We establish a geometric condition that determines when a type III von Neumann algebra arises from a foliation whose holonomy becomes affine with respect to a suitable transverse coordinate system. Under such an assumption the Godbillon-Vey class of the foliation becomes trivial in contrast to the case considered in Connes’s famous theorem.

We give a characterization theorem for non-degenerate plane foliations of degree different from 1 having a rational first integral. Moreover, we prove that the degree r of a non-degenerate foliation as above provides the minimum number, r+ 1, of points in the projective plane through which pass infinitely many algebraic leaves of the foliation.

Mutations in the gene encoding the ATP dependent chromatin-remodeling factor, CHD7 are the major cause of CHARGE (Coloboma, Heart defects, Atresia of the choanae, Retarded growth and development, Genital-urinary anomalies, and Ear defects) syndrome. Neurodevelopmental defects and a range of neurological signs have been identified in individuals with CHARGE syndrome, including developmental dela...

The cerebellum consists of a highly organized set of folia that are largely generated postnatally during expansion of the granule cell precursor (GCP) pool. Since the secreted factor sonic hedgehog (Shh) is expressed in Purkinje cells and functions as a GCP mitogen in vitro, it is possible that Shh influences foliation during cerebellum development by regulating the position and/or size of lobe...

In [Ch1], Chern gives a generalization of projective geometry by considering foliations on the Grassman bundle of p-planes Gr(p, R) → R by p-dimensional submanifolds that are integrals of the canonical contact differential system. The equivalence method yields an sl(n + 1, R)valued Cartan connection whose curvature captures the geometry of such foliation. In the flat case, the space of leaves o...

the geometry of a system of second order differential equations is the geometry of a semispray, which is a globally defined vector field on tm. the metrizability of a given semispray is of special importance. in this paper, the metric associated with the semispray s is applied in order to study some types of foliations on the tangent bundle which are compatible with sode structure. indeed, suff...

We establish a geometric condition that determines when a type III von Neumann algebra arises from a foliation whose holonomy becomes affine with respect to a suitable transverse coordinate system. Under such an assumption the Godbillon-Vey class of the foliation becomes trivial in contrast to the case considered in Connes’s famous theorem.

An important approach to establishing stochastic behavior of dynamical systems is based on the study of systems expanding a foliation and of measures having smooth densities along the leaves of this foliation [44], [46], [38]. We review recent results on this subject and present some extensions and open questions. 2000 Math. Subj. Class. Primary 34C29; Secondary 37D30.

We study transverse conformal Killing forms on foliations and prove a Gallot-Meyer theorem for foliations. Moreover, we show that on a foliation with C-positive normal curvature, if there is a closed basic 1-form φ such that ∆Bφ = qCφ, then the foliation is transversally isometric to the quotient of a q-sphere.