Non-closed Lie subgroups of Lie groups

We obtain several homotopy obstructions to the existence of non-closed connected Lie subgroupsH in a connected Lie groupG.First we show that the foliationF(G, H) onG determined byH is transversely complete [4]; moreover, forK the closure ofH inG, F(K, H) is an abelian Lie foliation [2].Then we prove that ₁(K) and ₁(H) have the same torsion subgroup, _n (K)= _n (H) for alln 2, and rank₁(K) — rank₁(H) > codimF(K, H). This implies, for instance, that a contractible (e.g. simply connected solvable) Lie subgroup of a compact Lie group must be abelian. Also, if rank₁(G) 1 then any connected invariant Lie subgroup ofG is closed; this generalizes a well-known theorem of Mal'cev [3] for simply connected Lie groups.Finally, we show that the results of Van Est on (CA) Lie groups [6], [7] provide many interesting examples of such foliations. Actually, any Lie group with non-compact centre is the (dense) leaf of a foliation defined by a closed 1-form. Conversely, when the centre is compact, the latter is true only for (CA) Lie groups (e.g. nilpotent or semisimple).     

Bott's vanishing theorem for regular Lie algebroids

Differential geometry has discovered many objects which determine Lie algebroids playing a role analogous to that of Lie algebras for Lie groups. For example:

--- differential groupoids,

--- principal bundles,

--- vector bundles,

--- actions of Lie groups on manifolds,

--- transversally complete foliations,

--- nonclosed Lie subgroups,

--- Poisson manifolds,

--- some complete closed pseudogroups.

We carry over the idea of Bott's Vanishing Theorem to regular Lie algebroids (using the Chern-Weil homomorphism of transitive Lie algebroids investigated by the author) and, next, apply it to new situations which are not described by the classical version, for example, to the theory of transversally complete foliations and nonclosed Lie subgroups in order to obtain some topological obstructions for the existence of involutive distributions and Lie subalgebras of some types (respectively).

     

Finslerian foliations of compact manifolds are Riemannian

We prove that a Finslerian foliation of a compact manifold is Riemannian.     

Foliation-Preserving Maps Between Solvmanifolds

For i=1,2, let _i be a lattice in a simply connected, solvable Lie group G _i , and let X _i be a connected Lie subgroup of G _i . The double cosets _i g X _i provide a foliation _i of the homogeneous space _i \G _i . Let f be a continuous map from ₁\G ₁ to ₂\G ₂ whose restriction to each leaf of ₁ is a covering map onto a leaf of ₂. If we assume that ₁ has a dense leaf, and make certain technical assumptions on the lattices ₁ and ₂, then we show that f must be a composition of maps of two basic types: a homeomorphism of ₁\G ₁ that takes each leaf of ₁ to itself, and a map that results from twisting an affine map by a homomorphism into a compact group. We also prove a similar result for many cases where G ₁ and G ₂ are neither solvable nor semisimple.     

Hyperfoliations on compact 3-manifolds with restrictions on the external curvature of leaves

C∞-foliations of codimension 1 on compact Riemannian 3-manifolds are studied. New classes of foliations, namely hyperbolic, elliptic, and parabolic foliations, are considered. Examples of such foliations are presented. In particular, aC∞-metric of nonnegative sectional curvature onS ³ such that the Reeb foliation is parabolic with respect to this metric is constructed. Analytic 3-manifolds with sectional curvature of constant sign admitting parabolic foliations are classified. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 651–659, May, 1998. The author wishes to express his thanks to Professor A. A. Borisenko for his supervision, and to Yu. A. Nikolaevskii for useful advice in the process of preparing the present paper.     

Determination of the top Baum-Bott number via classical intersection theory

We determine geometrically the number , associated to a smooth m-dimensional projective variety V^m, invariant by a one-dimensional holomorphic foliation of ⁿ , using polar divisors associated to the foliation and Bézouts theorem.     

A note on geodesic foliations on the torus

We look at geodesic foliations on the Lorentzian 2-tori which are lightlike on a proper subset. We prove that they do not exist if the torus is geodesically complete. We describe some properties of their orthogonal foliations and we give several new examples.      

Lie comodules and the constructions of Lie bialgebras

In this paper,we first give a direct sum decomposition of Lie comodules,and then accord- ing to the Lie comodule theory,construct some(triangular)Lie bialgebras through Lie coalgebras.     

Transverse Dimension of R~n Actions on Compact Foliated Manifolds

ＴｒａｎｓｖｅｒｓｅＤｉｍｅｎｓｉｏｎｏｆＲ￣ｎＡｃｔｉｏｎｓｏｎＣｏｍｐａｃｔＦｏｌｉａｔｅｄＭａｎｉｆｏｌｄｓ￥ＦｒａｎｃｉｓｃｏＪａｖｉｅｒＴＵＲＩＥＬ（ＧｅｏｍｅｔｒｉａＹＴｏｐｏｌｏｇｉａ，ＦａｃｕｌｔａｄｄｅＣｉｅｎｃｉａｓＡ．Ｐ．５９；...     

Residues of a Pfaff system relative to an invariant subscheme

In this paper we give a purely algebraic construction of the theory of residues of a Pfaff system relative to an invariant subscheme. This construction is valid over an arbitrary base scheme of any characteristic.

     

Characteristic Foliations of Spheres Embedded in the Standard Overtwisted Structure (R3, ζ1)

The characteristic foliation of a sphere embedded in the standard tight contact structure (R³, ₀) is unique up to isotopy. We show that any Morse-Smale foliation on the sphere with null Euler class, is, up to isotopy, the characteristic foliation of a sphere embedded in the standard overtwisted contact structure (R³, ₁). We thus have a new way of looking at the two standard structures as opposites in the world of contact structures.     

On the existence of stable compact leaves for transversely holomorphic foliations

We prove that a transversely holomorphic foliation on a compact manifold exhibits some compact leaf with finite holonomy group, provided that the set of compact leaves is not a zero measure set. A similar result is stated for groups of complex diffeomorphisms and periodic orbits.     

Foliations on the tangent bundle of Finsler manifolds

Let M be a smooth manifold with Finsler metric F,and let T M be the slit tangent bundle of M with a generalized Riemannian metric G,which is induced by F.In this paper,we prove that (i) (M,F) is a Landsberg manifold if and only if the vertical foliation F V is totally geodesic in (T M,G);(ii) letting a:= a(τ) be a positive function of τ=F 2 and k,c be two positive numbers such that c=2 k(1+a),then (M,F) is of constant curvature k if and only if the restriction of G on the c-indicatrix bundle IM (c) is bundle-like for the horizontal Liouville foliation on IM (c),if and only if the horizontal Liouville vector field is a Killing vector field on (IM (c),G),if and only if the curvature-angular form Λ of (M,F) satisfies Λ=1-a 2/R on IM (c).     

A Nilpotent Ideal in the Lie Rings with Automorphism of Prime Order

We improve the conclusion in Khukhro's theorem stating that a Lie ring (algebra) L admitting an automorphism of prime order p with finitely many m fixed points (with finite-dimensional fixed-point subalgebra of dimension m) has a subring (subalgebra) H of nilpotency class bounded by a function of p such that the index of the additive subgroup |L: H| (the codimension of H) is bounded by a function of m and p. We prove that there exists an ideal, rather than merely a subring (subalgebra), of nilpotency class bounded in terms of p and of index (codimension) bounded in terms of m and p. The proof is based on the method of generalized, or graded, centralizers which was originally suggested in [E. I. Khukhro, Math. USSR Sbornik 71 (1992) 51–63]. An important precursor is a joint theorem of the author and E. I. Khukhro on almost solubility of Lie rings (algebras) with almost regular automorphisms of finite order.     

Completing the conformal boundary of a simply connected Lorentz surface

This paper extends Kulkarni's conformal boundary for a simply connected Lorentz surface to a compact conformal boundary . The procedure used is analogous to Carathéodory's construction (in the definite metric setting) of prime ends from the accessible points of a bounded simply connected planar domain. The space of conformal boundary elements is homeomorphic to the circle, and contains Kulkarni's conformal boundary as a dense subspace.

     

Foliations of some 3-manifolds which fiber over the circle

We show that a hyperbolic punctured torus bundle admits a foliation by lines which is covered by a product foliation. Thus its fundamental group acts freely on the plane.

     

Harmonic splittings of surfaces

We give a harmonic maps proof of a theorem of Morgan-Otal and Skora, conjectured by Shalen: any minimal, small action of a higher-genus surface group on a real tree is dual to the lift of a measured foliation.     

Log canonical foliation singularities on surfaces

We give a classification of the dual graphs of the exceptional divisors on the minimal resolutions of log canonical foliation singularities on surfaces. As an application, we show the set of foliated minimal log discrepancies for foliated surface triples satisfies the ascending chain condition and a Grauert–Riemenschneider–type vanishing theorem for foliated surfaces with special log canonical foliation singularities.