A smooth distribution, invariant under a dynamical system, integrates to give an invariant foliation, unless certain resonance conditions are present.

Abstract. We describe recent work due to Niall Ó Murchadha and the author (grqc/9706046, Phys. Rev. D57, 4728 (1998)) on the late time behaviour of the maximal foliation of the extended Schwarzschild geometry which results from evolving a time symmetric slice into the past and future, with time running equally fast at both spatial ends of the manifold. We study the lapse function of this foliat...

We construct an example of a Hölder continuous vector field on the plane which is tangent to all foliations in a continuous family of pairwise distinct C foliations. Given any 1 ≤ r < ∞, the construction can be done in such a way that each leaf of each foliation is the graph of a Cr function from R to R. We also show the existence of a continuous vector field X on R and two foliations F and G o...

The volume of a k-dimensional foliation F in a Riemannian manifold Mn is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing the construction by Gluck and Ziller in [4], “singular” foliations by 3-spheres are constructed on round spheres S, as well as a singular foliation by 7-spheres on S, which minimize v...

This work investigates some global questions about cosmological spacetimes with two dimensional spherical, plane and hyperbolic symmetry containing matter. The result is, that these spacetimes admit a global foliation by prescribed mean curvature surfaces, which extends at least towards a crushing singularity. The time function of the foliation is geometrically defined and unique up to the choi...

Given a foliation S of a manifold M, a distribution Z in M transveral to S and a Poisson bivector Π on M we present a geometric method of reducing this operator on the foliation S along the distribution Z. It encompasses the classical ideas of Dirac (Dirac reduction) and more modern theory of J. Marsden and T. Ratiu, but our method leads to formulas that allow for an explicit calculation of the...

In a recent paper, general solutions for the vacuum wave functionals in the Schrödinger picture were given for a variety of classes of curved spacetimes. Here, we describe a number of simple examples which illustrate how the presence of spacetime boundaries influences the vacuum wave functional and how physical quantities are independent of the choice of spacetime foliation used in the Schrödin...