A Examples and counter-examples of foliations A.1. Elmar Vogt A.1.1 Question : Let F be a compact foliation (i.e. a foliation with all leaves compact) of codimension 3 on a compact manifold. Let B be the first bad set of the Epstein hierarchy of F , i.e. B is the union of all leaves of F with infinite holonomy group. If B is not empty, can B be a manifold ? Remarks : In [44], arguments (due to ...

We prove that a one-dimensional foliation with generic singularities on a projective space, exhibiting a Lie group transverse structure in the complement of some codimension one algebraic subset is logarithmic, i.e., it is the intersection of codimension one foliations given by closed one-forms with simple poles. If there is only one singularity in a suitable affine space, then the foliation is...

We consider a partially hyperbolic diffeomorphism of a compact smooth manifold preserving a smooth measure. Assuming that the central distribution is integrable to a foliation with compact smooth leaves we show that this foliation fails to have the absolute continuity property provided that the sum of Lyapunov exponents in the central direction is not zero on a set of positive measure. We also ...

Here, a Finsler manifold $(M,F)$ is considered with corresponding curvature tensor, regarded as $2$-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of $M$ determined by the curvature are introduced and called $k$-nullity foliations of the curvature operator. It is shown that if the dimension of foliation is constant, then the distribution is involutive...

A parametric manifold can be viewed as the manifold of orbits of a (regular) foliation of a manifold by means of a family of curves. If the foliation is hypersurface orthogonal, the parametric manifold is equivalent to the 1-parameter family of hypersurfaces orthogonal to the curves, each of which inherits a metric and connection from the original manifold via orthogonal projections; this is th...

We obtain the following two results through foliation theoretic approaches including a review of Lawson’s construction of a codimension-one foliation on the 5-sphere: 1) The standard contact structure on the 5-sphere deforms to ‘Reeb foliations’. 2) We define a 5-dimensional Lutz tube which contains a plastikstufe. Inserting it into any contact 5-manifold, we obtain a contact structure which vi...

Abstract. A foliation F on a Riemannian manifold M is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold of M that intersects each leaf of F orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space M of noncompact type. These foliations are constructed as follows. Let Φ be an orthogonal subset of a set ...

Panax ginseng is one of the most important medicinal plants in Asia. Triterpene saponins, known as ginsenosides, are the major pharmacological compounds in P. ginseng. The present study was conducted to evaluate the changes in ginsenoside composition according to the foliation stage of P. ginseng cultured in a hydroponic system. Among the three tested growth stages (closed, intermediate, and op...

We prove that if the normal distribution of a singular riemannian foliation is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section), which meets every leaf orthogonally. In addition the set of regular points is open and dense in each section. This result generalizes a result of Boualem and solves a problem...

Introduction Despite their basal place in vertebrate evolution, little quantitative data is available on shark (chondrichthyan) brain organization & its implications for nervous system adaptations [1,2], including our own. The cerebellum appeared at the onset of the chondrichthyan radiation, with extreme variations in cerebellar morphology during subsequent evolution, highlighting their role as...