Despite the conservative circuitry of the cerebellum, there is considerable variation in the shape of the cerebellum among vertebrates. One aspect of cerebellar morphology that is of particular interest is the degree of folding, or foliation, of the cerebellum and its functional significance. Here, we present the first comprehensive analysis of variation in cerebellar foliation in birds with th...

A singular foliation on a complete riemannian manifold is said to be riemannian if every geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. In this paper we study singular riemannian foliations that have sections, i.e., totally geodesic complete immersed submanifolds that meet each leaf orthogonally and whose dimensions are the codimensions of th...

Let Γ be a nondegenerate geodesic in a compact Riemannian manifold M . We prove the existence of a partial foliation of a neighbourhood of Γ by CMC surfaces which are small perturbations of the geodesic tubes about Γ. There are gaps in this foliation, which correspond to a bifurcation phenomenon. Conversely, we also prove, under certain restrictions, that the existence of a partial CMC foliatio...

We formulate a scheme for a Lorentz invariant Bohmian quantum theory for N non-interacting Dirac particles involving as an additional beable a foliation of space-time. This theory, which is a generalization of Bohm’s theory for N Dirac particles, allows for an equivariant measure on the leafs of the foliation—and hence for a simple statistical analysis analogous to the equilibrium analysis of n...

Let F be a singular Riemannian foliation on a compact Riemannian manifold M . By successive blow-ups along the strata of F we construct a regular Riemannian foliation F̂ on a compact Riemannian manifold M̂ and a desingularization map ρ̂ : M̂ → M that projects leaves of F̂ into leaves of F . This result generalizes a previous result due to Molino for the particular case of a singular Riemannian folia...

Little is known about the genetic pathways and cellular processes responsible for regional differences in cerebellum foliation, which interestingly are accompanied by regionally distinct afferent circuitry. We have identified the Engrailed (En) homeobox genes as being crucial to producing the distinct medial vermis and lateral hemisphere foliation patterns in mammalian cerebella. By producing a...

We prove that a holomorphic foliation of codimension k which is transverse to the fibers of a fibration and has a compact leaf with finite holonomy group is a Seifert fibration, i.e., has all leaves compact with finite holonomy. This is the case for C1-small deformations of a foliation where the original foliation exhibits a compact leaf and the base B of the fibration satisfies H1(B,R) = 0 and...