Showing papers on "Distribution (differential geometry) published in 2010"

A geometric theory of thermal stresses

Abstract: In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change in temperature corresponds to a change in the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change in the material manifold, i.e., a change in the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configura...

A Three-Generation Calabi-Yau Manifold with Small Hodge Numbers

Abstract: We present a complete intersection Calabi-Yau manifold Y that has Euler number -72 and which admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z_12 and the non-Abelian dicyclic group Dic_3. The quotient manifolds have chi=-6 and Hodge numbers (h^11,h^21)=(1,4). With the standard embedding of the spin connection in the gauge group, Y gives rise to an E_6 gauge theory with 3 chiral generations of particles. The gauge group may be broken further by means of the Hosotani mechanism combined with continuous deformation of the background gauge field. For the non-Abelian quotient we obtain a model with 3 generations with the gauge group broken to that of the standard model. Moreover there is a limit in which the quotients develop 3 conifold points. These singularities may be resolved simultaneously to give another manifold with (h^11,h^21)=(2,2) that lies right at the tip of the distribution of Calabi-Yau manifolds. This strongly suggests that there is a heterotic vacuum for this manifold that derives from the 3 generation model on the quotient of Y. The manifold Y may also be realised as a hypersurface in the toric variety. The symmetry group does not act torically, nevertheless we are able to identify the mirror of the quotient manifold by adapting the construction of Batyrev.

Two-dimensional almost-Riemannian structures with tangency points

Abstract: Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. We study the relation between the topological invariants of an almost-Riemannian structure on a compact oriented surface and the rank-two vector bundle over the surface which defines the structure. We analyse the generic case including the presence of tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper provides a classification of oriented almost-Riemannian structures on compact oriented surfaces in terms of the Euler number of the vector bundle corresponding to the structure. Moreover, we present a Gauss–Bonnet formula for almost-Riemannian structures with tangency points.

On the Support of Minimizers of Causal Variational Principles

Abstract: A class of causal variational principles on a compact manifold is introduced and analyzed both numerically and analytically. It is proved under general assumptions that the support of a minimizing measure is either completely timelike, or it is singular in the sense that its interior is empty. In the examples of the circle, the sphere and certain flag manifolds, the general results are supplemented by a more detailed and explicit analysis of the minimizers. On the sphere, we get a connection to packing problems and the Tammes distribution. Moreover, the minimal action is estimated from above and below.

Points and Curves in the Monster Tower

Abstract: Cartan introduced the method of prolongation which can be applied either to manifolds with distributions (Pfaffian systems) or integral curves to these distributions. Repeated application of prolongation to the plane endowed with its tangent bundle yields the Monster tower, a sequence of manifolds, each a circle bundle over the previous one, each endowed with a rank 2 distribution. In an earlier paper (2001), the authors proved that the problem of classifying points in the Monster tower up to symmetry is the same as the problem of classifying Goursat distribution flags up to local diffeomorphism. The first level of the Monster tower is a three-dimensional contact manifold and its integral curves are Legendrian curves. The philosophy driving the current work is that all questions regarding the Monster tower (and hence regarding Goursat distribution germs) can be reduced to problems regarding Legendrian curve singularities.

On the local structure of Lorentzian Einstein manifolds with parallel distribution of null lines

Abstract: We study transformations of coordinates on a Lorentzian Einstein manifold with a parallel distribution of null lines and show that the general Walker coordinates can be simplified. In these coordinates, the full Lorentzian Einstein equation is reduced to equations on a family of Einstein Riemannian metrics.

Smooth distributions are finitely generated

Abstract: A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.

Integrable GL(2) geometry and hydrodynamic partial differential equations

Abstract: This article is a local analysis of integrable GL(2)-structures of degree 4. A GL(2)-structure of degree n corresponds to a distribution of rational normal cones over a manifold M of dimension (n+1). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These GL(2)-structures are important because they naturally arise from a certain family of second-order hyperbolic PDEs in three variables that are integrable via hydrodynamic reduction. Familiar examples include the wave equation, the first flow of the dKP equation, and the Boyer--Finley equation. The main results are a structure theorem for integrable GL(2)-structures, a classification for connected integrable GL(2)-structures, and an equivalence between local integrable GL(2)-structures and Hessian hydrodynamic hyperbolic PDEs in three variables. This yields natural geometric characterizations of the wave equation, the first flow of the dKP hierarchy, and several others. It also provides an intrinsic, coordinate-free infrastructure to describe a large class of hydrodynamic integrable systems in three variables.

Bi-paracontact structures and Legendre foliations

Abstract: We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold $(M,\eta)$, then under some natural assumptions of integrability, $M$ carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures are defined on a contact manifold, then $M$ admits an almost bi-paracontact structure. We define a canonical connection on an almost bi-paracontact manifold and we study its curvature properties, which resemble those of the Obata connection of an anti-hypercomplex (or complex-product) manifold. Further, we prove that any contact metric manifold whose Reeb vector field belongs to the $(\kappa,\mu)$-nullity distribution canonically carries an almost bi-paracontact structure and we apply the previous results to the theory of contact metric $(\kappa,\mu)$-spaces.

Maximally homogeneous para-CR manifolds of semisimple type

Abstract: An almost para-CR structure on a manifold $M$ is given by a distribution $HM \subset TM$ together with a field $K \in \Gamma({\rm End}(HM))$ of involutive endomorphisms of $HM$. If $K$ satisfies an integrability condition, then $(HM,K)$ is called a para-CR structure. The notion of maximally homogeneous para-CR structure of a semisimple type is given. A classification of such maximally homogeneous para-CR structures is given in terms of appropriate gradations of real semisimple Lie algebras.

Bi-paracontact structures and Legendre foliations

Abstract: We study almost bi-paracontact structures on contact manifolds. We prove that if an almost bi-paracontact structure is defined on a contact manifold (M, η), then under some natural assumptions of integrability M carries two transverse bi-Legendrian structures. Conversely, if two transverse bi-Legendrian structures are defined on a contact manifold, then M admits an almost bi-paracontact structure. We define a canonical connection on an almost bi-paracontact manifold and we study its curvature properties, which resemble those of the Obata connection of a para-hypercomplex (or complex-product) manifold. Further, we prove that any contact metric manifold whose Reeb vector field belongs to the (κ, μ)-nullity distribution canonically carries an almost bi-paracontact structure and we apply the previous results to the theory of contact metric (κ, μ)-spaces.

Integrability of weak distributions on Banach manifolds

Abstract: This paper concerns the problem of integrability of non closed distributions on Banach manifolds. We introduce the notion of weak distribution and we look for conditions under which these distributions admit weak integral submanifolds. We give some applications to Banach Lie algebroid and Banach Lie-Poisson manifold. The main results of this paper generalize the works presented in [ChSt], [Nu] and [Gl].

Gravity-fed drip irrigation design procedure for a single-manifold subunit

Abstract: A simple procedure was developed for the design of low-cost, gravity-fed, drip irrigation single-manifold subunits in hilly areas with laterals to one or both sides of the manifold. The allowable pressure head variation in the manifold and laterals is calculated individually for different pressure zones, and the manifold subunit design is divided into independent processes for laterals and manifold. In the manifold design, a two-stage optimal design method is used. In the first design stage, the pipe cost is minimized and a set of optimal manifold pipe diameters is obtained. In the second design stage, a partial list of available diameters is prepared based on the calculated optimal diameters, and the lengths for available diameters and pressure head of every lateral location along the manifold are calculated. The size of each of the pressure sections is determined according to the pressure head distribution along the manifold. Using the proposed methodology, the minimum manifold pipe cost is obtained, and the target emission uniformity is satisfied for gravity-fed drip irrigation subunits.

Contact structures of arbitrary codimension and idempotents in the Heisenberg algebra

Abstract: A contact manifold is a manifold equipped with a distribution of codimension one that satisfies a `maximal non-integrability' condition. A standard example of a contact structure is a strictly pseudoconvex CR manifold, and operators of analytic interest are the tangential Cauchy-Riemann operator and the Szego projector onto its kernel. The Heisenberg calculus is the natural pseudodifferential calculus developed originally for the analysis of these operators. We introduce a `non-integrability' condition for a distribution of arbitrary codimension that directly generalizes the definition of a contact structure. We call such distributions polycontact structures. We prove that the polycontact condition is equivalent to the existence of generalized Szego projections in the Heisenberg calculus, and explore geometrically interesting examples of polycontact structures.

Global stability of special multi-flags

Abstract: In this paper, we study special multi-flags on manifolds. A special multiflag is a certain nested sequence of subbundles of the tangent bundle which are derived by Lie brackets. A property of a special multi-flag is characterized by the existence of a completely integrable subdistribution of corank one in the largest distribution in the sequence, which is a so-called covariant subdistribution. It is proved that a one-parameter deformation of a special multi-flag on a compact manifold can be described by a family of global diffeomorphisms of the underlying manifold, if the covariant subdistribution of the largest distribution in the flag is preserved.

Sub-Riemannian geodesics and heat operator on odd dimensional spheres

Abstract: In this article we study the sub-Riemannian geometry of the spheres $S^{2n+1}$ and $S^{4n+3}$, arising from the principal $S^1-$bundle structure defined by the Hopf map and the principal $S^3-$bundle structure given by the quaternionic Hopf map respectively. The $S^1$ action leads to the classical contact geometry of $S^{2n+1}$, while the $S^3$ action gives another type of sub-Riemannian structure, with a distribution of corank 3. In both cases the metric is given as the restriction of the usual Riemannian metric on the respective horizontal distributions. For the contact $S^7$ case, we give an explicit form of the intrinsic sub-Laplacian and obtain a commutation relation between the sub-Riemannian heat operator and the heat operator in the vertical direction.

On the number of weakly Noetherian constants of motion ofnonholonomic systems

Abstract: We develop a method to give an estimate on the number of functionally independent constants of motion of a nonholonomic system with symmetry which have the so called 'weakly Noetherian' property [22]. We show that this number is bounded from above by the corank of the involutive closure of a certain distribution on the constraint manifold. The effectiveness of the method is illustrated on several examples.

Humans Learn Using Manifolds, Reluctantly

Abstract: When the distribution of unlabeled data in feature space lies along a manifold, the information it provides may be used by a learner to assist classification in a semi-supervised setting. While manifold learning is well-known in machine learning, the use of manifolds in human learning is largely unstudied. We perform a set of experiments which test a human's ability to use a manifold in a semi-supervised learning task, under varying conditions. We show that humans may be encouraged into using the manifold, overcoming the strong preference for a simple, axis-parallel linear boundary.

Tracy-Widom GUE law and symplectic invariants

Abstract: We establish the relation between two objects: an integrable system related to Painleve II equation, and the symplectic invariants of a certain plane curve \Sigma_{TW} describing the average eigenvalue density of a random hermitian matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This explains directly how the Tracy-Widow law F_{GUE}, governing the distribution of the maximal eigenvalue in hermitian random matrices, can also be recovered from symplectic invariants.

System for improving gas distribution in an intake manifold

Abstract: A system for improving distribution of gases within an intake manifold of an engine is presented. The system may be used to improve engine air-fuel control. In one example, turbulence of gases entering an intake manifold is increased.

Nonlinear embedding towards articulated spine shape inference using higher-order MRFs

Abstract: In this paper we introduce a novel approach for inferring articulated spine models from images. A low-dimensional manifold embedding is created from a training set of prior mesh models to establish the patterns of global shape variations. Local appearance is captured from neighborhoods in the manifold once the overall representation converges. Inference with respect to the manifold and shape parameters is performed using a Markov Random Field (MRF). Singleton and pairwise potentials measure the support from the data and shape coherence from neighboring models respectively, while higher-order cliques encode geometrical modes of variation for local vertebra shape warping. Optimization of model parameters is achieved using efficient linear programming and duality. The resulting model is geometrically intuitive, captures the statistical distribution of the underlying manifold and respects image support in the spatial domain. Experimental results on spinal column geometry estimation from CT demonstrate the approach's potential.

The Geometry of the Osculating Nilpotent Group Structures of the Heisenberg Calculus

Abstract: We explore the geometry that underlies the osculating nilpotent group structures of the Heisenberg calculus. For a smooth manifold $M$ with a distribution $H\subseteq TM$ analysts use explicit (and rather complicated) coordinate formulas to define the nilpotent groups that are central to the calculus. Our aim in this paper is to provide insight in the intrinsic geometry that underlies these coordinate formulas. First, we introduce `parabolic arrows' as a generalization of tangent vectors. The definition of parabolic arrows involves a mix of first and second order derivatives. Parabolic arrows can be composed, and the group of parabolic arrows can be identified with the nilpotent groups of the (generalized) Heisenberg calculus. Secondly, we formulate a notion of exponential map for the fiber bundle of parabolic arrows, and show how it explains the coordinate formulas of osculating structures found in the literature on the Heisenberg calculus. The result is a conceptual simplification and unification of the treatment of the Heisenberg calculus.

Screen conformal einstein lightlike hypersurfaces of a lorentzian space form

Abstract: In this paper, we study the geometry of lightlike hypersur- faces of a semi-Riemannian manifold. We prove a classiflcation theorem for Einstein lightlike hypersurfaces M of a Lorentzian space form subject such that the second fundamental forms of M and its screen distribution S(TM) are conformally related by some non-vanishing smooth function.

Lightlike real hypersurfaces with totally umbilical screen distributions

Abstract: In this paper, we study the geometry of lightlike real hyper- surfaces of an indefinite Kaehler manifold. The main result is a character- ization theorem for lightlike real hypersurfaces M of an indefinite complex space form ¯ M(c) such that the screen distribution is totally umbilic.

Manifold and distribution manifold assembly for air-entrained material

Abstract: A manifold and distribution manifold assembly is described for use in dividing the flow of an air-entrained material from a primary distribution line into a plurality of secondary distribution lines. The manifold is formed of a single molded piece thereby reducing the overall tooling cost. One application of such a distribution manifold assembly is in an air seeder.

The asymptotic expansion of Tracy-Widom GUE law and symplectic invariants

Abstract: We establish the relation between two objects: an integrable system related to Painleve II equation, and the symplectic invariants of a certain plane curveTW. This curve describes the average eigenvalue density of a random hermitian matrix spectrum near a hard edge (a bound for its maximal eigenvalue). This shows that the s ! −1 asymptotic expansion of Tracy-Widow law FGUE(s), governing the distribution of the maximal eigenvalue in hermitian random matrices, is given by symplectic invariants.

Pseudoharmonic maps and vector fields on CR manifolds

Abstract: Building on the work by J. Jost and C.-J. Xu (32), and E. Barletta et al. (3), we study smooth pseudoharmonic maps from a compact strictly pseudoconvex CR manifold and their generalizations e.g. pseudoharmonic unit tangent vector fields. The purpose of this paper is to study several analogs to differential geometric objects appearing in Riemannian geometry and admitting a treatment based on elliptic theory e.g. the Laplace-Beltrami operator (cf. (40)), harmonic maps among Riemannian manifolds (cf. (49)), and harmonic vector fields (regarded as smooth maps of a Riemannian manifold into the total space of the tangent bundle endowed with the Sasaki metric, cf. (51 )a nd (52)). We obtain the following results. Boundary values of Bergman-harmonic maps � :� ! S from a smoothly bounded strictly pseudoconvex domain � � C n into a Riemannian manifoldS are shown to be pseudoharmonic maps, provided their normal derivatives vanish. We prove that @b-pluriharmonic maps are pseudoharmonic maps. A pseudoharmonic map� : M ! Sfrom a compact strictly pseudoconvex CR manifold into a sphere is shown either to link or to meet any codimension 2 totally geodesic sphere in S � . Also we prove that a smooth vector field X : M ! T ðMÞ from a strictly pseudoconvex CR manifold M is a pseudoharmonic map if and only if X is parallel (with respect to the Tanaka-Webster connection) along the maximally complex, or Levi, distribution. We start a theory of pseudoharmonic vector fields i.e. unit vector fields X 2 U ðM;� Þ which are critical points of the energy functional EðX Þ¼ 1 R M traceG� ð� HXS� Þ� ^ð d� Þ n relative to variations through unit vector fields. Any such critical point X is shown to satisfy the nonlinear subelliptic systembX þk r H Xk 2 X ¼ 0 .A lsoinfX2U ðM;� ÞEðX Þ¼ n VolðM;� Þ yet E is unbounded from above. We establish first and second variation formulae for E : U ðM;� Þ!½ 0; þ1Þ and give applications.

Distribution manifold with multiple dispensing needles

Abstract: A distribution manifold having a manifold and a removable cartridge with a plurality of needle tubes extending from the removable cartridge is disclosed. The distribution manifold can be used with a pre-metered coating system to apply coating material to a substrate.

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, that was first introduced by two of the authors; and the 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.