in a leaf of the foliation (M, F).
A vertical-horizontal homotopy (v.h.h.) is a piecewise smooth map H : I1 ×
I2 → M , where I1 = I2 = [0, 1], such that for any (s, t) ∈ I1 × I2 the curve
H|I1×{t} is horizontal and the curve H|{s}×I2 is vertical. The pair of the curves
(H|I1×{0}, H|{0}×I2) is called the base of the v.h.h. H. Two paths (σ, h) with the
common origin σ(0) = h(0), where σ is a horizontal path and h is a vertical one,
are called an admissible pair of paths.
A distribution M transversal to a foliation (M, F) is called an Ehresmann
connection for (M, F) if for any admissible pair of paths (σ, h) there exists a
v.h.h. with a base (σ, h).
Let M be an Ehresmann connection for the foliation (M, F). Then for any
admissible pair of the paths (σ, h) there exists the unique v.h.h. H with the base
(σ, h). We say that σ̃ := H|I1×{1} is the result of the translation of the path σ
along h with respect to the Ehresmann connection M. It is denoted by σ
h→> σ̃.
In the similar way, we define the v.h.h. and translations for the cases when
I1 and I2 are replaced with half-intervals.
Let (M, F) be a Riemannian foliation of codimension q. Then there exists a
bundle-like metric gM on M relatively (M, F). Denote by M the q-dimensional
distribution complementary (by orthogonality) to TF on the Riemannian man-
ifold (M, gM ). It is known that the completeness of the Riemannian metric gM
guarantees that the distribution M is an Ehresmann connection for the foliation
(M, F).
2.2. Holonomy Groups of Foliations with Ehresmann Connections
Let (M, F) be a foliation with an Ehresmann connection M. Take any point
x ∈ M . Denote by Ωx the set of horizontal curves with the origin at x. An action
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of the fundamental group π1(L, x) of the leaf L = L(x) on the set Ωx is defined
in the following way:
Φx : π1(L, x)× Ωx → Ωx : ([h], σ) 7→ σ̃,
where [h] ∈ π1(L, x), and σ̃ is the result of the translation of σ ∈ Ωx along h
relatively M. Let KM(L, x) be the kernel of the action Φx, i.e.,
KM(L, x) = {α ∈ π1(L, x) |α(σ) = σ,∀σ ∈ Ωx}.
The quotient group HM(L, x) = π1(L, x)/KM(L, x) is the M-holonomy group of
the leaf L [1]. Due to the linear connectedness of the leaves, the M-holonomy
groups at different points of the same leaf are isomorphic.
Let Γ(L, x) be the germinal holonomy group of the leaf L. Then there exists
a unique group epimorphism χ : HM(L, x) → Γ(L, x) satisfying the equality
χ ◦ µ = ν, (1)
where µ : π1(L, x) → HM(L, x) is the quotient map and ν([h]) :=<h> is a germ
of the holonomy diffeomorphism of a transverse q-dimensional disk along the loop
h at the point x.
Let us emphasize that the M-holonomy group HM(L, x) has a global character
in contrast to the germinal holonomy group Γ(L, x) which has a local-global
character: global along the leaves and local along the transverse directions.
Lemma 1. Let (M, F) be a foliation with an Ehresmann connection M. If
there exists a leaf L0 with the trivial M-holonomy group HM(L0, x0), then for any
other leaf L there is a horizontal curve σ such that x0 = σ(0) ∈ L0, y0 = σ(1) ∈ L,
and there is defined a regular covering map fσ : L0 → L which takes a point
x ∈ L0 to the point y := σ̃(1), where h is a path in L0 connecting x0 with x and
σ
h−→> σ̃. Moreover, the group of the deck transformations of fσ : L0 → L is
isomorphic to the M-holonomy group HM(L, y0) of the leaf L.
P r o o f. The existence of a horizontal curve σ connecting any two leaves
of a foliation with an Ehresmann connection was shown in [1]. The triviality of
the M-holonomy group HM(L0, x0) of the leaf L0 implies independence of the
definition of fσ(x) from the choice of the path h connecting x0 with x. Using
the Ehresmann connection M it is not difficult to check that fσ : L0 → L is the
covering map.
Consider an element g = µ([h]) ∈ HM(L, y0), where [h] ∈ π1(L, y0). Take an
arbitrary point z ∈ f−1
σ (y0) and put g(z) := ĥ(1), where ĥ is the path with the
origin at z = ĥ(0) covering the path h via fσ : L0 → L. Thus, an action of the
group HM(L, y0) on the fibre f−1
σ (y0) is defined. It is easy to check that HM(L, y0)
acts on f−1
σ (y0) simply transitively and the group HM(L, y0) is isomorphic to the
group of the deck transformations of fσ : L0 → L.
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3. Two Graphs of a Foliation with an Ehresmann Connection
3.1. Graph GM(F )
The graph GM(F ) of a foliation (M, F) with an Ehresmann connection was
introduced by the author in [31] (see also [32–34].
Let (M, F) be a foliation of arbitrary dimension k on an n-manifold M , and
q = n− k be the codimension of this foliation. Suppose that the foliation (M, F)
admits an Ehresmann connection M.
Consider the set Ωx of the M-horizontal curves with the origin at x ∈ M .
Take any points x and y in a leaf L of (M, F). Introduce an equivalence
relation ρ on the set A(x, y) of the vertical paths in L connecting x with y. The
paths h and f from A(x, y) are called ρ-equivalent if the loop h ·f−1 generates the
trivial element of the M-holonomy group HM(L, x). In other words, the paths h
and f are ρ-equivalent iff they define the same translations of the M-horizontal
curves from Ωx relatively to the Ehresmann connection M. The ρ-equivalence
class containing h is denoted by {h}.
The set of the ordered triplets (x, {h}, y), where x and y are the points of an
arbitrary leaf L of the foliation (M, F), and {h} is a class of the ρ-equivalent paths
from x to y in L, is called the graph of the foliation (M, F) with the Ehresmann
connection M and is denoted by GM(F).
A chart (U,ϕ) of the manifold M is said to be M-fibred with the center at the
point x if:
1) ϕ(U) = Rn, ϕ(x) = {0} ∈ Rn, and ϕ maps each connected component of
the intersection U ∩ Lα of U with an arbitrary leaf Lα, which is called the local
leaf in U , onto some leaf of the trivial foliation F = {Rk × {c} | c ∈ Rq} of the
coordinate space Rn;
2) if Lx is the local leaf in U containing x, then for any z ∈ Lx the submanifold
Dz := ϕ−1({0}×Rq) is a q-dimensional transverse disk at the point z formed by
the points of some smooth curves from Ωz;
3) if h is an arbitrary path in the local leaf Lx, then for any admissible pair of
the paths (σ, h) such that σ(0) = h(0) = z and σ(I1) ⊂ Dz, the v.h.h. H with the
base (σ, h) is equal to ϕ−1 ◦H0, where H0 is the standard v.h.h. of the product
Rn = Rp ×Rq with the base (ϕ ◦ σ, ϕ ◦ h).
It is not difficult to show that at any point x there exists an M-fibred chart.
Similarly to the proof of the theorem on the continuity of foliations from [18],
it is easy to show that for any point (a, {h}, b) of the graph GM(F) there exist
M-fibred charts (U,ϕ) and (V, ψ) with the centers at a and b, respectively, and
the transversal disks Da and Db having the following properties:
1) for any c ∈ Rq, the local leaves ϕ−1(Rp × {c}) and ψ−1(Rp × {c}) belong
to the same leaf of the foliation (M, F);
2) if σ ∈ Ωa, and σ(I) ⊂ Da, with σ
h−→> σ̃, then σ̃ ∈ Ωb σ̃(I) ⊂ Db.
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Define an open neighbourhood Vz of a point z = (a, {h}, b) in GM(F) using
the M-fibred charts (U,ϕ) and (V, ψ) indicated above. Let x be a point in U and
Lx be a local leaf in U containing x. Then there exists an M-curve σ : I → Da
connecting a with x0 := Lx ∩Da. Let h
σ−→> h̃ and y0 := h̃(1). Take any point
y from the local leaf passing through y0. Connect x with x0 in Lx by a path tx,
and y with y0 in Ly by a path ty. Put ĥ := tx · h̃ · t−1
y . Consider the set Vz,h of all
points ẑ := {(x, {ĥ}, y)} obtained. The contractibility of the local leaves implies
that ẑ is independent from the choice of the paths connecting tx and ty. If h′ is
another path from {h}, then the result of the translation of σ along h′ coincides
with the curve σ̃ obtained by translating σ along h. The definition of Vz is given
by Vz := Vz,h.
A coordinate map χz : Vz → Rn+p is given by the equality
χz(x, {ĥ}, y) = (ϕ(x), pr ◦ ψ(y)),
where pr : Rn ∼= Rp × Rq → Rp is the canonical projection onto the first factor.
The pair (Vz, χz) is a coordinate chart, and the family of the charts {(Vz, χz) | z ∈
GM(F)} is an atlas of the manifold of dimension 2n− q, which defines a smooth
structure on GM(F).
The maps
p1 : GM(F) → M : (x, {h}, y) 7→ x, p2 : GM(F) → M : (x, {h}, y) 7→ y
are called the canonical projections. The graph GM(F), equipped with the binary
operation (y, {h1}, z)∗(x, {h2}, y) := (x, {h1 ·h2}, z) and the canonical projections
p1 and p1, becomes a smooth M-holonomy groupoid.
In [31] (see also [33] and [34]), the author proved the following properties of
the graph GM(F), its canonical projections and the induced foliation:
F := {p−1
1 (Lα) |Lα ∈ (F)}.
Proposition 1. 1. The graph GM(F) of a foliation (M, F) with an Ehresmann
connection M equipped with the smooth structure as indicated above becomes a
Hausdorff manifold. The canonical projections p1 and p2 determine locally trivial
fibrations with the common typical fibre Y .
2. For any point x ∈ M there is defined a regular covering map px : Y → L(x),
and the group of its deck transformations is isomorphic to the M-holonomy group
HM(L, x).
3. The diagonal action of the M-holonomy group HM(L, x) on the product
Y × Y is free and properly discontinuous, so it defines a regular covering whose
deck transformation group Ψ is isomorphic to HM(L, x). Moreover, the base of
this covering (Y × Y )/Ψ is diffeomorphic to the leaf L := p−1
1 (L).
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4. The map Φ : GM(F) → G(F) : (x, {h}, y) → (x,< h >, y) is an epimor-
phism of groupoids.
R e m a r k 3. For a foliation (M, F) with an Ehresmann connection M the
existence of a manifold Y , which satisfies the statement 2 in Proposition 1, was
proved [1] in another way.
3.2. Winkelnkemper’s Criterion
Let (M, F) be a smooth foliation and G(F) be its graph (the definition of which
was reminded in Introduction). As it is known, the topological space of G(F) is
not Hausdorff in general. We remark that the criterion of the property of the
graph G(F) to be Hausdorff proved by Winkelnkemper [25] can be reformulated
as follows.
Proposition 2. The topological space of the graph G(F) of the foliation (M, F)
is Hausdorff iff the holonomy pseudogroup of this foliation is quasi analytic.
4. Quasi Analyticity of Holonomy Pseudogroups
4.1. Criterion of Isomorphism between Holonomy Groups of Foliation
with an Ehresmann Connection
Proposition 3. Let (M, F) be a foliation with an Ehresmann connection M.
Then the group epimorphism χ : HM(L, x) → Γ(L, x) satisfying equality (1)
is the group isomorphism if and only if the holonomy pseudogroup H(M, F) is
quasianalytic.
P r o o f. Use the notations from Sec. 2. As proved by Blumenthal and Hebda
that if a foliation (M, F) is formed by the fibres of the submersion p : M → B,
then M is an Ehresmann connection for (M, F) iff M is an Ehresmann connection
for the submersion p : M → B. Let Ui and a submersion fi : Ui → Vi belong
to the N -cocycle defining the foliation (M, F). Consider an admissible pair of
the paths σ, h in Ui such that the translation of σ along h is realized in Ui, and
σ
h−→> σ̃. Then σ̃ is the M-horizontal lift of the path fi ◦ σ into the point h(1)
of relativity fi.
Consider any σ ∈ Ωx and [h] ∈ π1(L, x). Let σ
h−→> σ̃. Then Φx([h], σ) =
σ̃. Cover the loop h(t), t ∈ I2, by a finite chain of the fibred neighbourhoods
U1, . . . , Uk from the N -cocycle defining the foliation (M, F). Let fi : Ui → Vi be
the corresponding submersions and 0 = t0 < t1 < . . . < tk = 1 be the division of
the segment I1 such that h([ti−1, ti]) ⊂ Ui, ∀i = 1, . . . , k. Let us follow to ([17],
Appendix D) and consider Vi as a transversal q-dimensional disk embedded to Ui
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and h(ti) ∈ Vi ⊂ Ui. Then a composition γ(i+1)i ◦ γi(i−1), i = 1, . . . , k − 1, is the
local holonomic diffeomorphism along the path h|[ti−1,ti+1] of a neighbourhood of
the point h(ti−1) in Vi−1 to the conforming neighbourhood of the point h(ti+1) in
Vi+1. A local diffeomorphism γ := γ1k ◦γk(k−1) ◦ . . .◦γ32 ◦γ21 from the holonomy
pseudogroup H(M, F) is defined at some neighbourhood of the point v = f1(x)
belonging to V1 ⊂ N .
A set of the germs {γ}v at v ∈ N of the local diffeomorphisms γ obtained as
shown above, when [h] runs over π1(L, x), is a group which can be interpreted as
the germinal holonomy group Γ(L, x) of the leaf L.
Assume that the holonomy pseudogroup H(M, F) is quasi analytic. To prove
that χ is a group isomorphism, it is sufficient to show that any element [h] ∈
π1(L, x) from the kernel Ker(ν) of the epimorphism ν : π1(L, x) → Γ(L, x)
belongs to the kernel KM(L, x) of the epimorphism µ : π1(L, x) → HM(L, x), i.e.,
each curve σ ∈ Ωx is fixed by the action of [h] via the map Φx. Let h
σ|[0,s]−→> hs,
∀s ∈ [0, 1], xs = hs(0).
Suppose that [h] ∈ Ker(ν), which is equivalent to the triviality of the germ
of the diffeomorphism γ at v, i.e., {γ}v = {idV1}v. Therefore there exists the
number δ > 0 such that f1 ◦ σ|[0,δ] is a curve in the neighbourhood V ⊂ V1 of v,
where γ|V = idV . Thus, σ̃|[0,δ] = σ|[0,δ]. Moreover, for any s ∈ [0, δ] the path hs
belongs to Ker(ν).
Consider the set
A := {a ∈ I1 |hs ∈ Ker(ν), ∀s ∈ [0, a]}.
Then [0, δ] ⊂ A, and hence A is a nonempty set. Applying the previous arguments
to the path [hδ] and repeating them, we can see that A is an open subset of I1.
Let us show that A is a closed subset of I1. In the opposite case, there exists
the number ε > 0, ε /∈ A, such that [0, ε) ⊂ A. Hence, σ(s) = σ̃(s) for any
s ∈ [0, ε). Due to the continuity of the paths σ and σ̃, the equality σ(ε) = σ̃(ε)
is valid, so hε is a loop at the point xε. There is a neighbourhood Uj containing
xε from the N -cocycle defining the foliation (M, F). Let fj : Uj → Vj be a
submersion from this N -cocycle and vε = fj(xε) ∈ Vj . By covering the curve
hε(t), t ∈ I2, by a finite chain of fibred neighbourhoods from N -cocycle, in the
same way as above, we get a local holonomic diffeomorphism γε, γε(vε) = vε of
some neighbourhood Vε of vε belonging to the holonomy pseudogroup H(M, F).
The choice of ε implies the existence of an open subset W ⊂ Vε such that γ|W =
idW , and vε ∈ W, where W is the closure of W in N. Due to quasi analyticity of
the pseudogroup H(M, F), the equality γε = idD is valid in the entire connected
domain of D = D(γε) of γε containing vε. As Vε ⊂ D, so γε|Vε = idVε . Therefore
hε ∈ Ker(ν), hence ε ∈ A. The contradiction with the assumption shows that A
is a closed subset of I1. As I1 is connected, the nonempty open-closed subset A
coincides with it, i.e., A = I1.
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Thus, Ker(ν) ⊂ KM(L, x) = Ker(µ), therefore equality (1) implies the trivi-
ality of the kernel of the epimorphism χ : HM(L, x) → Γ(L, x), i.e., χ is a group
isomorphism.
Let us show the converse statement. Assume that χ : HM(L, x) → Γ(L, x)
is a group isomorphism for any point x ∈ M . Then, according to Prop. 1, the
map β : GM(F) → G(F) is a diffeomorphism. Therefore the property of the
graph GM(F) to be Hausdorff implies the same property of G(F). Hence, by
Winkelnkemper’s criterion (Prop. 2), the holonomy pseudogroup of the foliation
(M, F) must be quasi analytic.
4.2. Proof of Theorem 1
Let (M, F) be any locally stable compact foliation. As well known, on M there
is a complete bundle-like metric with respect to (M, F). Therefore, according to
([17], Appendix D, Prop. 2.6), the holonomy pseudogroup of the Riemannian
foliation is complete. It is also quasi analytic because its elements are local
isometries.
On the other hand, suppose that the holonomy pseudogroup H = H(M, F)
of a compact foliation (M, F) is complete and quasi analytic. Assume that there
exists a leaf L = L(x) with infinite holonomy group. Then there is a submersion
fi : Ui → Vi from the N -cocycle defining (M, F) such that x ∈ Ui, and the
group of germs of local diffeomorphisms from the stationary pseudogroup Hw =
{h ∈ H |h(w) = w} at w = fi(x) is infinite. The completeness of the holonomy
pseudogroup H implies the existence of an open neighbourhood U of w in Vi on
which every h ∈ Hw is defined.
By the theorem of Epstein–Millett–Tischler [13], the union of all leaves with
trivial holonomy is a dense Gδ-subset in M . Thus there is a leaf L0 without
holonomy intersecting the open subset f−1
i (U). Let y ∈ L0 ∩ f−1
i (U) and v =
fi(y) ∈ U .
Notice that the leaf L0 of (M, F) is compact if and only if the orbit H · v of
the point v is finite. Thus the orbit Hw · v is finite. Hence there is an infinite
sequence of the elements {gn}, n ∈ N, from Hw belonging to different germs at
w such that gn(v) = v. By the definition of v, for each element gn there exists
a neighbourhood Wn, where gn|Wn = idWn . Therefore the quasi analyticity of
H implies gn|U = idU . This contradicts to the property of gn to define different
germs at w. Hence the holonomy group Γ(L, x) is finite.
Thus, all leaves have finite holonomy groups and (M, F) is a locally stable
foliation.
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4.3. Proof of Theorem 2
Let (M, F) be a compact foliation with quasi analytic holonomy pseudogroup
admitting an Ehresmann connection M. Use the notations introduced in Sec. 2.
By the theorem of Epstein–Millett–Tischler [13], the union of all leaves without
holonomy is the Gδ-subset of the manifold M . Therefore the foliation (M, F)
has a leaf L0 = L0(x0), x0 ∈ M , with the trivial holonomy group Γ(L0, x0).
According to Prop. 3, due to the quasi analyticity of the holonomy pseudogroup
of the foliation (M, F), for any point x ∈ M the map χ : HM(L, x) → Γ(L, x) is
a group isomorphism. Hence the leaf L0 = L0(x0) has the trivial M-holonomy
group.
By Lemma 1, for any leaf L = L(x), there exists a regular covering map f :
L0 → L, with the group of deck transformations isomorphic to the M-holonomy
group HM(L, x) of the leaf L. As both leaves L0 and L are compact, so f : L0 → L
is a finitely sheeted covering. It implies the finiteness of the group HM(L, x).
Therefore the holonomy group Γ(L, x), which is isomorphic to HM(L, x), is also
finite.
Thus, according to the Reeb theorem on the local stability of a compact leaf
with finite holonomy group, all the leaves of (M, F) are locally stable.
Let us prove the converse statement. Suppose that a compact foliation (M, F)
of codimension q is locally stable. As well known, in this case (M, F) is the Rie-
mannian foliation. Hence its holonomy pseudogroup H(M, F) consists of local
isometries. Therefore H(M, F) is quasi analytic. Moreover, there exists a com-
plete bundle-like metric g relatively (M, F) (see, for instance, [30]). Hence the
q-dimensional orthogonal distribution M is an Ehresmann connection for the
foliation (M, F) [1].
5. Proof of Theorems 3–7
5.1. Proof of Theorem 3
As it was shown in ([36], Prop. 2), the completeness of a foliation admitting
a transverse rigid geometry implies the existence of an Ehresmann connection
for this foliation. Therefore it is sufficient to observe that the holonomy pseu-
dogroup of every foliation mentioned in Theorem 3 is quasi analytic and to apply
Theorem 2.
5.2. Proof of Theorem 4
Suppose that a foliation (M, F) satisfies conditions 1) and 2) of Theorem 2.
Assume that there exists a compact leaf L′ with the finite germinal holonomy
group Γ(L′, x′). According to [13], there is a leaf L0 having the trivial germinal
holonomy group. As the conditions of Prop. 3 are satisfied, χ : HM(L, x) →
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Γ(L, x) is the group isomorphism for any x ∈ M . Hence the M-holonomy group
of the leaf L0 is also trivial. By Lemma 1, there exists a regular covering map
f0 : L0 → L′, and the group of deck transformations of this map is isomorphic to
the group HM(L′, x′) ∼= Γ(L′, x′). Therefore the group HM(L′, x′) is finite and the
leaf L0 is compact. With accordance to Lemma 1, the leaf L0 covers each leaf L of
this foliation, and the group of the deck transformations is HM(L, x) ∼= Γ(L, x).
Hence every leaf L is compact and has finite germinal holonomy group Γ(L, x).
If there exists a compact leaf L′ with the finite fundamental group π1(L′, x′),
then, by analogy, we can show that all leaves have the same compact universal
covering space. Therefore each leaf L is compact and it has the finite fundamental
group.
5.3. Proof of Theorem 5
By Theorems 1 and 2, conditions 1), 2) and 3) are equivalent.
We remark that Winkelnkemper’s criterion for the graph G(F) of a foliation
(M, F) to be Hausdorff [25], which was reformulated by us as Prop. 2, implies
the equivalence of 3) and 4).
Suppose that 4) is true. As 4) is equivalent to 3), it follows from the proof of
Theorem 2 that the map χ : HM(L, x) → Γ(L, x) is the group isomorphism for
every point x ∈ M , i.e. 4) ⇒ 5). Using Prop. 1 it is not difficulty to show that
5) ⇔ 6).
Assume that 6) holds. According to Prop. 1, the M-holonomy groupoid
GM(F) is always Hausdorff. The isomorphism of the holonomy groupoids GM(F)
and G(F) is the diffeomorphism between them. Therefore the topological space
of G(F) is also Hausdorff. By Prop. 2, it is equivalent to the quasi analyticity
of the holonomy pseudogroup of (M, F), i.e., 6) implies 4). Thus, the first six
conditions are equivalent.
Now suppose that 6) is valid, i.e., G(F) ∼= GM(F). According to the first
statement of Prop. 1, the canonical projection p1 : G(F) → M is a locally trivial
fibration with the standard fibre Y . Consequently, any fibre p−1
1 (x) over x ∈ M
is diffeomorphic to Y . By the mentioned above result of [13], there exists a leaf
L0 = L0(x0) with the trivial germinal holonomy group. By the definition of
the graph G(F), the manifold p−1
1 (x0) is diffeomorphic to a leaf L0 of (M, F).
Due to the compactness of the foliation (M, F), the leaf L0 is compact. Hence
Y ∼= p−1
1 (x0) is compact. Therefore each fibre p−1
1 (x) is also compact, i.e., 6)
implies 7).
Notice that an arbitrary leaf L = p−1
1 (L), where L = L(x), of the induced
foliation is diffeomorphic to the quotient manifold (p−1
1 (x) × p−1
1 (x))/Ψ, and
the group Ψ is isomorphic to the holonomy group Γ(L, x) of the leaf L. The
quotient map p−1
1 (x)× p−1
1 (x) → L is a regular covering mapping with the group
Journal of Mathematical Physics, Analysis, Geometry, 2013, vol. 9, No. 3 417
N.I. Zhukova
of the deck transformations Ψ . If condition 7) is valid, then any leaf L of the
induced foliation (G(F),F) is compact since it is the image of the compact space
p−1
1 (x)× p−1
1 (x) under a continuous map. Thus, 7) ⇒ 8).
To prove the implication 8)⇒ 1), suppose that the induced foliation (G(F),F)
is compact, i.e., each its leaf L is compact. Note that the map f : L → L × L,
taking a point z = (x, {h}, y) from L to the point (x, y) ∈ L × L, is a regular
covering map with the group of the deck transformations isomorphic to the holon-
omy group Γ(L, x) of the leaf L. The map f : L → L × L is a finitely sheeted
covering, because it is a covering map of one compact manifold onto another
compact manifold. Hence every leaf L is compact with the finite holonomy group
Γ(L, x). Therefore, in conformity with Reeb’s theorem, the foliation (M, F) is
locally stable, i.e., 8) ⇒ 1).
The equivalence of conditions 9)–16) and 1) follows from the works given in
Introduction.
5.4. Proof of Theorem 6
Being compact, the foliation (M, F) does not admit an attractor. According
to our result ([37], Theorem 2), in this case the conformal foliation (M, F) of
codimension q > 2 must be a compact Riemannian foliation. Therefore it is
locally stable.
5.5. Proof of Theorem 7
Consider a conformal foliation (M, F) of codimension q > 2 on a compact
manifold M . It follows from Theorem 4 proved by us in [37] that if there exists a
compact leaf with a finite germinal holonomy group, then (M, F) is a Riemannian
foliation. Due to the compactness of M , there is a complete bundle-like metric
g with respect to (M, F). Therefore, the orthogonal q-dimensional distribution
M is an Ehresmann connection for (M, F). Thus, the required assertion follows
from Theorem 2.
Anknowledgements. I am grateful to Yu.A. Kordyukov for useful remarks.
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