Distribution (differential geometry)

About: Distribution (differential geometry) is a research topic. Over the lifetime, 911 publications have been published within this topic receiving 10149 citations.

Lipschitz distributions and Anosov flows

Abstract: We show that if a distribution is locally spanned by Lipschitz vector fields and is involutive a.e., then it is uniquely integrable giving rise to a Lipschitz foliation with leaves of class C1,Lip. As a consequence, we show that every codimension-one Anosov flow on a compact manifold of dimension > 3 such that the sum of its strong distributions is Lipschitz, admits a global cross section. The main purpose of this paper is to generalize the theorem of Frobenius on integrability of smooth vector distributions and to give an application of the theorem to the question of existence of global cross sections to Anosov flows. Accordingly, the paper is divided into two parts, A and B. A. Integrability of Lipschitz distributions Let M be a C∞ n-dimensional Riemannian manifold equipped with a Lebesgue measure. Definition 1. We will say that a distribution (or plane field) E on M is Lipschitz if it is locally spanned by Lipschitz continuous vector fields. Recall that a map f between metric spaces (M1, d1) and (M2, d2) is called Lipschitz continuous (or simply Lipschitz) if there is a constant C > 0 such that d2(f(p), f(q)) ≤ Cd1(p, q), for all p, q ∈ M1. By saying that a vector field X on M is Lipschitz we mean that in some (and therefore in any) coordinate system, X can be written in the form X = n ∑

Nonholonomic Lorentzian Geometry on Some ℍ-Type Groups

Abstract: We consider examples of ℍ-type Carnot groups whose noncommutative multiplication law gives rise to a smooth 2-step bracket generating distribution of the tangent bundle. In the contrast with the previous studies we furnish the horizontal distribution with the Lorentzian metric, which is nondegenerate metric of index 1, instead of a positive definite quadratic form. The causal character is defined. We study the reachable set by timelike future directed curves. The parametric equations of geodesics are obtained.

Higher-order mechanical systems with constraints

Abstract: A general mathematical theory covering higher-order mechanical systems subject to constraints of arbitrary order (i.e., depending on time, positions, velocities, accellerations, and higher derivatives) is presented, including higher-order holonomic systems as a particular case. Within differential geometric setting on higher-order jet bundles, the concept of a mechanical system (not necessarily regular, or Lagrangian) is introduced to be a class of 2-forms equivalent with a dynamical form. Dynamics are then represented by means of corresponding exterior differential systems. Higher-order constraint structure on a fibered manifold is defined to be a submanifold endowed with a distribution (canonical distribution, higher-order Chetaev bundle). With help of a constraint structure a constraint force is naturally introduced. Higher-order mechanical systems subject to different kinds of higher-order constraints are then geometrically characterized and their dynamics are studied from a geometrical point of view....

An intrinsic definition of the Colombeau generalized functions

Abstract: A slight modification of the definition of the Colombeau generalized functions allows to have a canonical embedding of the space of the distributions into the space of the generalized functions on a C∞ manifold. The previous attempt in [5] is corrected, several equivalent definitions are presented.

How to determine a quantum state by measurements: The Pauli problem for a particle with arbitrary potential.

Abstract: The problem of reconstructing a pure quantum state ??> from measurable quantities is considered for a particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution ??(x,t)?2 has been measured at time t, and let it have M nodes. It is shown that after measuring the time evolved distribution at a short-time interval ?t later, ??(x,t+?t)?2, the set of wave functions compatible with these distributions is given by a smooth manifold M in Hilbert space. The manifold M is isomorphic to an M-dimensional torus, TM. Finally, M additional expectation values of appropriately chosen nonlocal operators fix the quantum state uniquely. The method used here is the analog of an approach that has been applied successfully to the corresponding problem for a spin system.