Dynamics of conformal foliations

Nina I. Zhukova 1
  • 1 National Research University Higher School of Economics, Laboratory of Dynamical Systems and Applications, 25/12, Bolshaya Pecherskaya St., Russia
Nina I. Zhukova
  • Corresponding author
  • National Research University Higher School of Economics, Laboratory of Dynamical Systems and Applications, 25/12, Bolshaya Pecherskaya St., Nizhny Novgorod, 603155, Russia
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors of conformal foliations. Results on strong transversal equivalence of conformal foliations are also presented. Connections with works of other authors are indicated. Examples of conformal foliations with exceptional, exotic and regular minimal sets which are attractors are constructed.

1 Introduction

The purpose of this article is to review the author's results on the existence and structure of minimal sets and attractors conformal foliations. The results are illustrated by various examples. Connections with works of other authors are indicated.

Remind that two Riemannian metrics h and g on a manifold M are called conformally equivalent if there exists a positive smooth function f on M with h = f g. A conformal equivalence class [g] of Riemannian metrics on M is called a conformal structure on M, and the pair (M,[g]) is said to be a conformal manifold.

A group of conformal transformations of a Riemannian manifold (M,g) is called inessential if it is a group of isometries of a Riemannian manifold (M,h) with some h โˆˆ [g]. Otherwise, it is called essential.

Lichnerowicz conjectured that for n โ‰ฅ 3 every n-dimensional compact Riemannian manifold admitting an essential group of conformal transformations is conformally equivalent to the standard n-dimensional sphere ๐•Šn. The articles by M. Obata [22], D.V. Alekseevskii [1,2], J. Ferrand [13] and others are devoted to this conjecture. It was also established that if a group of conformal transformations of a non-compact Riemannian manifold M is essential then M is conformally equivalent to the n-dimensional Euclidean space. In 1996 J. Ferrand [13] gave a complete proof of the Lichnerowicz conjecture, including the case of non-compact manifolds.

C. Tarquini [26] and then C. Tarquini and C. Frances [14] posed the following question about conformal foliations:

Is every codimension q โ‰ฅ 3 conformal foliation on a compact manifold either a Riemannian foliation or a (Con f (๐•Šq),๐•Šq)-foliation?

C. Frances and C. Tarquini called the question a foliated analog of the Lichnerowicz conjecture. For q โ‰ฅ 3 a conformal foliation is a (Con f (๐•Šq), ๐•Šq)-foliation if and only if it is transversally conformally flat. C. Frances and C. Tarquini [14] gave a positive answer to this question under some additional assumptions. We proved that in general the answer to the C. Frances and C. Tarquini question is positive (Theorem 7).

Our main results included in this article are as follows:

  1. 1)Different interpretations for the germ holonomy group of a leaf of a conformal foliation (Theorem 2).
  2. 2)A criterion for a conformal foliation to be Riemannian (Theorem 3).
  3. 3)The existence of an attractor which is a minimal set for every non-Riemannian conformal foliation of codimension q โ‰ฅ 3 (Theorem 4).
  4. 4)A structure theorem for conformal and transversely similar foliations on closed manifolds (Theorems 7 and 8).
  5. 5)Generalization of the results of P. Molino on the closures of leaves of Riemannian foliations (Theorem 10) and equivalence of completeness and existence of an Ehresmann connection for a conformal foliation (Theorem 9).
  6. 6)The existence of a global transitive attractor for every complete non-Riemannian conformal foliation of codimension q โ‰ฅ 3 (Theorem 12) and description of the structure of such foliations (Theorem 13).
  7. 7)Properties of global holonomy groups of conformal foliations (Theorems 13, 14.)
  8. 8)The global holonomy group defined up to conjugacy in the group Con f (๐•Šq) is a complete invariant of the class of strong transversely equivalent conformal foliations (Theorem 15). An analogous result for strong transversely equivalent similar foliations (Theorem 16).
  9. 9)The existence of a two-dimensional suspension conformal foliation in every class of transversely equivalent non-Riemannian conformal foliations (Theorem 18).

In Section 3.5 we compare our results with some results of B. Deroin and V. Kleptsyn [11].

Section 7 contains examples of conformal foliations with global attractors. In particular, we construct a conformal foliation with two global attractors, one of which is a transitive attractor but not a minimal set (Example 22).

Notations In this article we denote by M a smooth manifold of dimension n and by Cโˆž(M) the space of smooth functions on M. Let ๐”›(M) be the Cโˆž(M)-module of smooth vector fields on M. Also, we denote by ๐”›c(M) the Cโˆž(M)-module of compactly supported smooth vector fields on M.

2 Conformal foliations and associated constructions

2.1 Conformal foliations

First, we recall some basic notions.

A diffeomorphism f : N1 โ†’ N2 of Riemannian manifolds (N1,g1) and (N2,g2) is called conformal if there exists a smooth function ฮป on N1 with f*g2 = ฮป g1. A conformal diffeomorphism f from a Riemannian manifold (N,g) to itself is also called a conformal transformation. A conformal transformation f of (N,g) is said to be a similarity, if f*g2 = ฮป g for a constant ฮป.

Conformal foliations were studied by some authors as foliations admitting a transverse conformal structure.

Definition 1

A codimension q โ‰ฅ 3 smooth foliation (M,F) is called conformal, if there are:

  • a possibly disconnected Riemannian manifold (N,g);
  • an open cover {Ui |i โˆˆ J} of M;
  • submersions fi : Ui โ†’ Vi with connected fibers, Vi โŠ‚ N,
such that: if Ui โˆฉ Uj โ‰  โˆ…, then there exists a conformal diffeomorphism ฮณij : fj(Ui โˆฉ Uj) โ†’ fi(Ui โˆฉ Uj) satisfying the condition fi = ฮณij โ—‹ fj on Ui โˆฉ Uj.

Definition 2

A maximal (with respect to inclusion) N-cocycle {Ui, fi,ฮณij}i,jโˆˆJ, satisfying the above properties, determines a new topology on M, whose base is the set of fibers of all submersions fi. This topology is called the leaf topology and denoted by ฯ„.

Path-connected components of the topological space (M,ฯ„) form a partition F = {Lฮฑ |ฮฑ โˆˆ A} of M, and (M,F) is called the conformal foliation with leaves Lฮฑ determined by the N-cocycle {Ui, fi,ฮณij}i,jโˆˆJ.

It is also said that (M,F) is modeled on the conformal geometry (N,[g]).

Definition 3

If each ฮณij is a local similar transformation of (N,g), then (M,F) is called a transversely similar Riemannian foliation.

Definition 4

If each ฮณij is an isometry, then (M,F) is called a Riemannian foliation.

Let Con f (๐•Šq) be the Lie group of all conformal transformations of the q-dimensional sphere ๐•Šq.

Definition 5

If a foliation (M,F) is defined by the N-cocycle {Ui, fi,ฮณij}i,jโˆˆJ, where N = ๐•Šq and each ฮณij is the restriction of a transformation f โˆˆ Con f (๐•Šq), then (M,F) is referred to as a (Con f (๐•Šq),๐•Šq)-foliation.

A conformal transformation of the complex plane is conformal in the above sense. Therefore, every transversely complex analytic foliation of complex codimension one is a conformal foliation of real codimension two in the sense of Definition 1. The converse is also true, i.e., every conformal foliation of codimension two can be considered as a transversely complex analytic foliation of complex codimension one.

Conformal foliations of codimension two have been studied by several authors, using methods of complex analysis. For example, in [6] P. Baird and M. Eastwood gave a detailed twistor description of conformal foliations of codimension two in the 3-dimensional Euclidean space. Holonomy diffeomorphisms of these foliations preserve the conformal equivalence class of Riemannian metrics on orthogonal complements to their leaves.

Since every 2-dimensional manifold is locally conformally flat, every conformal foliation of codimension two is transversely conformally flat. A transversal conformal manifold (N,[g]) is locally conformally flat if and only if, in codimension q > 3, its Weyl conformal curvature tensor vanishes or, in codimension q = 3, its Schouten tensor vanishes.

Like S. Morita [21], in contrast to the case of conformal foliations of codimension two, we study conformal foliations of codimension q, q โ‰ฅ 3, by differential geometric methods, considering them as Cartan foliations.

2.2 The foliated bundle over a conformal foliation

The construction of the foliated bundle is essentially used by us. Foliated bundles were introduced in works of P. Molino, F. Kamber and Ph. Tondeur.

Recall that a foliation admitting an effective Cartan geometry of type (G,H) [10] as a transverse structure is said to be a Cartan foliation of the same type. Remark that any conformal foliation may be considered as a Cartan foliation of type (G,H), where G = Con f (๐•Šq) is the full group of conformal transformations of ๐•Šq and H is the stabilizer of G at an arbitrary point b โˆˆ ๐•Šq. Let ๐”ค and ๐”ฅ be the Lie algebras of the Lie groups G and H, respectively. Then we have the following statement [27].

Proposition 1

Let (M,F) be a conformal foliation of codimension q โ‰ฅ 3. Then there are the following objects:

  1. 1a principal H-bundle ฯ€ : โ„› โ†’ M;
  2. 2an H-invariant foliation (โ„›,โ„ฑ) which is mapped by ฯ€ into (M,F);
  3. 3a ๐”ค-valued 1-form ฯ‰ on โ„›, satisfying the properties:
    1. (i)ฯ‰ (A*) = A for any A โˆˆ ๐”ฅ, where A* is the fundamental vector field corresponding to A;
    2. (ii)Ra*ฯ‰=AdG(aโˆ’1)ฯ‰for any a โˆˆ H, where AdG is the adjoint representation of the Lie group G in its Lie algebra ๐”ค;
    3. (iii)the Lie derivative LX ฯ‰ vanishes for every vector field X tangent to the leaves of (โ„›,โ„ฑ).

The H-bundle ฯ€ : โ„› โ†’ M is said to be foliated. The foliation (โ„›,โ„ฑ) is called the lifted foliation, and (โ„›,โ„ฑ) is a transversely parallelizable foliation, i.e., an e-foliation.

2.3 Holonomy groups of a conformal foliation

We consider conformal foliations as Cartan foliations and use the construction of the principal foliated bundle. Due to this, we can apply the results of our previous work [27].

In foliation theory, the germ holonomy groups are usually used. By analogy with [27], we give the following interpretations for the germ holonomy group of a leaf of a conformal foliation.

As above, let G = Con f (๐•Šq) be the Lie group of conformal transformations of the sphere ๐•Šq and let H be the stabilizer of G at an arbitrary point in ๐•Šq.

Theorem 2

Let (M,F) be a conformal foliation of codimension q โ‰ฅ 3, โ„› be the foliated H-bundle over (M,F) with the projection ฯ€ : โ„› โ†’ M and (โ„›,โ„ฑ) be the lifted foliation. Then, for every leaf L = L(x), x โˆˆ M, of (M,F) the restriction ฯ€ |โ„’ : โ„’ โ†’ L to the leaf โ„’ = โ„’ (u), u โˆˆ ฯ€โˆ’1(x), of the lifted foliation (โ„›,โ„ฑ) is a regular covering map, and the germ holonomy group ฮ“(L,x) of L is isomorphic to each of the following groups:

  1. (i)the subgroup H(โ„’) := {a โˆˆ H |Ra(โ„’) = โ„’ } of H;
  2. (ii)the group of deck transformations of the covering ฯ€ |โ„’ : โ„’ โ†’ L.

Note that the group H(โ„’) is defined up to conjugacy in the group H.

3 Minimal sets and attractors of conformal foliations

3.1 Minimal sets of foliations

A saturated set of a foliation is a union of leaves. Recall that a minimal set of a foliation (M,F) is a nonempty closed saturated subset of M without proper subsets satisfying these properties.

Existence of minimal sets and description of their structure are one of the central problems in the foliation theory. A. H. Aranson, V. Z. Grines [4, 5] and J. Levitt [19] obtained a topological classification of nontrivial minimal sets of flows and foliations on closed surfaces of genus p โ‰ฅ 2. Minimal sets of Riemannian foliations were studied by P. Molino [20], A. Haefliger [15] and E. Salem [23] without using the term minimal set.

Every foliation on a compact manifold has a minimal set. This is wrong for non-compact manifolds. Foliations without minimal sets (on non-compact manifolds) are constructed in the works of J.-C. Beniere and G. Meigniez [7], T. Inaba [16] and M. Kulikov [18]. Moreover, J.-C. Beniere and G. Meigniez proved the existence of foliations without minimal sets on every non-compact manifold of dimension n โ‰ฅ 2, different from surfaces of finite kind [7].

A minimal set โ„ณ is called regular, if it is a submanifold of M. Let {Ui|i โˆˆ N} be an at most countable locally finite open cover of M by foliated coordinate neighborhoods. Let Ti be a submanifold in Ui, which intersects transversely each leaf of the induced foliation (Ui,FUi), and let T = โŠ”iโˆˆNTi be the complete transversal. A minimal set โ„ณ, having no interior points, is called:

  • exceptional if โ„ณ โˆฉ T is a Cantor set;
  • exotic if โ„ณ โˆฉ T is not a totally disconnected topological space.

In Section 7 we construct examples of conformal foliations with exotic, exceptional, and regular minimal sets.

3.2 A criterion for a conformal foliation to be Riemannian

Definition 6

The holonomy group of a leaf L of a conformal foliation is said to be relatively compact or inessential if the corresponding subgroup H(โ„’) of the Lie group H is relatively compact.

Since H(โ„’) is defined up to conjugacy in H, then this definition is correct.

We established the following criterion for a conformal foliation to be Riemannian [29, Theorem 3].

Theorem 3

Let (M,F) be a codimension q โ‰ฅ 3 conformal foliation modeled on a conformal geometry (N,[g]). Then there exists a Riemannian metric d โˆˆ [g] such that (M,F) is a Riemannian foliation modeled on (N,d) if and only if every holonomy group of this foliation is relatively compact, i.e. inessential.

Remark 1

According to [31, Theorem 3], Theorem 3 holds for transversely similar Riemannian foliations of codimension q โ‰ฅ 2. Note that it is true also for q = 1.

3.3 The existence of attractors of conformal foliations

Let (M,F) be a foliation.

Definition 7

A nonempty closed saturated subset โ„ณ of M is called an attractor of (M,F) if there exists a saturated open neighborhood Attr(โ„ณ) of โ„ณ such that the closure of every leaf belonging to Attr(โ„ณ) \ โ„ณ contains โ„ณ.

If, in addition, Attr(โ„ณ) = M, then the attractor โ„ณ is called global.

In the case when there exists a dense leaf in โ„ณ, the attractor โ„ณ is referred to be transitive.

Using the criterion for conformal foliation to be Riemannian (Theorem 3) and some results from local conformal geometry, we prove the following theorem on two alternative possibilities for conformal foliations.

Theorem 4

Let (M,F) be a codimension q โ‰ฅ 3 conformal foliation. Then:

  • either it is Riemannian,
  • or it has an attractor โ„ณ which is the closure of a leaf L with essential holonomy group,โ„ณ=Lยฏ , and a minimal set. Moreover, the restriction (๐’œttr(โ„ณ),F) of the foliation to the attraction basin is a (Con f (๐•Šq),๐•Šq)-foliation.

Recall that a foliation (M,F) is called proper if all its leaves are embedded submanifolds of M. A leaf L is called closed if L is a closed subset of M.

Corollary 5

Every proper non-Riemannian conformal foliation of codimension q โ‰ฅ 3 has a closed leaf with essential holonomy group that is an attractor.

Corollary 6

A non-Riemannian conformal foliation (M,F) of codimension q โ‰ฅ 3 has a minimal set that is an attractor of this foliation.

Remark 2

Theorem 4 is proved without assumption that the foliated manifold M is compact.

3.4 An analog of the Lichnerowicz conjecture for conformal foliations

The following theorem [28, Theorem 4] gives a positive answer to the C. Frances and C. Tarquini question stated in Introduction in the general case.

Theorem 7

Any codimension q โ‰ฅ 3 conformal foliation (M,F) on a compact manifold M is

  • either a Riemannian foliation;
  • or a (Con f (๐•Šq),๐•Šq)-foliation with finitely many minimal sets. All these sets are attractors given by the closures of leaves with essential holonomy group, and each leaf of F belongs to the basin of at least one of these attractors.

For the proof of Theorem 7, we applied the results of D.V. Alekseevskii [1] and J. Ferrand [12] concerning with local differential geometry as well as Riemannian geometry on non-Hausdorff manifolds.

Let us emphasize that transversely similar foliations form a subclass of conformal foliations. Since any Riemannian manifold of dimension 1 or 2 is locally conformally flat, every transversely similar Riemannian foliation of codimension q = 1 or 2 is a (Sim(๐”ผq),๐”ผq)-foliation. By using similar arguments as in the proof of [28, Theorem 4], we prove the following statement.

Theorem 8

Every transversely similar Riemannian foliation (M,F) of codimension q โ‰ฅ 1 on a compact manifold M is

  • either a Riemannian foliation
  • or a (Sim(๐”ผq),๐”ผq)-foliation with finitely many minimal sets. They are all attractors formed by the closures of leaves with essential holonomy group, and each leaf of the foliation belongs to the basin of at least one of them.

3.5 Comparison with the results B. Deroin and V. Kleptsyn [11]

Theorem 7 strengthens (in an appropriate smoothness class and for q โ‰ฅ 3) the first part of the main theorem of B. Deroin and V. Kleptsyn [11], which claims that every conformal foliation of arbitrary codimension q โ‰ฅ 1 on a compact manifold either admits a transversal invariant measure or has finitely many minimal sets satisfying some properties. As emphasized in the introduction of [11], E. Ghys conjectured that in the absence of attractors of the conformal foliation, the invariant transverse measure is given by a Riemannian metric. It follows from our Theorem 7 that this conjecture of E. Ghys holds for conformal foliations of codimension q โ‰ฅ 3.

It is known (see, for example [1]) that there are nontrivial second order conformal diffeomorphisms f of the sphere ๐•Šq, q โ‰ฅ 2, such that the differential f*x at a fixed point x is an orthogonal transformation of the tangent Euclidean vector space Tx๐•Šq at x. These conformal transformations are essential. The 1-dimensional conformal foliation (M,F) formed by the suspension of such a conformal diffeomorphism f does not admit a transversely invariant measure and does not have hyperbolic holonomy; therefore, (M,F) does not satisfy [11, Cor. 1.3]. Thus, unlike us, in [11] the authors do not consider conformal foliations with such holonomies.

Note that in [28, 29] we use the methods of local and global differential geometry, including foliated principal bundles, geometries on non-Hausdorff manifolds as well as Ehresmann connections for foliations. While B. Deroin and V. Kleptsyn apply methods of random dynamical systems, including Lyapunov exponents of harmonic measures. In [11], a Laplace operator along leaves is considered and the existence of a transverse invariant measure is investigated.

4 Completeness and Ehresmann connections for conformal foliations

4.1 Smooth distributions on manifolds

We will use the definition of a smooth distribution given by I. Androulidakis and G. Skandalis [3, ยง1.1]. Let us recall this definition.

Let ๐” be a Cโˆž(M)-submodule of ๐”›c(M) and let U be an open subset of M. Set iU : U โ†’ M the inclusion map. For any vector field X โˆˆ ๐”›(M) we denote X|U := X โ—‹ iU. The restriction of ๐” to U is the Cโˆž(M)-submodule of ๐”›c(M) generated by the vector fields f ยท X|U, where f โˆˆ Cโˆž(U) and X โˆˆ ๐”. This restriction is denoted by ๐”|U.

The module ๐” is said to be locally finitely generated if, for every x โˆˆ M there exist an open neighbourhood U of x and a finite number of vector fields ฮพ1,..., ฮพk belonging to ๐”›(M) such that ๐”|U = Cโˆž(U)c ยท ฮพ1 + ... + Cโˆž(U)c ยท ฮพk. It is said that the vector fields ฮพ1,...,ฮพk generate the restriction ๐”|U of ๐” to U.

Definition 8

([3]) A smooth distribution on M is a locally finitely generated Cโˆž(M)-submodule ๐” of the Cโˆž(M)-module ๐”›c(M). It is also denoted by the pair (M,๐”).

If ๐” is a constant rank distribution, then ๐” is a projective submodule of the Cโˆž(M)-module ๐”›c(M), i.e., the space of smooth sections of some vector bundle. We consider constant rank distributions.

Remark 3

A foliation (M,F) in the sense of [3] is a smooth involutive distribution on M, i.e. F is a locally finitely generated Cโˆž(M)-submodule of ๐”›c(M) for which [F,F] โŠ‚ F. Due to the Frobenius theorem, this definition is equivalent to the commonly used definition of Cโˆž-smooth foliation [25].

4.2 Completeness of conformal foliations

Let (M,F) be a smooth foliation of codimension q โ‰ฅ 1 on an n-dimensional manifold M. Let ๐” be a q-dimensional distribution on M. We say that ๐” is transverse to (M,F) if the distributions ๐” and T F define transverse subbundles of the tangent bundle to M.

Let (M,F) be a conformal foliation of codimension q โ‰ฅ 3 on a smooth n-dimensional manifold M. We keep notation introduced in Section 2.2. Denote by โ„› the foliated bundle over (M,F) with the projection ฯ€ : โ„› โ†’ M and by ฯ‰ the ๐”ค-valued 1-form on โ„›. Let (โ„›,โ„ฑ) be the lifted e-foliation. Consider a q-dimensional distribution ๐” on M. Denote by ฯ€*๐” the distribution on โ„› generated by vector fields Y โˆˆ ๐”›(โ„›) for which ฯ€*Y โˆˆ ๐”.

Definition 9

A conformal foliation (M,F) of codimension q โ‰ฅ 3 is referred to as complete, if there exists a smooth transverse q-dimensional distribution ๐” on M such that every vector field Y โˆˆ ฯ€* ๐”, for which ฯ‰ (Y) = const โˆˆ ๐”ค, is complete.

4.3 Ehresmann connection for foliations

The notion of an Ehresmann connection for foliations was introduced by R.A. Blumenthal and J.J. Hebda in [8]. R.A. Blumenthal and J.J. Hebda [8] considered a smooth distribution on a smooth manifold M as a smooth subbundle of the tangent bundle T M. We clarify the notion of Ehresmann connection for foliations due to the new definition of a distribution in sense I. Androulidakis and G. Skandalis which we have accepted.

Consider a smooth foliation (M,F) of codimension q, q โ‰ฅ 1. Let ๐” be a smooth q-dimansional distribution in sense I. Androulidakis and G. Skandalis, and ๐” is transverse to (M,F).

All maps considered here are assumed to be piecewise smooth. The integral curves of the foliation (M,F) are called vertical; the integral curves of the distribution ๐” are called horizontal. Emphasize that a vertical curve lies in some leaf of (M,F).

A map H : I1 ร— I2 โ†’ M, where I1 = I2 = [0,1], is called a vertical-horizontal homotopy if for each fixed t โˆˆ I2, the curve H|I1ร—{t} is horizontal, and for each fixed s โˆˆ I1, the curve H|{s}ร—I2 is vertical. The pair of curves (H|I1ร—{0}, H|{0}ร—I2) is called the base of H.

A pair of curves (ฯƒ,h) with a common starting point ฯƒ(0) = h(0), where ฯƒ : I1 โ†’ M is a horizontal curve, and h : I2 โ†’ M is a vertical curve, is called admissible. If for each admissible pair of curves (ฯƒ,h) there exists a vertical-horizontal homotopy with the base (ฯƒ,h), then the distribution ๐” is called an Ehresmann connection for the foliation (M,F). Note that there exists an at most one vertical-horizontal homotopy with a given base. Let H be a vertical-horizontal homotopy with the base (ฯƒ,h). We say that ฯƒหœ=H|I1ร—{1} is the result of the transfer of the horizontal curve ฯƒ along the vertical curve h with respect to the Ehresmann connection ๐”. Similarly the curve hหœ=H|{1}ร—I2 is called the transfer of the curve h along ฯƒ with respect to ๐”. Thus, an Ehresmann connection allows us to transfer horizontal curves along respective vertical curves.

The following theorem is proved in [29, Theorem 4].

Theorem 9

For any non-Riemannian conformal foliation (M,F) of codimension q โ‰ฅ 3 the following two conditions are equivalent:

  1. (i)There exists an Ehresmann connection for (M,F).
  2. (ii)The conformal foliation (M,F) is complete.

Remark 4

Like Theorem 8, Theorem 9 holds for transversely similar Riemannian foliations of codimension q โ‰ฅ 1.

5 Minimal sets and global attractors of conformal foliations

5.1 Minimal sets of Riemannian foliations

The following theorem was proved by us in [29, Theorem 1], see also [32]. For compact manifolds this theorem was earlier proved by P. Molino [20]. In the case when a bundle-like Riemannian metric on M with respect to (M,F) is complete, it was proved by E. Salem [23] without using the term โ€œminimal setโ€.

Theorem 10

Let (M,F) be a Riemannian foliation with an Ehresmann connection. Then the closureLยฏof any leaf L is a smooth embedded submanifold of M which is a minimal set of the foliation. The closures of leaves of the foliation (M,F) form a Riemannian foliation with singularities.

In particular, if (M,F) is a proper foliation, then all its leaves are closed, and the leaf space is a smooth q-dimensional orbifold.

5.2 Global attractors of a countable subgroup of Con f (๐•Šq)

Let ฮจ be a homeomorphism group of a topological space B. A nonempty closed invariant subset ๐’ฆ โŠ‚ B is called an attractor of ฮจ if there exists an open invariant subset W โŠ‚ B such that the orbit closure Cl(ฮจ.z) of any point z โˆˆ W \ ๐’ฆ contains ๐’ฆ. An attractor ๐’ฆ of ฮจ is global if W = B.

We recall that a minimal set of a homeomorphism group ฮจ of a topological space B is a nonempty closed invariant subset ๐’ฆ with respect to ฮจ containing no proper subsets with this property. A finite minimal set is called trivial.

Let ๐’ฆ be a minimal set of a diffeomorphism group ฮจ of a manifold B. The minimal set ๐’ฆ with empty interior is called exceptional if ๐’ฆ is a Cantor set, and exotic if ๐’ฆ is not a totally disconnected topological subspace of B.

It is well known that the conformal group Con f (๐•Šq), q โ‰ฅ 2, is isomorphic to the Mobius group Mobq(โ„) [24].

The limit set ฮ›(ฮจ) of an arbitrary subgroup ฮจ of Con f (๐•Šq) coincides with the intersection of the closures of all non-one-point orbits of this group, that is, ฮ›(ฮจ) = โˆฉ Cl(ฮจ.z), where z โˆˆ ๐•Šq and ฮจ.z โ‰  z. If the limit set ฮ›(ฮจ) is finite, then either it is empty or it consists of one or two points. In this case, the group ฮจ is called elementary.

Using some facts about subgroups of Con f (๐•Šq) [17], we get the following statement.

Proposition 11

Let ฮจ be a countable essential subgroup of Con f (๐•Šq). Then ฮจ has a global attractor which coincides with the limit set ฮ›(ฮจ) of ฮจ.

5.3 Global attractors of complete conformal foliations

The following theorem is the main result of our works [28] and [29]. Emphasize that the Ehresmann connection for (M,F) is one of the main technical tools used in the proof.

Theorem 12

A complete non-Riemannian conformal foliation (M,F) of codimension q, q โ‰ฅ 3, has a global attractor โ„ณ such that one of the following conditions holds:

  • โ„ณ is either a closed leaf or a union of two leaves.
  • โ„ณ is either a nontrivial minimal set, or contains a closed leaf. The induced foliation (M0,FM0) on M0 = M \ โ„ณ is Riemannian with Ehresmann connection. The closureLยฏin M of any L โŠ‚ M0is equal to โ„’ โˆช โ„ณ, where โ„’ is an embedded submanifold of M, and โ„’ coincides with the closure of L in M0.

Remark 5

Sufficient conditions for the existence of a global attractor for transversely similar pseudo-Riemannian foliations are obtained in [31].

5.4 The global holonomy group

The following theorem [29, Theorem 5] describes the global structure of the foliated manifold and establishes a relationship between the attractors of conformal and transversely similar foliations and the attractors of their global holonomy groups.

Theorem 13

Let (M,F) be a non-Riemannian complete conformal foliation (M,F). Then one of the following two statements holds.

  1. 1)The foliation (M,F) is a complete transversely similar foliation with the only minimal set โ„ณ, which is a global attractor and contains all leaves with essential holonomy group.
  2. 2)The foliation (M,F) is covered by a bundle : r:M^โ†’Sq , wheref:M^โ†’Mis a regular covering map. A group homomorphism
    ฯ‡:ฯ€1(M,x)โ†’Conโ€‰f(Sq)
    of the fundamental group ฯ€1(M,x) to the group Con f (๐•Šq) of conformal transformations of ๐•Šq is induced, and the group of covering transformations of the coveringf:M^โ†’Mis isomorphic to the group ฮจ := ฯ‡(ฯ€1(M,x)). Moreover, in this case one of the following two statements holds.
    1. (i)The foliation (M,F) has a global attractor โ„ณ which is either one closed leaf or two closed leaves.
    2. (ii)There exists a global attractor โ„ณ that coincides with the closure of a leaf with essential holonomy group, and โ„ณ = f (rโˆ’1(๐’ฆ)) where ๐’ฆ = ฮ›(ฮจ) is the limit set of ฮจ.

In both cases 1) and 2) the induced foliation (M0,FM0) on the complement M0 := M \ โ„ณ is a Riemannian foliation with Ehresmann connection. The closures of its leaves in M0form a Riemannian foliation with singularities, and the closureLยฏin M of a leaf L of (M0,FM0) equals ๐” โˆช โ„ณ, where ๐” is an embedded submanifold of M. The set ๐” coincides with the closure of the leaf L in M0.

Remark 6

Just like Theorem 8, the statement of the Theorem 13 holds for transversely similar Riemannian foliations of every codimension q โ‰ฅ 1.

Definition 10

The group ฮจ := ฯ‡(ฯ€1(M,x)) specified in Theorem 13, is called the global holonomy group of the conformal foliation (M,F).

The following statement is proved in a constructive way for q โ‰ฅ 3 [29, Theorem 7]. For q = 2, the proof is similar.

Theorem 14

Every countable subgroup ฮจ of Con f (๐•Šq), where q โ‰ฅ 2, can be realized as the global holonomy group of some two-dimensional conformal foliation (M,F) of codimension q.

If ฮจ is finitely generated, then such a foliation (M,F) exists on a closed manifold M of dimension (q + 2).

6 Strong transverse equivalence of foliations

6.1 Invariants with respect to strong transverse equivalence of conformal foliations

The notion of strong transverse equivalence of foliations was introduced and investigated by us in [30] under the name โ€œtransverse equivalenceโ€.

A continuous map p : X โ†’ Y has the covering homotopy property with respect to a topological space K if, for any continuous map G0 : K โ†’ X and any homotopy Ht : K โ†’ Y, t โˆˆ [0,1], such that p โ—‹ G0 = H0, there exists an extension of G0 to a homotopy Gt : K โ†’ X satisfying the equality p โ—‹ Gt = Ht, t โˆˆ [0,1].

We recall that a Serre fibration is a continuous surjective map having the covering homotopy property with respect to any finite polyhedron. It is known that for Serre fibrations it is possible to construct the exact homotopy sequence for a fibration. It is also well known that any locally trivial fibration is a Serre fibration. If a Serre fibration is a submersion, then it is called a smooth Serre fibration.

Definition 11
Foliations (M,F) and (Mโ€ฒ,Fโ€ฒ) are strong transversely equivalent if there exist a foliation (๐•„,๐”ฝ) and surjective submersions p : ๐•„ โ†’ M and pโ€ฒ : ๐•„ โ†’ Mโ€ฒ, which are Serre fibrations, with connected fibers such that
F:={pโˆ’1(L)โ€Š|โ€ŠLโˆˆF}={pโ€ฒโˆ’1(Lโ€ฒ)โ€Š|โ€ŠLโ€ฒโˆˆFโ€ฒ}.

In comparison with the notion of transverse equivalence introduced by P. Molino [20, p. 63], Definition 11 contains an additional requirement that the submersions p : ๐•„ โ†’ M and pโ€ฒ : ๐•„ โ†’ Mโ€ฒ are Serre fibrations, i.e., have the covering homotopy property. As shown in [30, Proposition 4.1], the strong transverse equivalence is indeed an equivalence relation.

Using this additional requirement, we proved that the strong transverse equivalence of foliations covered by fibrations can be realized by a foliation covered by a fibration. This fact was used in the proofs of the following theorems in [30]. As we have shown, in general, this is not true for transverse equivalent foliations in the sense of P. Molino.

Theorem 15

Two complete conformal, but not transversally similar foliations (M1, F1) and (M2, F2) of codimension q โ‰ฅ 3 are strong transversely equivalent if and only if their global holonomy groups ฮจ1and ฮจ2coincide (up to conjugacy in Con f (๐•Šq)).

Theorem 16

Two complete non-Riemannian transversally similar foliations (M1, F1) and (M2, F2) of codimension q โ‰ฅ 1 are strong transversely equivalent if and only if their global holonomy groups ฮจ1and ฮจ2coincide (up to conjugacy in Sim(๐”ผq)).

The structure Lie algebra ๐”ค0 of a complete non-Riemannian conformal foliation (M,F) of codimension q โ‰ฅ 3 is isomorphic to the Lie algebra of the Lie group ฮจยฏ , where ฮจยฏ is the closure of the global holonomy group ฮจ of this foliation in the Lie group Con f (๐•Šq). If (M,F) is a transversally similar Riemannian foliation of codimension q โ‰ฅ 1, then ๐”ค0 is the Lie algebra of the closure ฮจยฏ of ฮจ in the Lie group Sim(๐”ผq). In particular, the structure Lie algebra ๐”ค0 is equal to zero if and only if ฮจ is a discrete subgroup.

From Theorems 15 and 16 we obtain the following assertion.

Proposition 17

The Lie group ฮจ, the structure Lie algebra ๐”ค0and the limit set ฮ›(ฮจ) of the global holonomy group ฮจ of a complete non-Riemannian conformal foliation (M, F) are invariants with respect to the strong transverse equivalence.

6.2 Suspension foliations

A. Haefliger introduced the construction of a suspension foliation. Let B and T be smooth connected manifolds, and ฯ : ฯ€1(B,b) โ†’ Diff (T) be a group homomorphism. Let G := ฯ€1(B,b) and ฮจ := ฯ(G). Consider a universal covering map p^:B^โ†’B . Define a right action of the group G on the product B^ร—T as
ฮ˜:B^ร—Tร—Gโ†’B^ร—T,โ€Šโ€Šโ€Šโ€Šโ€Šโ€Šฮ˜(x,t,g)=(xg,ฯ(gโˆ’1)(t)),
where B^โ†’B^:xโ†’xg is the deck transformation of the covering p^ induced by an element g โˆˆ G, which acts on B^ on the right. The map p:M:=(B^ร—T)/Gโ†’B=B^/G is a locally trivial bundle over B with standard fiber T. It is associated with the principal bundle p^:B^โ†’B with the structural group G. Let ฮ˜g:=ฮ˜|B^ร—tร—g . Since
ฮ˜g(B^ร—t)=B^ร—ฯ(gโˆ’1)(t)โ€Šโ€Šโ€Šโ€Šโ€Šโ€Šโ€Šโ€Šโ€Šโˆ€tโˆˆT,
the action of G preserves the trivial foliation F:={B^ร—tโ€Š|โ€ŠtโˆˆT} on the product B^ร—T . Therefore the quotient map
f0:B^ร—Tโ†’(B^ร—T)/G=M
induces a smooth foliation F on M whose leaves are transversal to fibers of the bundle p : M โ†’ B. The pair (M,F) is called a suspension foliation and is denoted by Sus(T,B,ฯ). The group ฮจ := ฯ(G) of diffeomorphisms of the manifold T is said to be the structural group of the suspension foliation (M,F). When T is simply connected, ฮจ is called the global holonomy group of (M,F).

Using suspension foliations, we proved the following theorem. Note that the same method is used in the construction of Example 21.

Theorem 18

Every strong transverse equivalence class of a non-Riemannian conformal foliation contains a two-dimensional conformal foliation, which is a suspension foliation.

7 Examples

Let Bk be a smooth closed 3-manifold, which is homeomorphic to the connected sum โ™ฏi=1kS1ร—S2 of k copies of S1 ร— S2. Its fundamental group ฯ€1(Bk,b) = < g1,..., gk > is the free group of rank k.

Example 19

Recall first the definition of the Menger curve. Let S = I ร—I be the unit square. Let us divide it into 9 squares of size 1/3 ร— 1/3, then remove the open square (1/3,2/3) ร— (1/3,2/3). Then repeat this procedure with each of the remaining eight squares and continue by induction. As a result, a countable family of open squares will be removed from S and we get a compact subset of S, called the Sierpinski carpet. Let Q = I ร— I ร— I be the unit cube. Consider a copy ๐’ฎj of the Sierpinski carpet on each face Pj of Q,j=1,6ยฏ . Let pj : Q โ†’ Pj be the orthogonal projection. Then the setโ„ณ:=โˆฉpjโˆ’1(๐’ฎj)is called the Menger curve. As well-known, the topological dimension of the Menger curve โ„ณ is equal to 1.

For k โ‰ฅ 6 and m โ‰ฅ 3, consider the group
ฮ“km=<si,i=1,...,kโ€Š|โ€Šsim=1,[si,si+1]=1>.
Bourdon proved [9] that ifsinฯ€/k<1/m , then there is an exact representation ฮฑkm : ฮ“km โ†’ Con f (๐•Š2mโˆ’2) such that the group ฮจkm = ฮฑkm(ฮ“km) is a Fuchsian group (i.e. a finitely generated discrete subgroup of Con f (๐•Š2mโˆ’2)), and the limit set ฮ›(ฮจkm) of ฮจkm is homeomorphic to the Menger curve. Thus ฯˆi := ฮฑ (si) are generators of ฮจkm.

Using the fact that sinx < x for all x > 0, it is easy to check that the conditionsinฯ€/k<1/mholds ifk>ฯ€m . Letk(m)=[ฯ€m]be the integer part ofฯ€m . Then, for any fixed natural m โ‰ฅ 3, there exists a countable family of Fuchsian groups ฮจkm, k > k(m), such that ฮ›(ฮจkm) is homeomorphic to the Menger curve. For example k(3) = 5, k(4) = 6, k(5) = k(6) = 7, k(7) = k(8) = 8, k(9) = k(10) = 9.

Thus, ฮ›(ฮจkm) is an exotic minimal set of the group ฮจkm in S2mโˆ’2.

LetBk:=โ™ฏi=1kS1ร—S2be as above. Consider a group homomorphism ฯkm : ฯ€1 (Bk,b) โ†’ ฮจkm defined on generators by ฯkm(gi) := ฮฑkm(si), i=1,kยฏ . Let (Mkm,Fkm) := Sus(๐•Š2mโˆ’2,Bk,ฯkm), m โ‰ฅ 3, be the corresponding suspension foliation. Since ฮจkm is a subgroup of Con f (๐•Š2mโˆ’2), (Mkm,Fkm) is a complete conformal foliation of codimension q = 2m โˆ’ 2 โ‰ฅ 4. Hence, according to Theorem 13, โ„ณ := f (rโˆ’1(ฮ›(ฮจkm))) is an exotic minimum set and a global attractor of the foliation (Mkm,Fkm).

Example 20

Letโ„ฌ1+,โ€ฆ,โ„ฌk+ , โ„ฌ1โˆ’,โ€ฆ,โ„ฌkโˆ’be a finite set of disjoint closed topological balls with smooth boundary in the sphere ๐•Šq, and q โ‰  4. Let ฯˆi โˆˆ Con f (๐•Šq) be a conformal transformation such thatฯˆi(int(โ„ฌi+))=ext(โ„ฌiโˆ’) . The group ฮจ generated by ฯˆ1,...,ฯˆk is called the Schottky group. As well-known (see, for example, [17]), the Schottky group ฮจ is the free group of rank k, i.e. ฮจ =< ฯˆ1,...,ฯˆk >, and ฮจ has a minimal set ฮ›(ฮจ) homeomorphic to a Cantor subset of the segment [0,1]. Therefore, the topological dimension ฮ›(ฮจ) is equal to zero.

LetBk:=โ™ฏi=1kS1ร—S2be as above. Define a group isomorphism ฯk : ฯ€1 (Bk,b) โ†’ ฮจ, setting ฯk(gi) = ฯˆi,i=1,kยฏ . The suspension foliation (Mk,Fk) := Sus(๐•Šq,Bk,ฯk) is a complete conformal foliation and, according to Theorems 13, (Mk,Fk) has a global attractor, representing an exceptional minimal set.

Iffk:Mหœkโ†’Mkis the universal covering map, then the induced foliationFหœk=fk*Fkis given by the fibers of a locally trivial bundlerk:Mหœkโ†’Sq . As (Mk,Fk) is a suspended foliation, (Mk,Fk) admits an Ehresmann connection ๐”k.

LetMk0:=Mk\โ„ณ . Therefore, Mk0=fk(rkโˆ’1(Sq\ฮ›(ฮจ))) . Since the group ฮจ acts freely and properly discontinuously on the open dense subset ๐•Šq\ฮ›(ฮจ) in ๐•Šq, the setMk0is an open saturated dense submanifold of the compact manifold Mk. Hence, the leaf spaceMk0/FMk0is the smooth q-dimensional manifold which is homeomorphic to (๐•Šq \ ฮ›(ฮจ))/ฮจ. The induced foliation(Mk0,FMk0)is Riemannian, and the restrictionMk|Mk0is an Ehresmann connection for(Mk0,FMk0) . Therefore, Theorem 10 implies that(Mk0,FMk0)is formed by fibres of a Riemannian submersionh:Mk0โ†’Nk , where Nk as well as (๐•Šq \ ฮ›(ฮจ))/ฮจ is homeomorphic to the connected sum of k copies of ๐•Šqโˆ’1 ร— ๐•Š1.

Note that the closureLฮฑยฏin M of a leafLฮฑโŠ‚Mk0is equal to Lฮฑ โˆช โ„ณ, what agrees with Theorem 12.

Example 21

Let ๐”ผq be the q-dimensional Euclidean space and ๐”ผk, 2 โ‰ค k < q, be its k-dimensional subspace. Let ei,i=1,kยฏ , be a basis of vector space โ„k corresponding to ๐”ผk. Consider a subgroup ฮจ of the group of similarities Sim(๐”ผq) generated by transformations ฯˆj,j=1,k+1ยฏ , where ฯˆj(z) := z + ej forj=1,kยฏand ฯˆk+1(z) := ฮป ยท z for some number ฮป โˆˆ (0,1) and all z โˆˆ ๐”ผq. Note that ฮจ is essential group of similar transformations, and the limit set ฮ›(ฮจ) of ฮจ is equal to ๐”ผk. Moreover, ฮ›(ฮจ) is a minimal set and a global attractor of ฮจ.

Denote bySm2the smooth two-dimensional sphere with m, m := k + 1, handles. The fundamental group ofSm2is equal to{ai,biโ€Š|โ€Ši=1,mยฏ;โ€Ša1b1a1โˆ’1b1โˆ’1...ambmamโˆ’1bmโˆ’1=1} .

LetB:=Sm2 , T := ๐”ผq and define the group homomorphism ฯ : ฯ€1(B,b) โ†’ Sim(๐”ผq) by setting it on generators: ฯ(ai) := ฯˆi,i=1,mยฏ , ฯ(bi) := Id๐”ผqwhere Id๐”ผq is the identical map of ๐”ผq. Then(M,F):=Sus(Eq,Sm2,ฯ)is a suspension two-dimensional non-Riemannian transversely similar foliation. Let f : โ„2 ร— ๐”ผq โ†’ M be the universal covering map. Then (M,F) is covered by the trivial bundle r : โ„2 ร— ๐”ผq โ†’ ๐”ผq and has the global holonomy group ฮจ. By properties of suspension foliations, the (q + 2)-dimensional manifold M is the space of a locally trivial bundlep:Mโ†’Sm2with a standard fibre ๐”ผq over the baseSm2 . Therefore M is not compact.

Thus, we get a transversely similar foliation (M,F) of codimension q, q โ‰ฅ 3, with a regular minimal set โ„ณ := f (rโˆ’1(๐”ผk)), and โ„ณ is a global attractor and a minimal set of (M,F). According to [27, Theorem 9], โ„ณ and M are homotopy equivalent.

Let M0 := M \ โ„ณ and Lฮฑ be an arbitrary leaf in M0. Emphasize that the induced foliation (M0,FM0) is an improper Riemannian foliation without holonomy, admitting an Ehresmann connection. According to Theorem 12, the closureLฮฑยฏis equal to โ„’ฮฑ โˆช โ„ณ, where โ„’ฮฑ is the closure of Lฮฑ in M0. Note that if k < q โˆ’ 1, then M0is connected and dense in M. In the case when k = q โˆ’ 1, the submanifold M0is dense and has two connected components denoted byM0+andM0โˆ’ , hence โ„’ฮฑ belongs toM0+orM0โˆ’ , respectively. We emphasize that all โ„’ฮฑ are pairwise diffeomorphic, that is, โ„’ฮฑ does not depend on ฮฑ. Therefore, the closureLฮฑยฏof any leaf Lฮฑ โŠ‚ M0is also independent of ฮฑ.

Example 22

Fix a point a โˆˆ ๐•Šq. Let f : ๐•Šq โ†’ โ„q โˆช {โˆž} be the homeomorphism defined via the stereographic projection ๐•Šq \ {a} โ†’ โ„q, and f (a) = {โˆž}. Identify ๐•Šq with โ„q โˆช {โˆž} through f. The map f induces a group isomorphismfหœ:Confa(Sq)โ†’Sim(Eq)of the stationary subgroup Con fa(๐•Šq) of the group Con f (๐•Šq) at point a onto the similarity group Sim(๐”ผq).

Let ฮจ โŠ‚ Sim(๐”ผq) be the similarity group defined in Example 21. Henceฮจหœ:=fหœโˆ’1(ฮจ)is a countable subgroup of Con fa(๐•Šq), andฯˆjหœ:=fหœโˆ’1(ฯˆj) , j=1,k+1ยฏ , are generators ofฮจหœ .

LetB:=Sm2be as above, T := ๐•Šq and define the group homomorphism ฯ : ฯ€1(B,b) โ†’ Con f (๐•Šq) by setting it on generators:ฯ(ai):=ฯˆหœi,i=1,k+1ยฏ , ฯ(bi) := Id๐•Šq. Then we get the suspension foliation(M,F):=Sus(Sq,Sm2,ฯ)which is non-Riemannian conformal foliation of codimension q, andฮจหœis its global holonomy group.

The universal covering manifold for M is the product โ„2 ร— ๐•Šq. Denote by h : โ„2 ร— ๐•Šq โ†’ M the universal covering map. Let r : โ„2 ร— ๐•Šq โ†’ ๐•Šq be the canonical projection.

Since the global holonomy groupฮจหœof (M,F) has a fixed point a, the leaf L := h(rโˆ’1(a)) is a unique closed leaf and a global regular attractor of (M,F). Emphasize that L is diffeomorphic to the baseB:=Sm2 .

According to the definition of a limit set,ฮ›(ฮจหœ)=fโˆ’1(โ„kโˆช{โˆž}) . Therefore,ฮ›(ฮจหœ)is the canonically embedded k-dimensional sphere in ๐•Šq containing the point a. As it was shown in Example 21, ฮ›(ฮจ) is the global attractor of the respective transversely similar foliation, henceฮ›(ฮจหœ)is a global attractor of the conformal foliation (M,F). Sinceฮ›(ฮจหœ)contains a fixed point a,ฮ›(ฮจหœ)is not a minimal set. We emphasize that for every pointzโˆˆฮ›(ฮจหœ)\{a}the orbitฮจหœ.z . z is dense inฮ›(ฮจหœ) . This implies that
โ„ณ:=h(rโˆ’1(ฮ›(ฮจหœ))
is a global transitive regular attractor of the conformal foliation (M,F).

Thus we construct a conformal foliation having two global regular attractors L and โ„ณ, L โŠ‚ โ„ณ, such that L is a closed leaf and โ„ณ is transitive attractor which is not a minimal set.

Acknowledgements

The work was supported by the grant 17-11-01041 of the Russian Science Foundation with the exception of Sections 6 and 7. The work on Sections 6 and 7 is supported by Laboratory of Dynamical Systems and Applications NRU HSE, of the Ministry of science and higher education of the RF grant ag. N 075-15-2019-1931.

References

  • [1]โ†‘

    D.V. Alekseevskii (1972), Groups of conformal transformations of Riemannian spaces, Math. USSR-Sb., 18(2), 285โ€“301.

    • Crossref
    • Export Citation
  • [2]โ†‘

    D.V. Alekseevskii (1973), Sn and En are the only Riemannian spaces that admit an essential conformal transformation, Uspekhi Mat. Nauk, 28(5), 225โ€“226.

  • [3]โ†‘

    I. Androulidakis and G. Skandalis (2009), The holonomy groupoid of a singular foliation, J. Reine Angew. Math., 626, 1โ€“37.

  • [4]โ†‘

    S.Kh. Aranson, V.Z. Grines (1978), On the representation of minimal sets of currents on two-dimensional manifolds by geodesics, Math. USSR-Izv., 12(1), 103โ€“124.

    • Crossref
    • Export Citation
  • [5]โ†‘

    S.Kh. Aranson, V.Z. Grines (1986), Topological classification of flows on closed two-dimensional manifolds, Russian Math. Surveys, 41(1), 183โ€“208.

    • Crossref
    • Export Citation
  • [6]โ†‘

    P. Baird and M. Eastwood (2013), CR geometry and conformal foliations, Ann. Global Anal. Geom., 44(1), 73โ€“90.

    • Crossref
    • Export Citation
  • [7]โ†‘

    J.-C. Beniere, G. Meigniez (1999), Flows without minimal set, Erg. Th. and Dyn. Sys., 19(1), 1โ€“30.

  • [8]โ†‘

    R.A. Blumenthal, J.J. Hebda (1984), Ehresmann connections for foliations, Indiana Univ. Math. J., 33(4), 597โ€“611.

    • Crossref
    • Export Citation
  • [9]โ†‘

    M. Bourdon (1997), Sur la dimension de Hausdorff de lโ€™ensemble limite dโ€™une famille de sous-groupes convexes co-compacts, C. R. Acad. Sci. Paris Ser. I Math., 325(10), 1097โ€“1100.

    • Crossref
    • Export Citation
  • [10]โ†‘

    A. Cap, J. Slovak (2009), Parabolic Geometries I: Background and General Theory, AMS Publishing House, Math. Surveys Monogr., 154, 1โ€“628.

  • [11]โ†‘

    B. Deroin, V. Kleptsy(2007), Random Conformal Dynamical Systems. Geometric and functional analysis, 17(4), 1043โ€“1105.

    • Crossref
    • Export Citation
  • [12]โ†‘

    J. Ferrand (1977), Sur un lemme dโ€™Alekseevskii relatif aux transformations conformes, C. R. Acad. Sc. Paris, 284, 121โ€“123.

  • [13]โ†‘

    J. Ferrand (1996), The action of conformal transformations on a Riemannian manifold, Math. Ann., 304(2), 277โ€“291.

    • Crossref
    • Export Citation
  • [14]โ†‘

    C. Frances, C. Tarquini (2007), Autour du theoreme de Ferrand-Obata, Ann. Glob. Anal. Geom., 21(1), 51โ€“62.

  • [15]โ†‘

    A. Haefliger (1985), Pseudogroups of local isometries, Research Notes in Math., 131, 174โ€“197.

  • [16]โ†‘

    T. Inaba (1999), An example of a flow on a non-compact surface without minimal set, Erg. Th. and Dyn. Sys., 19(1), 31โ€“33.

    • Crossref
    • Export Citation
  • [17]โ†‘

    M. Kapovich (2007), Kleinian groups in higher dimensions, Geometry and dynamics of groups and spaces, Progr. Math., 265, Birkhauser, Basel 2007, 487โ€“564.

  • [18]โ†‘

    M.S. Kulikov (2004), Schottky-type groups and minimal sets of horocycle and geodesic flows, Sb. Math., 195(1), 35โ€“64.

    • Crossref
    • Export Citation
  • [19]โ†‘

    G. Levitt (1983), Foliations and laminations on hyperbolic surfaces, Topology, 22(2), 119โ€“135.

    • Crossref
    • Export Citation
  • [20]โ†‘

    P. Molino (1988), Riemannian Foliations, Progr. Math., 263, Birkhauser, Boston, MA.

  • [21]โ†‘

    S. Morita (1979), On characteristic classes of conformal and projective foliations, J. Math. Soc. Japan, 31(4), 693โ€“718.

    • Crossref
    • Export Citation
  • [22]โ†‘

    M. Obata (1971), The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom., 6(2), 247โ€“258.

    • Crossref
    • Export Citation
  • [23]โ†‘

    E. Salem (1988), Riemannian foliations and pseudogroups of isometries, Application D in: P. Molino, Riemannian foliations, Progr. Math., 263, Birkhauser, Boston, MA, 1988.

  • [24]โ†‘

    R.W. Sharpe (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Progpam, Grad. Texts in Math. 166, New York: Springer-Verlag.

  • [25]โ†‘

    I. Tamura (1992), Topology of foliations: An Introduction, Transl. of Math. Monographs, 97, Publisher: AMS.

  • [26]โ†‘

    C. Tarquini (2004), Feuilletages conformes, Ann. Inst. Fourier, 52(2), 453โ€“480.

  • [27]โ†‘

    N.I. Zhukova (2007), Minimal sets of Cartan foliations, Proc. Steklov Inst. Math., 256(1), 105โ€“135.

    • Crossref
    • Export Citation
  • [28]โ†‘

    N.I. Zhukova (2011), Attractors and an analog of the Lichnerowicz conjecture for conformal foliations, Siberian Math. J., 52(3), 436โ€“450.

    • Crossref
    • Export Citation
  • [29]โ†‘

    N.I. Zhukova (2012), Global attractors of complete conformal foliations. Sb. Math., 203(3), 380โ€“405.

    • Crossref
    • Export Citation
  • [30]โ†‘

    N.I. Zhukova (2015), Transverse Equivalence of Complete Conformal Foliations J. Math. Sci., 208(1), 115โ€“130.

    • Crossref
    • Export Citation
  • [31]โ†‘

    N.I. Zhukova (2018), The existence of attractors of Weyl foliations modelled on pseudo-Riemannian manifolds, JPCS, 990(1), 1โ€“15.

  • [32]โ†‘

    N.I. Zhukova (2018), The structure of Riemannian foliations with Ehresmann connection, Zh. Sredn. Mat. Obshch., 20(4), 395โ€“407 (in Russian).

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    D.V. Alekseevskii (1972), Groups of conformal transformations of Riemannian spaces, Math. USSR-Sb., 18(2), 285โ€“301.

    • Crossref
    • Export Citation
  • [2]

    D.V. Alekseevskii (1973), Sn and En are the only Riemannian spaces that admit an essential conformal transformation, Uspekhi Mat. Nauk, 28(5), 225โ€“226.

  • [3]

    I. Androulidakis and G. Skandalis (2009), The holonomy groupoid of a singular foliation, J. Reine Angew. Math., 626, 1โ€“37.

  • [4]

    S.Kh. Aranson, V.Z. Grines (1978), On the representation of minimal sets of currents on two-dimensional manifolds by geodesics, Math. USSR-Izv., 12(1), 103โ€“124.

    • Crossref
    • Export Citation
  • [5]

    S.Kh. Aranson, V.Z. Grines (1986), Topological classification of flows on closed two-dimensional manifolds, Russian Math. Surveys, 41(1), 183โ€“208.

    • Crossref
    • Export Citation
  • [6]

    P. Baird and M. Eastwood (2013), CR geometry and conformal foliations, Ann. Global Anal. Geom., 44(1), 73โ€“90.

    • Crossref
    • Export Citation
  • [7]

    J.-C. Beniere, G. Meigniez (1999), Flows without minimal set, Erg. Th. and Dyn. Sys., 19(1), 1โ€“30.

  • [8]

    R.A. Blumenthal, J.J. Hebda (1984), Ehresmann connections for foliations, Indiana Univ. Math. J., 33(4), 597โ€“611.

    • Crossref
    • Export Citation
  • [9]

    M. Bourdon (1997), Sur la dimension de Hausdorff de lโ€™ensemble limite dโ€™une famille de sous-groupes convexes co-compacts, C. R. Acad. Sci. Paris Ser. I Math., 325(10), 1097โ€“1100.

    • Crossref
    • Export Citation
  • [10]

    A. Cap, J. Slovak (2009), Parabolic Geometries I: Background and General Theory, AMS Publishing House, Math. Surveys Monogr., 154, 1โ€“628.

  • [11]

    B. Deroin, V. Kleptsy(2007), Random Conformal Dynamical Systems. Geometric and functional analysis, 17(4), 1043โ€“1105.

    • Crossref
    • Export Citation
  • [12]

    J. Ferrand (1977), Sur un lemme dโ€™Alekseevskii relatif aux transformations conformes, C. R. Acad. Sc. Paris, 284, 121โ€“123.

  • [13]

    J. Ferrand (1996), The action of conformal transformations on a Riemannian manifold, Math. Ann., 304(2), 277โ€“291.

    • Crossref
    • Export Citation
  • [14]

    C. Frances, C. Tarquini (2007), Autour du theoreme de Ferrand-Obata, Ann. Glob. Anal. Geom., 21(1), 51โ€“62.

  • [15]

    A. Haefliger (1985), Pseudogroups of local isometries, Research Notes in Math., 131, 174โ€“197.

  • [16]

    T. Inaba (1999), An example of a flow on a non-compact surface without minimal set, Erg. Th. and Dyn. Sys., 19(1), 31โ€“33.

    • Crossref
    • Export Citation
  • [17]

    M. Kapovich (2007), Kleinian groups in higher dimensions, Geometry and dynamics of groups and spaces, Progr. Math., 265, Birkhauser, Basel 2007, 487โ€“564.

  • [18]

    M.S. Kulikov (2004), Schottky-type groups and minimal sets of horocycle and geodesic flows, Sb. Math., 195(1), 35โ€“64.

    • Crossref
    • Export Citation
  • [19]

    G. Levitt (1983), Foliations and laminations on hyperbolic surfaces, Topology, 22(2), 119โ€“135.

    • Crossref
    • Export Citation
  • [20]

    P. Molino (1988), Riemannian Foliations, Progr. Math., 263, Birkhauser, Boston, MA.

  • [21]

    S. Morita (1979), On characteristic classes of conformal and projective foliations, J. Math. Soc. Japan, 31(4), 693โ€“718.

    • Crossref
    • Export Citation
  • [22]

    M. Obata (1971), The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom., 6(2), 247โ€“258.

    • Crossref
    • Export Citation
  • [23]

    E. Salem (1988), Riemannian foliations and pseudogroups of isometries, Application D in: P. Molino, Riemannian foliations, Progr. Math., 263, Birkhauser, Boston, MA, 1988.

  • [24]

    R.W. Sharpe (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Progpam, Grad. Texts in Math. 166, New York: Springer-Verlag.

  • [25]

    I. Tamura (1992), Topology of foliations: An Introduction, Transl. of Math. Monographs, 97, Publisher: AMS.

  • [26]

    C. Tarquini (2004), Feuilletages conformes, Ann. Inst. Fourier, 52(2), 453โ€“480.

  • [27]

    N.I. Zhukova (2007), Minimal sets of Cartan foliations, Proc. Steklov Inst. Math., 256(1), 105โ€“135.

    • Crossref
    • Export Citation
  • [28]

    N.I. Zhukova (2011), Attractors and an analog of the Lichnerowicz conjecture for conformal foliations, Siberian Math. J., 52(3), 436โ€“450.

    • Crossref
    • Export Citation
  • [29]

    N.I. Zhukova (2012), Global attractors of complete conformal foliations. Sb. Math., 203(3), 380โ€“405.

    • Crossref
    • Export Citation
  • [30]

    N.I. Zhukova (2015), Transverse Equivalence of Complete Conformal Foliations J. Math. Sci., 208(1), 115โ€“130.

    • Crossref
    • Export Citation
  • [31]

    N.I. Zhukova (2018), The existence of attractors of Weyl foliations modelled on pseudo-Riemannian manifolds, JPCS, 990(1), 1โ€“15.

  • [32]

    N.I. Zhukova (2018), The structure of Riemannian foliations with Ehresmann connection, Zh. Sredn. Mat. Obshch., 20(4), 395โ€“407 (in Russian).