Showing papers on "Distribution (differential geometry) published in 2008"

Branes and Quantization for an A-Model Complexification of Einstein Gravity in Almost Kahler Variables

Abstract: The general relativity theory is redefined equivalently in almost Kahler variables: symplectic form and canonical symplectic connection (distorted from the Levi-Civita connection by a tensor constructed only from metric coefficients and their derivatives). The fundamental geometric and physical objects are uniquely determined in metric compatible form by a (pseudo) Riemannian metric on a manifold enabled with a necessary type nonholonomic 2+2 distribution. Such nonholonomic symplectic variables allow us to formulate the problem of quantizing Einstein gravity in terms of the A-model complexification of almost complex structures on spacetime manifold, generalizing the Gukov-Witten method, see arXiv:0809.0305. Quantizing the complexified model, we derive a Hilbert space as a space of strings with two A-branes which for the Einstein gravity theory are nonholonomic because of induced nonlinear connection structures. Finally, we speculate on relation of such a method of quantization to curve flows and solitonic hierarchies defined by Einstein metrics on (pseudo) Riemannian spacetimes.

A Spinor Approach to Walker Geometry

Abstract: A four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes. This distribution has a natural characterization as a projective spinor field subject to a certain constraint. Spinors therefore provide a natural tool for studying Walker geometry, which we exploit to draw together several themes in recent explicit studies of Walker geometry and in other work of Dunajski [11] and Plebanski [30] in which Walker geometry is implicit. In addition to studying local Walker geometry, we address a global question raised by the use of spinors.

Smooth deformations of piecewise expanding unimodal maps

Abstract: In the space of $C^k$ piecewise expanding unimodal maps, $k\geq 1$, we characterize the $C^1$ smooth families of maps where the topological dynamics does not change (the "smooth deformations") as the families tangent to a continuous distribution of codimension-one subspaces (the "horizontal" directions) in that space. Furthermore such codimension-one subspaces are defined as the kernels of an explicit class of linear functionals. As a consequence we show the existence of $C^{k-1+Lip}$ deformations tangent to every given $C^k$ horizontal direction, for $k\ge 2$.

The geometric structures of the Weibull distribution manifold and the generalized exponential distribution manifold

Abstract: Investigating the geometric structures of the distribution manifolds is a basic task in information geometry. However, by so far, most works are on the distribution manifolds of exponential family. In this paper, we investigate two non-exponential distribution manifolds —the Weibull distribution manifold and the generalized exponential distribution manifold. Then we obtain their geometric structures.

On local geometry of rank 3 distributions with 6-dimensional square

Abstract: We solve the equivalence problem for rank 3 completely nonholonomic vector distributions with 6-dimensional square on a smooth manifold of arbitrary dimension n under very mild genericity conditions. The main idea is to consider the projectivization of the annihilator of a given 3-dimensional distribution. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. The problem of equivalence of distributions can be essentially reduced to the differential geometry of such curves. The class of all 3-distributions under consideration is split into a finite number of subclasses according to the Young diagram of their flags. The local geometry of distributions can be recovered from the properties of the symmetry group of so-called flat curves of flags associated with this Young diagram. In each subclass we describe the flat distribution and construct a canonical frame for any other distribution. It turns out that for n>6 in the most nontrivial case the symmetry algebra of the flat distribution can be described in terms of rational normal curves (their secants and tangential developables) in projective spaces and its dimension grows exponentially with respect to n.

Fluidic distribution system and related methods

Abstract: Embodiments of the present invention relate to a fluid distribution system. The system may include one or more electrochemical cell layers, a bulk distribution manifold having an inlet, a cell layer feeding manifold in direct fluidic contact with the electrochemical cell layer and a separation layer that separates the bulk distribution manifold from the cell feeding manifold, providing at least two independent paths for fluid to flow from the bulk distribution manifold to the cell feeding manifold.

A note on distribution spaces on manifolds 1

Abstract: Twenty-six difierent concrete representations of the space of vector valued distributions on a smooth manifold of dimension n are presented systematically, most of them new. In the particular case of representations as module homomorphisms acting on sections of the dual bundle resp. on n-forms, the continuity of these homomorphisms is already a consequence of their algebraic properties.

Killing and Geodesic Lightlike Hypersurfaces of Indefinite Sasakian Manifolds

Abstract: In this paper, we study a lightlike hypersurface of indefinite Sasakian manifold, tangent to the structure vector field x. Theorems on parallel and Killing distributions are obtained. Necessary and sufficient conditions have been given for lightlike hypersurface to be mixed totally geodesic, D-totally geodesic, D\perp-totally geodesic and D'-totally geodesic. We prove that, if the screen distribution of lightlike hypersurface M of indefinite Sasakian manifold is totally umbilical, the D\perp-geodesibility of M is equivalent to the D\perp-parallelism of the distribution T M\perp of rank 1 (Theorem \ref{Theoscre}). Finally, we give the D\perp-version (Theorem 4.22) of the Theorem 2.2 ([11], page 88).

An information geometry algorithm for distribution control

Abstract: In this paper, we consider the problem of distribution control from the viewpoint of information geometry. Different from most existing models used in stochastic control, it is assumed that the control input directly affects the distribution of the system output in probability sense. Here, we set up a new manifold (S), meanwhile the B-spline manifold (B) and the system output manifold (M) can be referred to as its submanifolds. We give an information geometrical algorithm which can be called as geodesic-projection algorithm using the properties of manifold. In the geodesic step, we can obtain the geodesic equation from the initial point V0 = (ω10, ω20, ··· , ω(n−1)0) to the specified point Vg = (ω1g, ω2g, ··· , ω(n−1)g) in B. This gives us an optimal trajectory for the points changing along in B. In the projection step, we project the sample points selected from the geodesic onto M. The coordinates of the projections in M give the trajectory of the control input u.

Integrability of exit times and ballisticity for random walks in Dirichlet environment

Abstract: We consider random walks in Dirichlet environment, introduced by Enriquez and Sabot in 2006. As this distribution on environments is not uniformly elliptic, the annealed integrability of exit times out of a given finite subset is a non-trivial property. We provide here an explicit equivalent condition for this integrability to happen, on general directed graphs. Such integrability problems arise for instance from the definition of Kalikow auxiliary random walk. Using our condition, we prove a refined version of the ballisticity criterion given by Enriquez and Sabot.

The curvature of contact structure on 3-manifolds

Abstract: We study the sectional curvature of plane distributions on 3-manifolds. We show that if the distribution is a contact structure it is easy to manipulate this curvature. As a corollary we obtain that for every transversally oriented contact structure on a closed 3-dimensional manifold $M$ there is a metric, such that the sectional curvature of the contact distribution is equal to -1. We also introduce the notion of Gaussian curvature of the plane distribution. For this notion of curvature we get the similar results.

An equivariant index formula for almost-CR manifolds

Abstract: We consider a consider the case of a compact manifold M, together with the following data: the action of a compact Lie group H and a smooth H-invariant distribution E, such that the H-orbits are transverse to E. These data determine a natural equivariant differential form with generalized coefficients J(E,X) whose properties we describe. When E is equipped with a complex structure, we define a class of symbol mappings in terms of the resulting almost-CR structure that are H-transversally elliptic whenever the action of H is transverse to E. We determine a formula for the H-equivariant index of such symbols that involves only J(E,X) and standard equivariant characteristic classes. This formula generalizes the formula given in arXiv:0712.2431 for the case of a contact manifold.

Myller configurations in finsler spaces. applications to the study of subspaces and of torse forming vector fields

Abstract: In this paper we define a Myller configuration in a Finsler space and use some special configurations to obtain results about Finsler subspaces. Let F n =(M, F ) be a Finsler space, with M a real, differentiable manifold of dimension n. Using the pull back bundle (π∗TM, π, TM) of the tangent bundle (TM, π, M) by the mapping π = π/TM and the Cartan Finsler connection of a Finsler space, we obtain an orthonormal frame of sections of π∗TM along a regular curve in TM and a system of invariants, geometrically associated to the Myller configuration. The fundamental equations are written in a very simple form and we prove a fundamental theorem. Important lines in a Finsler subspace are defined like special lines in a Myller configuration, geometrically associated to the subspace: auto parallels, lines of curvature, asymptotes. Torse forming vector fields with respect to the Cartan Finsler connection are characterized by means of the invariants of the Frenet frame of a versor field along a curve, and the new notion of torse forming vector fields in the sense of Myller is introduced. The particular cases of concurrence and parallelism in the sense of Myller are completely studied, for vector fields from the distribution T m of the Myller configuration and also from the normal distribution T p.

On existence of canonical screens for coisotropic submanifolds

Abstract: In this paper we study coisotropic lightlike submanifolds of a semi- Riemannian manifold. For a large variety of this class of submanifolds, we prove two theorems on the existence of integrable canonical screen distribution and canonical null transversal bundle subject to some reasonable geometric conditions.

Classification of a class of minimal semi-Einstein submanifolds with an integrable conullity distribution

Abstract: A full local classification and a geometric description of normally flat minimal semi-Einstein submanifolds of Euclidean spaces having multiple principal curvature vectors and an integrable conullity distribution are presented. Bibliography: 30 titles.

Covariance fields

Abstract: We introduce and study covariance fields of distributions on a Riemannian manifold At each point on the manifold, covariance is defined to be a symmetric and positive definite (2,0)-tensor Its product with the metric tensor specifies a linear operator on the respected tangent space Collectively, these operators form a covariance operator field We show that, in most circumstances, covariance fields are continuous We also solve the inverse problem: recovering distribution from a covariance field Surprisingly, this is not possible on Euclidean spaces On non-Euclidean manifolds however, covariance fields are true distribution representations

Conic Distributions and Accessible Sets

Abstract: Motivated by nonlinear control theory, we introduce the notion of conic distributions on a smooth manifold. We study topological and smoothness aspects of the set of accessible points associated with a conic distribution. We introduce the notion of abnormal paths and study their relation to boundary points of the accessible set. In particular, we provide sufficient conditions for the accessible set to be a maximal integral of the smallest integrable vector distribution containing the conic distribution. Under rather strong conditions, we are able to prove that the accessible set has the structure of a `manifold with corners'.

Dynamical systems and nilpotent sub-Riemannian geometry

Abstract: We consider dynamical systems determined by distributions of real analytic vector fields, the equations of motion, and the associated gauge transformation are presented in detail. A gradation of the associated Lie algebras leads to the consideration of polynomial vector fields. The minimization of kinetic energy action in the class of horizontal curves associated with the distribution is formulated as a sub-Riemannian geodesic problem. Normal geodesics are fully described for the so-called Gaveau–Brockett distribution. The exponential mapping, the unitary spheres, and the wave fronts are calculated for particular cases.

Covariance of centered distributions on manifold

Abstract: We define and study a family of distributions with domain complete Riemannian manifold. They are obtained by projection onto a fixed tangent space via the inverse exponential map. This construction is a popular choice in the literature for it makes it easy to generalize well known multivariate Euclidean distributions. However, most of the available solutions use coordinate specific definition that makes them less versatile. %We propose improvements in two directions. We define the distributions of interest in coordinate independent way by utilizing co-variant 2-tensors. Then we study the relation of these distributions to their Euclidean counterparts. In particular, we are interested in relating the covariance to the tensor that controls distribution concentration. We find approximating expression for this relation in general and give more precise formulas in case of manifolds of constant curvature, positive or negative. Results are confirmed by simulation studies of the standard normal distribution on the unit-sphere and hyperbolic plane.

Comparing and interpolating distributions on manifold

Abstract: We are interested in comparing probability distributions defined on Riemannian manifold. The traditional approach to study a distribution relies on locating its mean point and finding the dispersion about that point. On a general manifold however, even if two distributions are sufficiently concentrated and have unique means, a comparison of their covariances is not possible due to the difference in local parametrizations. To circumvent the problem we associate a covariance field with each distribution and compare them at common points by applying a similarity invariant function on their representing matrices. In this way we are able to define distances between distributions. We also propose new approach for interpolating discrete distributions and derive some criteria that assure consistent results. Finally, we illustrate with some experimental results on the unit 2-sphere.

Nonlinear Connections and Description of Photon-like Objects

Abstract: This paper aims to present a general idea for description of spatially finite physical objects with a consistent nontrivial translational-rotational dynamical structure and evolution as a whole, making use of the mathematical concepts and structures connected with the Frobenius integrability/nonintegrability theorems given in terms of distributions on manifolds with corresponding curvature defined by the Nijenhuis operator The idea is based on consideration of {\it nonintegrable} subdistributions of some appropriate completely integrable distribution (differential system) on a manifold and then to make use of the corresponding curvatures as generators of measures of interaction, ie of energy-momentum exchange among the physical subsystems mathematically represented by the nonintegrable subdistributions The concept of photon-like object is introduced and description of such objects in these terms is given

Complex vector fields and hypoelliptic partial differential operators

Abstract: We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for $CR$ manifolds and Hormander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of $(0,1)$ vector fields satisfies a subelliptic estimate. v2: minor revision, to appear in Ann. Inst. Fourier

Nonlinear Connections and Description of Photon-like Objects

Abstract: This paper aims to present a general idea for description of spatially finite physical objects with a consistent nontrivial translational-rotational dynamical structure and evolution as a whole, making use of the mathematical concepts and structures connected with the Frobenius integrability/nonintegrability theorems given in terms of distributions on manifolds with corresponding curvature defined by the Nijenhuis operator. The idea is based on consideration of nonintegrable subdistributions of some appropriate completely integrable distribution (differential system) on a manifold and then to make use of the corresponding curvatures as generators of measures of interaction, i.e. of energy-momentum exchange among the physical subsystems mathematically represented by the nonintegrable subdistributions. The concept of photon-like object is introduced and description of such objects in these terms is given.

Multi-zone pressure control system

Abstract: A pressure control system that controls the pressure of a fluid in a plurality of zones includes a distribution manifold, at least one main manifold connected to the distribution manifold, and at least one disposable manifold connected to the distribution manifold and the main manifold. The disposable manifold is adapted to be replaced independent of the distribution manifold and the main manifold, and is connected to each zone and to at least one vacuum source. The distribution manifold distributes the fluid to the plurality of zones, so as to cause flow of the fluid into and out of a measurement chamber located within each zone. The main manifold includes, for each zone, a pressure sensor configured to measure pressure in the measurement chamber in that zone, and a control valve configured to regulate the flow of the fluid through that zone.

The Three Gap Theorem and Riemannian Geometry

Abstract: The classical Three Gap Theorem asserts that for a natural number n and a real number p, there are at most three distinct distances between consecutive elements in the subset of [0,1) consisting of the reductions modulo 1 of the first n multiples of p. Regarding it as a statement about rotations of the circle, we find results in a similar spirit pertaining to isometries of compact Riemannian manifolds and the distribution of points along their geodesics.