Some problems concerning to Liouville distribution and framed f -structures are studied on the normal bundle of the lifted Finsler foliation to its normal bundle. It is shown that the Liouville distribution of transversally Finsler foliations is an integrable one and some natural framed f(3, ε)structures of corank 2 exist on the normal bundle of the lifted Finsler foliation.

Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction σ|Y of the holomorphic symplectic form induces a rank one foliation on Y . We investigate situations where this foliation has compact leaves; in such cases we obtain a space of leaves Y/F which has dimension 2n − 2 and admits a holomorphic symplectic form.

We show that any foliation transverse to a C 1 nonsingular flow φ on a closed 3-manifold can be detected algorithmically. We use this to describe a procedure that, for any δ > 0, will determine whether or not there is a foliation whose tangent space is bounded away from the tangent space to φ by a distance of δ.

The cerebellum has a conserved foliation pattern and a well-organized layered structure. The process of foliation and lamination begins around birth. β-catenin is a downstream molecule of Wnt signaling pathway, which plays a critical role in tissue organization. Lack of β-catenin at early embryonic stages leads to either prenatal or neonatal death, therefore it has been difficult to resolve its...

Let M be a smooth manifold and let F be a codimension one, C foliation on M , with isolated singularities of Morse type. The study and classification of pairs (M,F) is a challenging (and difficult) problem. In this setting, a classical result due to Reeb [Reeb] states that a manifold admitting a foliation with exactly two centertype singularities is a sphere. In particular this is true if the f...

We show the existence of weak logarithmic models for any (dicritical or not) holomorphic foliation F of (C, 0) without saddlenodes in its desingularization. The models are written in terms of a representative set of separatrices, whose equisingularity types are controlled by the Milnor number of the foliation.

In many aspects the most complicated foliated manifolds are those with nonvanishing Godbillon-Vey class. We argue that they probably do not appear in physics and that is due to gauge symmetry which prevents the foliation from becoming “too wild”; that means that the foliation does not develop resilient leaves which, at least in codim-1, by Duminy’s theorem are responsible for the nontriviality ...

We consider an integrable Hamiltonian system with n-degrees of freedom whose first integrals are invariant under the symplectic action of a compact Lie group G. We prove that the singular Lagrangian foliation associated to this Hamiltonian system is symplectically equivalent, in a Gequivariant way, to the linearized foliation in a neighborhood of a compact singular non-degenerate orbit. We also...

We introduce examples of hyperpolar actions on noncompact symmetric spaces that induce a regular foliation. We study some properties of these actions. Finally, we show that any hyperpolar action on a noncompact symmetric space that induces a regular foliation is one of these examples.