Basic automorphism of Cartan foliations covered by fibrations

The basic automorphism group ${A}_B(M,F)$ of a Cartan foliation $(M, F)$ is the quotient group of the automorphism group of $(M, F)$ by the normal subgroup, which preserves every leaf invariant. For Cartan foliations covered by fibrations, we find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in their basic automorphism groups. Estimates of the dimension of these groups are obtained. For some class of Cartan foliations with integrable an Ehresmann connection, a method for finding of basic automorphism groups is specified.


Introduction
One of the main objects associated with a geometric structure on a smooth manifold is its automorphism group. In the introduction to the monograph by S. Kobayashi [11], it was emphasized that the existence of a structure of a finite-dimensional Lie group in the group of automorphisms of a manifold with a geometric structure is one of the central problems in differential geometry.
As is known, the solved Hilbert's 5-th problem is devoted to finding conditions under which a topological group admits the structure of a Lie group [16]. It is known from the numerous works of E. Cartan, R. Mayer, H. Steenrod, K. Nomizu, S. Kabayashi, S. Ehresmann and other authors that the automorphism groups of many geometric structures are Lie groups of transformations (see overview [7]).
Let (M, F ) be a smooth foliation. Recall that geometry structure on the manifold M is called transverse to (M, F ) if it is a invariant with respect to local holonomic diffeomorphisms. Another, equivalent definition of a transverse geometric structure, which is represented by Cartan geometry, is given in Section 3. Morphisms are understood as local diffeomorphisms mapping leaves onto leaves and preserving transverse geometries (the precise definition see in Section 3). Let us denote by CF the category of Cartan foliations. This paper is devoted to the investigation of automorphism groups of Cartan foliations, i.e. foliations that admit Cartan geometries as transverse structures. The study of Cartan foliations is motivated by the fact that such broad classes of foliations as parabolic, conformal, projective, pseudo-Riemannian, Lorentzian, Weyl, transverse homogeneous foliations and foliations with transverse linear connection belong to Cartan foliations. Therefore, the investigation of Cartan foliations allows us to study the general properties of these foliations from a single point of view, while many authors study them separately. Let  We study the groups of basic automorphisms A B (M, F ) of Cartan foliations (M, F ) covered by fibration and find sufficient conditions for the existence of a structure of a finite-dimensional Lie group in the group A B (M, F ). J. Leslie [12] was the first who solved a similar problem for smooth foliations on compact manifolds and considered an application to foliations with transverse G-structures. For foliations with complete transversely projectable affine connection, this problem was raised by I.V. Belko [2]. Foliations (M, F ) with effective transverse rigid geometries were investigated by N.I. Zhukova [19] where an algebraic invariant g 0 = g 0 (M, F ), called the structural Lie algebra of (M, F ), was constructed and it was proved that g 0 = 0 is a sufficient condition for the existence of a unique Lie group structure in the basic automorphism group of this foliation. In [15], the existence of a Lie group structure was investigated in the basic automorphism groups of Cartan foliations modeled on inefficient Cartan geometries.

Main results
Among the Cartan foliations, foliations covered by fibrations are distinguished. The following theorem describes the global structure of Cartan foliations covered by fibrations. (1) there exists a regular covering map κ : M → M such that the induced foliation F is made up of fibres of the locally trivial bundle r : M → B over a simply connected manifold B, and ξ induces on B a Cartan geometry η that is locally isomorphic to ξ; (2) an epimorphism χ : π 1 (M, x) → Ψ, x ∈ M, of the fundamental group π 1 (M, x) onto a subgroup Ψ of the automorphism Lie group Aut(B, η) of the Cartan manifold (B, η) is determined; (3) the group of deck transformations of the covering κ : M → M is isomorphic to the group Ψ. We give a detailed proof of the following theorem, formulated without a proof in the work [15,Prop. 8]. Theorem 2 establishes a connection between the basic automorphism group A B (M, F ) of a Cartan foliation (M, F ) covered by fibration and its global holonomy group Ψ. The following theorem specifies a method for computing basic automorphism groups for Cartan foliations with an integrable Ehresmann connection. Using Theorem 3, we construct an example of computing the basic automorphism group of some conformal foliation of an arbitrary codimension q, where q ≥ 3, on a (q + 1)-dimensional manifold in Section 7.2. Some other examples are constructed in [15].

The category of Cartan foliations
The category of Cartan geometries Let G and H be Lie groups with the Lie algebras g and h relatively. Let H be a closed subgroup of G. A Cartan geometry of type (G, H) on the smooth manifold N is a principal H-bundle P (N, H) with a gvalued 1-form ω on P satisfying the following conditions: (c 1 ) the map ω u : T u P → g is an isomorphism of vector spaces for every u ∈ P ; (c 2 ) ω(A * ) = A for every A ∈ h, where A * is the fundamental vector field determined by A; is the adjoint representation of the Lie subgroup H of G in the Lie algebra g.
The g-valued form ω is called a Cartan connection form. This Cartan geometry is denoted by ξ = (P (N, H), ω). The pair (N, ξ) is called a Cartan manifold.
Maximal normal subgroup K of the group G belonging to H is called the kernel of pair (G, H). We denote the Lie algebra of the group K by k. The Cartan geometry ξ = (P (M, H), ω) of type (G, H) is called effective if the kernel K of the pair (G, H) is trivial. Further, we assume that all Cartan geometries under consideration are effective.
The projection γ is called an automorphism of the Cartan manifold (N, ξ). Denote by Aut(N, ξ) the full automorphism group of a Cartan foliation (N, ξ) and by Aut(ξ) the full automorphism group of a Cartan geometry ξ. Let A(P, ω) := {Γ ∈ Dif f (P ) | Γ * ω = ω} be the automorphism group of the parallelizable manifold (P, ω), which is known to be a Lie group, and dim(A(P, ω)) ≤ dim P.
Remark, that H} is a closed Lie subgroup of the Lie group A(P, ω). Therefore, Aut(ξ) is a Lie group, and due to the effictivity of a Cartan geometry ξ, there exists a Lie group isomorphism Cartan foliations Let N be a smooth q-dimensional manifold, the connectivity of which is not assumed. Let M be a smooth n-dimensional manifold, where 0 < q < n.
Two N-cocycles are called equivalent if there exists an N-cocycle containing both of these cocycles. Let respectively. All objects belonging to ζ ′ are distinguished by prime. Let f : M → M ′ be a smooth map which is a local isomorphism in the foliation category Fol. Hence for any x ∈ M and y := f (x) there exist neighborhoods U k ∋ x and U ′ s ∋ y from ζ and ζ ′ respectively, a diffeomorphism ϕ : Further we shall use the following notations: We say that f preserves transverse Cartan geometry if every such diffeomorphism . This means the existence of isomorphism Φ : P k → P ′ s in the category Car with the projection ϕ such that the following diagram is commutative. We emphasize that the indicated above isomorphism Φ : P k → P ′ s is unique if the transverse Cartan geometries are effective. The introduced concept is well defined, i. e., it does not depend of the choice of neighborhoods U k and U ′ k from the cocycles ζ and ζ ′ . The piecewise smooth integral curves of the distribution M are said to be horizontal, and the piecewise smooth curves in the leaves are said to be vertical. A piecewise smooth mapping H of the square I 1 × I 2 to M is called a vertical-horizontal homotopy if the curve H| {s}×I 2 is vertical for any fixed s ∈ I 1 and the curve H| I 1 ×{t} is horizontal for any fixed t ∈ I 2 . In this case, the pair of paths (H| I 1 ×{0} , H|

Classes of foliations covered by fibrations
(G, X)-foliations with an Ehresmann connection Let X be a smooth connected manifold and G be the Lie group of diffeomorphisms of X. Recall that the action of a group G on a manifold X is called quasi-analytically if for any open subset U ⊂ X and an element g ∈ G the equality g| U = id U implies g = e, where e = id X .
Assume that a Lie group G of diffeomorphisms of a manifold X acts on N quasi- . If, moreover, (X, ξ) is a Cartan manifold and the group G is a subgroup of the automorphism Lie group Aut(X, ξ), then (M, F ) is a Cartan (G, X)-foliation. It follows from [7, Section VI] that every Cartan (G, X)-foliations with Ehresmann connections is a foliation covered by fibration.
Cartan foliation with a vanishing transverse curvature Let (M, F ) be a Cartan foliation of type (G, H) with an Ehresmann connection. As is known [1, Section VI], if the transverse curvature of (M, F ) vanishes, then foliation (M, F ) is covered by fibration. Consequently, all the obtained results are valid for Cartan foliations with zero transverse curvature that admiting an Ehresmann connection.

Foliations with an integrable Ehresmann connection Recall that an Ehresmann connection M for a foliation (M, F ) is called integrable if the distribution M is integrable
i.e. if there exists the foliation such that T F t = M. Accoding to Kashiwabara's theorem [10], foliations with an integrable Ehresmann connection are covered by fibrations.

Suspended foliations
The construction of a suspension foliation was proposed by A. Haefliger and described in detail in [18]. Note that suspension foliations form a class of foliations with integrable Ehresmann connection and are covered by fibrations.
Cartan foliation of codimension q = 1 Any smooth one-dimensional distribution is integrable, so a Cartan foliation (M, F ) of codimension q = 1 with an Ehresmann connection is covered by fibration.  Let κ : K → K be the universal covering map, where K and K are smooth manifolds. By analogy with Theorem 28.10 in [4], it is easy to show that for any h ∈ Dif f (K) there exists h ∈ Dif f ( K) lying over h. For an arbitrary covering map the same statement is incorrect, in general. It is not difficult to prove the following criterion for the existence of lifts of arbitrary diffeomorphisms with respect to regular covers. Proposition 1. Let κ : K → K be a smooth regular covering map with the deck transformation group Γ. A diffeomorphism h ∈ Dif f ( K) lies over some diffeomorphism h ∈ Dif f (K) if and only if it satisfies the equality h • Γ = Γ • h.

Proof of Theorem 1
Suppose that a Cartan foliation (M, F ) modeled on an effective Cartan geometry ξ = (P (N, H), ω) is covered by a fibration r : M → B, where κ : M → M is the universal covering map. The fibration r : M → B has connected fibres and simply connected space M . Therefore, due to the application of the exact homotopic sequence for this fibration we obtain that the base manifold B is also simply connected.
For an arbitrary point b ∈ B take y ∈ r −1 (b) and x = κ(y). Without loss generality, we assume that there is a neighbourhood Let us fix points x 0 ∈ M and y 0 ∈ κ −1 (x 0 ) ∈ M. Then the fundamental group π 1 (M, x 0 ) acts on the universal covering space M as the deck transformation group G ∼ = π 1 (M, x 0 ) of κ. Since G preserves the inducted foliation ( M, F ) formed by fibres of the fibration r : M → B, then every ψ ∈ G defines ψ ∈ Dif f (B) satisfying the relation r • ψ = ψ • r. The map χ : G → Ψ : ψ → ψ is a group epimorphism and the statement (2) is proved.
Observe that G is a subgroup of the automorphism group Aut( M, F ) of ( M , F ) in the category CF. Therefore Ψ is a subgroup of the automorphism group Aut(B, η) in the category of Cartan geometries Car. The kernel ker(χ) of χ determines the quotient manifold M := M /ker(χ) with the quotient map κ : M → M and the quotient group G := G/ker(χ) such that M ∼ = M/ G. The quotient map κ : M → M is the required regular covering map, with G acts on M as a deck transformation group. The map G → Ψ : ψ · ker(χ) → χ( ψ), ψ ∈ G, is a group isomorphism. Thus the statment (3) is proved.

The associated foliated bundle
Let (M, F ) be a Cartan foliation modeled on Cartan geometry ξ = (P (N , H), ω) of type (G, H). Then there exists a principal H-bundle with the projection π : R → M, the H-invariant foliation (R, F ) and the g-valued H-equivariant 1-form β on R which satisfy the following conditions: (ii) the mapping β u : T u R → g ∀u ∈ R is surjective, and ker(β u ) = T u F ; (iii) the foliation (R, F ) is transversely parallelizable; (iv) the Lie derivative L X β is equal to zero for every vector field X tangent to the foliation (R, F ). If the lifted foliation (R, F ) is formed by fibres of the locally trivial fibration π b : R → W , then W = R/F is a smooth manifold , and a g-valued 1-form β such that π * b β := β and locally free action of the Lie group H on W are induced. In this case, (W, β) is a parallelizable manifold and A(W, β) is the Lie group of its automorphisms that acts freely on W . Further, as above, by A H (W, β) we denote the closed Lie subgroup of A(W, β) formed by transformations commuting with the induced action of the Lie group H on W .
Remark that the maps are regular covering maps with the deck transformation groups Γ, Γ and Γ, respectively, which are isomorphic to the relevant groups G, G and G, i.e. Γ ∼ = G, Γ ∼ = G and Γ ∼ = G. According to the conditions of Theorem 2, Ψ is a discrete subgroup of the Lie group Aut(B, η). Let N(Ψ) be the normalizer of Ψ in the Lie group Aut(B, η) ∼ = A H (W, β). Hence, N(Ψ) is a closed Lie subgroup of the Lie group Aut(B, η) and the quotient group N(Ψ)/Ψ is also a Lie group.
Let π : R → M be the projection of the foliated bundle over (M, F ). Due to the discreteness of the global holonomy group Ψ, the lifted foliation (R, F ) is formed by fibres of some locally trivial fibration π b : R → W , which is called the basic fibration.
Observe that there exists a map τ : W → W satisfying the equality τ • π b = θ •π b . It is easy to show that τ : W → W is a regular covering map with the deck transformations group Φ, Φ ⊂ A H ( W , β), which is naturally isomorphic to each of the groups Ψ, G and Γ.
Denote by η = (P (B, H), ω) the Cartan geometry with the projection p : P → B onto B determined in the proof of Theorem 1. Remark that W = P is the space of the H-bundle of the Cartan geometry η. Since In this case, by the same way as in the proof of Theorem 1, the Cartan geometry η is induced on B such that (M, F ) becomes an (Aut(B, η), B)-foliation.
2. Let Ψ be the global holonomy group of this foliation. Suppose now that the normalizer N(Ψ) is equal to the centralizer Z(Ψ) of the group Ψ in the group Aut(B, η).
Let h be any element from N(Ψ)/Ψ. Since N(Ψ) = Z(Ψ), we have the following chain of equalities

Example of finding a basic automorphism group
Let S q be a q-dimensional standard sphere, where q ≥ 3. We identify S q with R q ∪ {∞}, where {∞} is the point at infinity. Define the transformation ψ : S q ∼ = R q ∪ {∞} → S q by equality ψ(z) = λz for any z ∈ S q ∼ = R q ∪ {∞}, where λ is a real number, and 0 < λ < 1. We denote by Conf (S q ) the Lie group of all conformal transformations of the sphere S q .
Let Ψ =< ψ > be the subgroup of the group Conf (S q ) generated by ψ, and Ψ is isomorphic to the group of integers Z. Define the action of the group Z on the product of manifolds R 1 × S q by the equality n(t, z) = (t − n, ψ n (z)) for any n ∈ Z, (t, z) ∈ R 1 ×Z. This action is free and properly discontinuous. Therefore, the manifold of orbits M = R 1 × Z S q is defined. Denote by f : R 1 × S q → M the quotient map. Fix a point (t 0 , z 0 ) ∈ R 1 × S q , put x 0 = f (t 0 , z 0 ) ∈ M. Then the fundamental group π 1 (M, x 0 ) acts on the universal covering space R 1 ×S q as the deck transformation group G ∼ = π 1 (M, x 0 ) of f . Since the action G preserves the structure of the product R 1 × S q , then two foliations (M, F ) and (M, F t ), covered by trivial fibrations pr 2 : R 1 × S q → S q and pr 1 : R 1 × S q → R 1 respectively, are defined. Let us denote by χ : R 1 → S 1 = R 1 /Z and ν : S q → S q /Ψ the quotient maps onto the orbit spaces. Let r : M → M/F be the quotient map onto the leaf space. Observations show that the topological spaces M/F and S q /Ψ are homeomorphic and satisfy the commutative diagram where p : M → S 1 is the projection of the locally trivial fibration transforming the orbit Z.(t, z) of a point (t, z) ∈ R 1 × S q , considered as a point from M, into the orbit Z.t of a point t ∈ R 1 , considered as a point of the circle S 1 . Since the manifold M is the space of a locally trivial fibration p : M → S 1 over the circle S 1 with a compact standard fiber S q , then M is compact.
The distribution M tangent to (M, F t ), is an integrable Ehresmann connection for the foliation (M, F ). The foliation (M, F ) has two compact leaves L 1 and L 2 which are diffeomorphic to the circle S 1 . Every other leaf L of (M, F ) is diffeomorphic to R 1 , and its closure L is equal to the union L ∪ L 1 ∪ L 2 . We emphasize that (M, F ) is a proper conformal foliation, which can be regarded as a Cartan foliation of type (G, H), where G = Conf (S q ) and H is a stationary subgroup of the group Conf (S q ) at some point in S q .
As is known, H ∼ = CO(q)⋉R q is a semidirect product of a conformal group CO(q) ∼ = R + ×O(q) and a normal abelian subgroup R q . Note that Ψ is the global holonomy group of the foliation (M, F ), and Ψ is a discrete subgroup of the Lie group Conf (S q ).
The direct check shows that the normalizer of the group Ψ is equal to N(Ψ) = R + × O(q), and N(Ψ) coincides with the centralizer Z(Ψ). Applying Theorem 3, we obtain that the group of basic automorphisms A B (M, F ) is a Lie group isomorphic to the quotient group N(Ψ)/Ψ ∼ = U(1) × O(q), where U(1) ∼ = S 1 . Thus, the Lie group of basic automorphisms A B (M, F ) is isomorphic to the product of Lie groups U(1)×O(q).