We prove that a Morse type codimension one holomorphic foliation is not transverse to a sphere in the complex affine space. Also we characterize the variety of contacts of a linear foliation with concentric spheres.

The cerebellum consists of an intricate array of lobules that arises during the process of foliation. Foliation not only increases surface area, but may also facilitate organization of cerebellar neural circuitry. Defects in cerebellar foliation are associated with a number of diseases. Yet, little is known about how foliation, a process involving large-scale and simultaneous movement of severa...

Let F be a codimension one foliation on a closed manifold M which admits a transverse dimension one Riemannian foliation. Then any continuous leafwise harmonic functions are shown to be constant on leaves.

We prove that a codimension one foliation of an orientable paracompact manifold has at most a countable number of nonorientable leaves. We also give an example of a codimension one foliation of a compact orientable manifold with an infinite number of nonorientable leaves. Introduction. The purpose of this paper is to investigate the occurrence of nonorientable leaves in a codimension one foliat...

We study some kind of rigidity property for dicritical foliation in C 2 thanks to a new meromorphic function associated to any dicriti-cal foliation playing the role of equation of separatrix. To be more specific, we compute infinitesimal obstructions for realizing the holon-omy pseudo-group of a dicritical foliation F on any source manifold of a blowing-up morphism with same dual tree as the d...

In this paper we reduce the Poincaré Problem for foliations in P to a problem of postulation of plane curves of degree m − 1, with m denoting the degree of the foliation. In the cases in which we can assure a solution for the Poincaré Problem the bound for the degree of the first integral depends only on the degree of the foliation. An intermediate result gives a solution for Painlevé’s Problem...

15 3. Stable and unstable filtrations 17 3.1. Existence and subfoliation 17 3.2. Absolute continuity 19 4. Central Foliations 21 4.1. Normal hyperbolicity 21 4.2. Integrability of the central foliation and dynamical coherence 23 4.3. Smoothness of central leaves via normal hyperbolicity 25 4.4. Robustness of the central foliation 26 5. Intermediate Foliations 27 5.1. Nonintegrability of interme...

The results of Biswas (2000) are extended to the situation of transversely projective foliations. In particular, it is shown that a transversely holomorphic foliation defined using everywhere locally nondegenerate maps to a projective space CPn, and whose transition functions are given by automorphisms of the projective space, has a canonical transversely projective structure. Such a foliation ...

Let F be a foliation of the modular curve, given by the vertical trajectories of the Hecke eigenform. It is shown that F is either a Strebel or a pseudo-Anosov foliation or else can be reduced to the above foliations. An application of the result is discussed.