Showing papers on "Distribution (differential geometry) published in 1996"

The basic Laplacian of a Riemannian foliation

Abstract: We study the basic Laplacian on Riemannian foliations by writing the basic Laplacian in terms of the orthogonal projection from square-integrable forms to basic square-integrable forms. Using a geometric interpretation of this projection, we relate the ordinary Laplacian to the basic Laplacian. Among other results, we show the existence of the basic heat kernel and establish estimates for the eigenvalues of the basic Laplacian. Introduction. Let M be a compact oriented manifold and let be a transver- sally oriented foliation on M. A foliation is a Riemannian foliation if there exists a Riemannian metric on M with the property that the distance from one leaf of to another is locally constant; such a metric is called a bundle-like metric for . Associated to are the space of basic forms: Ω B(M )= Ω B ( M , )= Ω ( M ): i(X) =0 , i ( X ) d = 0 for all X Γ(T ) , where i(X) is the interior product with the vector field X and Γ(T ) denotes the sections of the distribution T associated to . The exterior derivative d maps basic forms to basic forms; let dB denote d restricted to Ω B (M). The basic Laplacian is the operator ΔB = dB B + BdB on basic forms, where B is the adjoint of dB on Ω B (M). The analytic and geometric properties of this operator have been studied by several researchers. In (5), the basic Laplacian was studied as an operator on basic functions (i.e., functions that are constant on leaves of ), and the author proved the existence of the heat kernel in this case. In (13), the existence of the heat kernel on basic forms was proved for the case where the mean curvature form of the foliation is basic. There are also "basic" Hodge theorems, for example (6) and (10). However, the proof of the Hodge theorem in (6) does not yield various estimates that are important in applications, while the theorem proved in (10) has the same restriction as the results in (13), namely that the authors require the mean curvature form to be basic. In this paper, we study the basic Laplacian on forms, without any restriction on the mean curvature. We prove the existence and uniqueness of the heat kernel for ΔB on forms for any Riemannian foliation, and we write down an explicit formula for the heat kernel. We also present a proof of the Hodge theorem for

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 ∑

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.

An Application of Convex Integration to Contact Geometry

Abstract: We prove that every closed, orientable 3-manifold M admits a parallelization by the Reeb vector fields of a triple of contact forms with equal volume form. Our proof is based on Gromov's convex integration technique and the h-principle. Similar methods can be used to show that M admits a parallelization by contact forms with everywhere linearly independent Reeb vector fields. We also prove a generalization of this latter result to higher dimensions. If M is a closed (2n + l)-manifold with contact form w whose contact distribution ker w admits k everywhere linearly independent sections, then M admits k + 1 linearly independent contact forms with linearly independent Reeb vector fields.

Heated distribution manifold

Abstract: A distribution manifold assembly for carrying a fluid includes a manifold body having at least one fluid channel running therethrough, and at least one infrared radiation source dimensioned and disposed to apply infrared radiation to the external surface of the manifold body. If desired, a control means controls the application of infrared radiation by within the fluid channel remains substantially constant. An insulation layer can surround at least a portion of the manifold body, and this insulation layer is disposed so as not to obstruct completely the infrared radiation source. In a second embodiment, the manifold assembly has a primary manifold body having at least one fluid channel running therethrough, and first and second surfaces so that there are first and second elongate hollows. First and second infrared radiation sources are contained within the elongate hollows. A method of heating a fluid distribution manifold having an internal fluid channel is also provided.

Differential-Geometric Structures On Manifolds

Abstract: In the present chapter we provide most of the prerequisites for reading the rest of the book. In the first two sections we present a review of vector bundles and introduce the main differential operators: Lie derivative, exterior differential, linear connection, general connection. Distributions on manifolds (known as non-holonomic spaces in classical terminology) are then introduced and studied by using both methods of vector fields and of differential 1-forms. We give here the characterization for the existence of a transversal distribution to a foliation, which is found to be very useful in Chapters 4 and 5 for a general study of lightlike submanifolds. In the last two sections we deal with semi-Riemannian manifolds and lightlike manifolds. While the geometry of a semi-Riemannian manifold is fully developed by using the Levi-Civita connection we stress the role of the radical distribution in studying the geometry of a lightlike manifold. The main formulas and results are expressed by using both the invariant form and the index form.

A Symmetric Product for Vector Fields and its Geometric Meaning

Abstract: We introduce the notion of geodesic invariance for distributions on manifolds with a linear connection. This is a natural weakening of the concept of a totally geodesic foliation to allow distributions which are not necessarily integrable. To test a distribution for geodesic invariance, we introduce a symmetric, vector field valued product on the set of vector fields on a manifold with a linear connection. This product serves the same purpose for geodesically invariant distributions as the Lie bracket serves for integrable distributions. The relationship of this product with connections in the bundle of linear frames is also discussed. As an application, we investigate geodesically invariant distributions associated with a left-invariant affine connection on a Lie group.

The Circle as a Fermionic Distribution

Abstract: Let us consider the free loop space of a manifold. It is very well known that the equivariant cohomology of the free loop spce is related to the index theorem for a finite dimensional Dirac operator. The reader can see [Bis85], [Bis86], [GJP90], [JP90] for instance.

Higher-order constrained systems on fibered manifolds: An exterior differential systems approach

Abstract: Some recent results on higher-order Lagrangean systems are presented. The concept of higher-order Lagrangean system as a Lepagean two-form defined on a certain jet prolongation of a fibered manifold over a one-dimensional base is recalled. The dynamics then can be defined by a distribution (the Euler-Lagrange distribution) which generally is of non-constant rank. This approach leads to a natural geometric definition of regularity and a geometric classification of constrained systems. Since a Lagrangean system is understood as a class of equivalent Lagrangians (which can be of different orders), the theory, including a Hamilton formulation, is independent on the choice of a particular Lagrangian for the Lagrangean system under consideration. Relations to the symplectic, presymplectic, cosymplectic and precosymplectic geometry are discussed.

A non-linear method for the calculation of the loss tangent distribution function

Abstract: A procedure to obtain the tangent distribution function from internal friction peaks in solids is described. The method was applied to simulated data in order to show that the distribution can reproduce accurately the internal friction data in transition region as well as in most of the terminal region. Finally, the procedure is applied to experimental data in glass transition peaks of polystyrene.

Notes on the integrability of exit times from unbounded domains of Brownian motion

Abstract: The integrability and the tail distribution of the first exit time from unbounded domain of Brownian motions will be con- sidered. They are characterized by the growth order of the first eigen values of the intersection of domains and sphere with radius r and quasi- hyperbolic distance.

Lightlike Hypersurfaces of Semi-Riemannian Manifolds

Abstract: Here we develop a theory on the differential geometry of a lightlike hypersurface M of a proper semi-Riemannian manifold \(\bar M\) For this purpose, we introduce a non-degenerate screen distribution and construct the corresponding lightlike transversal vector bundle tr(TM) of M, consistent with the well-known theory of Riemannian submanifolds. This enables one to define the induced geometrical objects such as linear connection, second fundamental form, shape operator, etc., and to obtain the Gauss-Codazzi equations leading to the Fundamental Theorem of lightlike hyper-surfaces. It is noteworthy that the second fundamental form (and, therefore, the results on totally geodesic and totally umbilical lightlike hypersurfaces) is independent of the choice of a screen distribution.

CR Lightlike Submanifolds of Indefinite Kaehler Manifolds

Abstract: In this Chapter, we study Cauchy Riemann (CR) lightlike hypersurfaces and submanifolds (in general) of indefinite Hermitian and Kaehler manifolds. We prove that a lightlike real hypersurface M of an indefinite Hermitian manifold is a CR manifold and show that the integrability of all distributions of M is characterized by both second fundamental forms of M and its screen distribution S(TM). Finally, we study the geometry of various foliations on a CR lightlike submanifold and the existence of CR lightlike products of \(\bar M\).

Robust estimation of the location of a vertical tangent in distribution

Abstract: It is shown that the location of the set of m + 1 observations with minimal diameter, within local data, is a robust estimator of the location of a vertical tangent in a distribution function. The rate of consistency of these estimators is shown to be the same as that of asymptotically efficient estimators for the same model. Robustness means (1) only properties of the distribution local to the vertical tangent play a role in the asymptotics, and (2) these asymptotics can be proven given approximate information about just two parameters, the shape and quantile of the vertical tangent.

Theory of connections on graded principal bundles

Abstract: The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded section and, further, that the sheaf of vertical derivations on such a bundle coincides with the graded distribution induced by the action of the structure graded Lie group. This result leads to a natural definition of the graded connection in terms of graded distributions; its relation with Lie superalgebra-valued graded differential forms is also exhibited. Finally, we define the curvature for the graded connection; we prove that the curvature controls the involutivity of the horizontal graded distribution corresponding to the graded connection.

Glassy dynamics of driven elastic manifolds

Abstract: We study the low-temperature dynamics of an elastic manifold driven through a random medium. For driving forces well below the zero- temperature depinning force, the manifold advances via thermally activated hops over the energy barriers separating favorable metastable states. We develop a scaling theory of the thermally activated dynamics (creep) and find a nonlinear glassy response for the driven manifold, {upsilon}{approximately}exp(-const{times}F{sup - {mu}}). We consider an exactly solvable 1-D model for random driven dynamics which exhibits a creep-like velocity-force characteristic. We discuss a microscopic mechanism for the creep motion and show that the distribution of waiting times for the hopping processes scales as a power law. This power-law distribution naturally yields an exponential response for the creep of the manifold.