Algèbre de Lie des champs feuilletés d'une extension d'un feuilletage de Lie (original) (raw)
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Riemannian foliation with dense leaves on a compact manifold
In this paper, we show that if G = Lie(G) is the Lie structural algebra of a Riemannian foliation with dense leaves (M, F) on a compact manifold M, there exists a representation ρ : H 0 → Dif f (V) where V is an open subset of G such as: (a) There exists a biunivocal correspondence between the Lie subal-gebras of G invariant by Ad (ρa(v)) −1 .v for every (a, v) ∈ H 0 × V and F extensions. (b) An extension is a Lie foliation if the subalgebra corresponding is an ideal of G. (c) Every extension F ′ of F is a Riemannian foliation and there exists a common bundle-like metric for the foliations F and F ′. (d) If F H is an extension of F corresponding to a subalgebra H of G, then to isomorphism nearly of Lie algebras we have ℓ(M, F H) = {u ∈ H ⊥ /∀ (h, a, v) ∈ H×H 0 ×V , [u, h] = 0 and Ad (ρa(v)) −1 .v (u) = u}.
Ends of leaves of Lie foliations
Journal of The Mathematical Society of Japan, 2005
Let G be a simply connected Lie group and consider a Lie G foliation F on a closed manifold M whose leaves are all dense in M. Then the space of ends E (F) of a leaf F of F is shown to be either a singleton, a two points set, or a Cantor set. Further if G is solvable, or if G has no cocompact discrete normal subgroup and F admits a transverse Riemannian foliation of the complementary dimension, then E (F) consists of one or two points. On the contrary there exists a Lie f SL(2, R) foliation on a closed 5-manifold whose leaf is diffeomorphic to a 2-sphere minus a Cantor set. 1. Introduction. Let G be a connected Lie group. A Lie G foliation F on a closed manifold M is a foliation locally modelled on the geometry (G, G). That is, F is a foliation defined by distinguished charts taking values in G, with transition functions the restrictions of the left translations by elements of G. See Section 2 for the precise definition. Lie G foliations form a very special class of foliations and satisfy various strong properties. For example, any left invariant Riemannian metric of G gives rise to a metric on the normal bundle of F , invariant by the holonomy pseudogroup; that is, F is a Riemannian foliation. Each leaf of F has trivial holonomy and they are mutually Lipshitz diffeomorphic. Conversely by the work of P. Molino [14], the study of Riemannian foliations reduces to that of Lie foliations. See [15] for detailed accounts. Classical examples of Lie G foliations are: Example 1.1. a) Let H be a Lie group admitting a (e.g. surjective) homomorphism f : H → G, and let Γ be a uniform lattice of H. Then the right action of N = Ker(f) on the quotient space Γ \ H gives rise to a Lie G foliation, with leaves diffeomorphic to (Γ ∩ N) \ N. b) Assume G is compact, let B be a closed manifold, and let ϕ : π 1 (B) → G be a group homomorphism. The suspension gives a fibration G → M → B together with a Lie G foliations transverse to the fibers; the leaf is the Ker(ϕ)-normal covering of B. Only very few essentially different examples are known, and in these examples G is solvable ([13]). This shortage of examples makes the study of Lie foliations difficult. Little is known
In this paper we show that the transverse Levi civita connection of a Riemannian foliation having dense leaves and admitting a flag of ex- tension on a compact manifold M is integrable and if additionally the fundamental group of M is finished then the foliation of the closure of leaf F♮ of lifted foliation F♮ of F on the orthonormal transverse frame bundleM♮ is defined by the connection of transverse Levi civita ! T · In summary, the horizontal spaces of the transverse Levi civita connec- tion ! T of a Riemannian foliation having dense leaves and admitting a flag of extension on a compact manifold M having a finite fundamen- tal group is the leaves of the closure F♮ of lifted foliation F♮ of F on the orthonormal transverse frame bundle M♮· We also establish that the structural Lie algebra of such Riemannian foliation is an abelian Lie algebra·
Foliated Lie systems: Theory and applications
arXiv (Cornell University), 2019
A stratified Lie system is a nonautonomous system of first-order ordinary differential equations on a manifold M described by a t-dependent vector field X = r α=1 g α X α , where X 1 ,. .. , X r are vector fields on M spanning an r-dimensional Lie algebra that are tangent to the strata of a stratification F of M while g 1 ,. .. , g r : R × M → R are functions depending on t that are constant along integral curves of X 1 ,. .. , X r for each fixed t. We analyse the particular solutions of stratified Lie systems and how their properties can be obtained as generalisations of those of Lie systems. We illustrate our results by studying Lax pairs and a class of t-dependent Hamiltonian systems. We study stratified Lie systems with compatible geometric structures. In particular, a class of stratified Lie systems on Lie algebras are studied via Poisson structures induced by r-matrices.
Leaves of foliations with a transverse geometric structure of finite type
Publicacions Matemàtiques, 1989
ROBERT A. WOLAK In Chis short note we find some conditions which ensure that a G foliation of finite type with all leaves compact is a Riemannian foliation or equivalently the space of leaves of such a foliation is a Satake manifold. A particular attention is paid to transversely affine foliations. We present several conditions such ensure completeness of these foliations .