Showing papers on "Distribution (differential geometry) published in 2009"

Co-clustering on manifolds

Abstract: Co-clustering is based on the duality between data points (e.g. documents) and features (e.g. words), i.e. data points can be grouped based on their distribution on features, while features can be grouped based on their distribution on the data points. In the past decade, several co-clustering algorithms have been proposed and shown to be superior to traditional one-side clustering. However, existing co-clustering algorithms fail to consider the geometric structure in the data, which is essential for clustering data on manifold. To address this problem, in this paper, we propose a Dual Regularized Co-Clustering (DRCC) method based on semi-nonnegative matrix tri-factorization. We deem that not only the data points, but also the features are sampled from some manifolds, namely data manifold and feature manifold respectively. As a result, we construct two graphs, i.e. data graph and feature graph, to explore the geometric structure of data manifold and feature manifold. Then our co-clustering method is formulated as semi-nonnegative matrix tri-factorization with two graph regularizers, requiring that the cluster labels of data points are smooth with respect to the data manifold, while the cluster labels of features are smooth with respect to the feature manifold. We will show that DRCC can be solved via alternating minimization, and its convergence is theoretically guaranteed. Experiments of clustering on many benchmark data sets demonstrate that the proposed method outperforms many state of the art clustering methods.

Intrinsic mean shift for clustering on Stiefel and Grassmann manifolds

Abstract: The mean shift algorithm, which is a nonparametric density estimator for detecting the modes of a distribution on a Euclidean space, was recently extended to operate on analytic manifolds. The extension is extrinsic in the sense that the inherent optimization is performed on the tangent spaces of these manifolds. This approach specifically requires the use of the exponential map at each iteration. This paper presents an alternative mean shift formulation, which performs the iterative optimization “on” the manifold of interest and intrinsically locates the modes via consecutive evaluations of a mapping. In particular, these evaluations constitute a modified gradient ascent scheme that avoids the computation of the exponential maps for Stiefel and Grassmann manifolds. The performance of our algorithm is evaluated by conducting extensive comparative studies on synthetic data as well as experiments on object categorization and segmentation of multiple motions.

Cones over pseudo-Riemannian manifolds and their holonomy

Abstract: By a classical theorem of Gallot (1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone

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 of temperature corresponds to a change of 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 of the material manifold, i.e. a change of the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configuration for a given temperature distribution, a change of temperature will change the equilibrium configuration. We obtain the explicit form of the governing partial differential equations for this equilibrium change. We also show that geometric linearization of the present nonlinear theory leads to governing equations that are identical to those of the classical linear theory of thermal stresses.

On local geometry of non-holonomic rank 2 distributions

Abstract: In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally non-holonomic rank 2 distributions in R5. We solve the analogous problem for germs of generic rank 2 distributions in Rn for n > 5. We use a completely different approach based on the symplectification of the problem. The main idea is to consider a special odd-dimensional submanifold WD of the cotangent bundle associated with any rank 2 distribution D. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution D. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. Using the classical theory of curves in projective spaces, we construct the canonical frame of the distribution D on a certain (2n - 1)-dimensional fiber bundle over WD with the structure group of all M¨obius transformations, preserving 0. The paper is the detailed exposition of the constructions and the results, announced in the short note (B. Doubrov and I. Zelenko, C. R. Math. Acad. Sci. Paris, Ser. I (8) 342 (2006) 589�594).

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.

G2 geometry and integrable systems

Abstract: We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. We also relate various real forms of the Toda equations to minimal surfaces in quadrics of arbitrary signature. In the case of the Hitchin component for PSL(3,R) we provide a new proof of the relation to convex RP²-structures and hyperbolic affine spheres. For PSp(4,R) we prove such representations are the monodromy for a special class of projective structure on the unit tangent bundle of the surface. We prove these are isomorphic to the convex-foliated projective structures of Guichard and Wienhard. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, work which traces back to Cartan. Nurowski showed that there is an associated signature (2,3) conformal structure. We clarify this as a relationship between a parabolic geometry associated to the split real form of G₂ and a conformal geometry with holonomy in G₂. Moreover in terms of the conformal geometry we prove this distribution is the bundle of maximal isotropics corresponding to the annihilator of a spinor satisfying the twistor-spinor equation. The moduli space of deformations of a compact coassociative submanifold L in a G₂ manifold is shown to have a natural local embedding as a submanifold of H2(L,R). We consider G2-manifolds with a T^4-action of isomorphisms such that the orbits are coassociative tori and prove a local equivalence to minimal 3-manifolds in R^{3,3} = H²(T⁴,R) with positive induced metric. By studying minimal surfaces in quadrics we show how to construct minimal 3-manifold cones in R^{3,3} and hence G₂-metrics from equations that are a set of affine Toda equations. The relation to semi-flat special Lagrangian fibrations and the Monge-Ampere equation is explained.

On the stabilization problem for nonholonomic distributions

Abstract: Let $M$ be a smooth connected and complete manifold of dimension $n$, and $\Delta$ be a smooth nonholonomic distribution of rank $m\leq n$ on $M$. We prove that, if there exists a smooth Riemannian metric on $\Delta$ for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of $\Delta$ on $M$. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using specific results of nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the corresponding value function is semiconcave and is a viscosity solution of a Hamilton-Jacobi equation, and establish fine properties of optimal trajectories.

Two-dimensional almost-Riemannian structures with tangency points

Abstract: Two dimensional almost-Riemannian geometries are metric structures on surfaces defined locally by a Lie bracket generating pair of vector fields. We study the relation between the topology of an almost-Riemannian structure on a compact oriented surface and the total curvature. In particular, we analyse the case in which there exist tangency points, i.e. points where two generators of the distribution and their Lie bracket are linearly dependent. The main result of the paper is a characterization of trivializable oriented almost-Riemannian structures on compact oriented surfaces in terms of the topological invariants of the structure. Moreover, we present a Gauss- Bonnet formula for almost-Riemannian structures with tangency points.

On semi-parallel lightlike hypersurfaces of indefinite Kenmotsu manifolds

Abstract: We study semi-parallel lightlike hypersurfaces of an indefinite Kenmotsu manifold, tangent to the structure vector field. Some Theorems on parallel and semi-parallel vector field, geodesibility of lightlike hypersurfaces are obtained. The geometrical configuration of such lightlike hypersurfaces is established. We prove that, in totally contact umbilical lightlike hypersurfaces of an indefinite Kenmotsu manifold which has constant \({\overline{\phi}}\)-holomorphic sectional curvature c, tangent to the structure vector field and such that its distribution is parallel, the parallelism and semi-parallelism notions are equivalent.

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.

Sheaves of nonlinear generalized functions and manifold-valued distributions

Abstract: This paper is part of an ongoing program to develop a theory of generalized differential geometry. We consider the space G[X, Y ] of Colombeau generalized functions defined on a manifold X and taking values in a manifold Y . This space is essential in order to study concepts such as flows of generalized vector fields or geodesics of generalized metrics. We introduce an embedding of the space of continuous mappings C(X, Y ) into G[X, Y ] and study the sheaf properties of G[X, Y ]. Similar results are obtained for spaces of generalized vector bundle homomorphisms. Based on these constructions we propose the definition of a space D 0 [X, Y ] of distributions on X taking values in Y . D 0 [X, Y ] is realized as a quotient of a certain subspace of G[X, Y ].

Natural gradient-projection algorithm for distribution control

Abstract: In this paper, we use an information geometric algorithm to solve the distribution control problem. Here, we consider the distribution of the output determined by the control input only. We set up two manifolds that are formed by the B-spline functions and the system output probability density functions, and we call them the B-spline manifold(B) and the system output manifold(M), respectively. Moreover, we call the new designed algorithm natural gradient-projection algorithm. In the natural gradient step, we use natural gradient algorithm to obtain an optimal trajectory of the weight vector on the B-spline manifold from the viewpoint of information geometry. In the projection step, we project the selected points on B onto M. The coordinates of the projections on M give the trajectory of the control input u. Copyright © 2008 John Wiley & Sons, Ltd.

Conveyance system including opposed fluid distribution manifolds

Abstract: A fluid conveyance system for thin film material deposition includes a first fluid distribution manifold and a second distribution manifold. The first fluid distribution manifold includes an output face that includes a plurality of elongated slots. The plurality of elongated slots includes a source slot and an exhaust slot. The second fluid distribution manifold includes an output face that includes a plurality of openings. The plurality of openings include a source opening and an exhaust opening. The second fluid distribution manifold is positioned spaced apart from and opposite the first fluid distribution manifold such that the source opening of the output face of the second fluid distribution manifold mirrors the source slot of the output face of the first fluid distribution manifold and the exhaust opening of the output face of the second fluid distribution manifold mirrors the exhaust slot of the output face of the first fluid distribution manifold.

Fluid distribution manifold including bonded plates

Abstract: A fluid distribution manifold includes a first plate and a second plate. At least a portion of at least the first plate and the second plate define a relief pattern. A metal bonding agent is disposed between the first plate and the second plate such that the first plate and the second plate form a fluid flow directing pattern defined by the relief pattern.

Normal forms for lagrangian distributions on 5-dimensional contact manifolds

Abstract: Article history: A contact distribution C on a manifold M determines a symplectic bundle C → M.I n this paper we find normal forms for its lagrangian distributions by classifying vector fields lying in C. Such vector fields are divided into three types and described in terms of the simplest ones (characteristic fields of 1st order PDE's). After having established the equivalence between parabolic Monge-Ampere equations (MAE's) and lagrangian distributions in terms of characteristics, as an application of our results we give normal forms for parabolic MAE's. © 2008 Elsevier B.V. All rights reserved.

Axially symmetric cosmological models in a scalar tensor theory based on Lyra manifold

Abstract: Axially symmetric cosmological models are obtained in a scalar tensor theory proposed by Sen (Z. Phys. 149:311, 1957) based on Lyra manifold with time dependent β in the presence of string source, perfect fluid distribution, dust distribution and thick domain walls. Some physical and geometrical properties of these models are discussed.

Leafwise 2-jet cohomology on foliated Finsler manifolds

Abstract: The tangent space of a Finsler manifold (M;F) is a Rieman- nian manifold with respect to the induced Sasaki-Finsler metric and ad- mits a natural foliated structure FV given by the vertical distribution (A. Bejancu and H.R. Farran, 2006). It is known that the Levi-Civita connec- tion on the slit tangent space TM 0 induces a connection in the structural bundle. In this paper the Vaisman connection on (TM 0 ;FV ) is introduced; this induces connections on the structural and the transversal bundles. It is shown that the Levi-Civita and the Vaisman connections induce the same connections in the structural bundle ifi (M;F) is a Landsberg man- ifold and that (TM 0 ;FV ) is a Reinhart space ifi (M;F) is a Rieman- nian manifold. Further, on the foliated manifold (TM 0 ;FV ) is introduced and studied the space J fl;2g (TM 0 ) of leafwise 2-jets. A decomposition of this space is obtained, and the 1-dimensional Cech cohomology group of (TM 0 ;FV ) with coe-cients in the sheaf of basic functions is expressed in terms of flelds of leafwise 2-jets. We deflne the leafwise Mastrogiacomo cohomology group with respect to the connection r induced in the struc- tural bundle by a connection on (TM 0 ;FV ) ; as well it is shown that the cohomology group is isomorphic with the 1-dimensional Cech cohomology group of TM 0 with coe-cients in the sheaf ›r of germs of functions f on TM 0 , which satisfy rdf = 0. In particular, for the 4-dimensional 4-root space, it is proved that the sheaf ›r is isomorphic with the sheaf of basic functions on TM 0 .

Screen Integrable Lightlike Hypersurfaces of Indefinite Sasakian Manifolds

Abstract: We investigate lightlike hypersurfaces of indefinite Sasakian manifolds, tangent to the structure vector field ξ and whose screen distribution is integrable. We prove some results on parallel vector fields and on a leaf of the integrable distribution \(D_{0} \perp \langle\xi\rangle\) of this class. A theorem on a geometrical configuration of the screen distribution is obtained. We show that any totally contact umbilical leaf of a screen integrable distribution of a lightlike hypersurface is an extrinsic sphere.

Three-dimensional cr-submanifolds in the nearly kahler 6-sphere with one-dimensional nullity

Abstract: In this paper, we study certain three-dimensional CR-submanifolds M of the nearly Kahler 6-dimensional sphere S6(1). It is well known that there does not exist a three-dimensional totally geodesic proper CR-submanifold in S6(1). In this paper we obtain a classification of the 3-dimensional CR-submanifolds which are the closest possible to totally geodesic submanifolds, i.e. those that admit a one-dimensional nullity distribution.

Einstein half lightlike submanifolds of codimension 2

Abstract: In this paper we study the geometry of Einstein half light like submanifolds M of a Lorentz manifold ((c), ) of constant curvature c, equipped with an integrable screen distribution on M such that the induced connection is a metric connection and the operator is a screen shape operator.

Characteristic vector fields of generic distributions of corank 2

Abstract: We study generic distributions D ⊂ T M of corank 2 on manifolds M of dimension n ⩾ 5 . We describe singular curves of such distributions, also called abnormal curves. For n even the singular directions (tangent to singular curves) are discrete lines in D ( x ) , while for n odd they form a Veronese curve in a projectivized subspace of D ( x ) , at generic x ∈ M . We show that singular curves of a generic distribution determine the distribution on the subset of M where they generate at least two different directions. In particular, this happens on the whole of M if n is odd. The distribution is determined by characteristic vector fields and their Lie brackets of appropriate order. We characterize pairs of vector fields which can appear as characteristic vector fields of a generic corank 2 distribution, when n is even.

Geometry of Control-Affine Systems

Abstract: Motivated by control-affine systems in optimal control theory, we introduce the notion of a point-ane distribution on a manifold X - i.e., an affine distribution F together with a distinguished vector field contained in F. We compute local invariants for point-affine distributions of constant type when dim(X) = n, rank(F) = n 1, and when dim(X) = 3, rank(F) = 1. Unlike linear distributions, which are characterized by integer- valued invariants - namely, the rank and growth vector - when dim(X) 4, we find local invariants depending on arbitrary functions even for rank 1 point-affine distributions on manifolds of dimension 2.

3-structures with torsion

Abstract: We find conditions which ensure the integrability of the canonical 3-dimensional distribution V spanned by the Reeb vector fields of an almost 3-contact manifold, showing by an explicit counterexample that the normality of the structures does not necessarily imply the integrability of V . Then we focus on those almost 3-contact metric manifolds for which V is integrable and we define an appropriate notion of almost 3-contact metric connection with torsion. The geometry of an almost 3-contact manifold with torsion is then studied and put in relation with the well-known HKT-geometry.

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.

Distributions that are convolvable with generalized Poisson kernel of solvable extensions of homogeneous Lie groups

Abstract: In this paper, we characterize the class of distributions on an homogeneous Lie group $\fN$ that can be extended via Poisson integration to a solvable one-dimensional extension $\fS$ of $\fN$ To do so, we introducte the $s'$-convolution on $\fN$ and show that the set of distributions that are $s'$-convolvable with Poisson kernels is precisely the set of suitably weighted derivatives of $L^1$-functions Moreover, we show that the $s'$-convolution of such a distribution with the Poisson kernel is harmonic and has the expected boundary behaviour Finally, we show that such distributions satisfy some global weak-$L^1$ estimates

An Equivariant Index Formula for Almost CR Manifolds

Abstract: We 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 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 and standard equivariant characteristic forms. This formula generalizes the formula given in [10] for the case of a contact manifold.

A geometric approach to energy shaping

Abstract: In this thesis is initiated a more systematic geometric exploration of energy shaping. Most of the previous results have been dealt with particular cases and neither the existence nor the space of solutions has been discussed with any degree of generality. The geometric theory of partial differential equations originated by Goldschmidt and Spencer in late 1960s is utilized to analyze the partial differential equations in energy shaping. The energy shaping partial differential equations are described as a fibered submanifold of a k-jet bundle of a fibered manifold. By revealing the nature of kinetic energy shaping, similarities are noticed between the problem of kinetic energy shaping and some well-known problems in Riemannian geometry. In particular, there is a strong similarity between kinetic energy shaping and the problem of finding a metric connection initiated by Eisenhart and Veblen. We notice that the necessary conditions for the set of so-called λ-equation restricted to the control distribution are related to the Ricci identity, similarly to the Eisenhart and Veblen metric connection problem. Finally, the set of λ-equations for kinetic energy shaping are coupled with the integrability results of potential energy shaping. This gives new insights for answering some key questions in energy shaping that have not been addressed to this point. The procedure shows how a poor design of closed-loop metric can make it impossible to achieve any flexibility in the character of the possible closed-loop potential function. The integrability results of this thesis have been used to answer some interesting questions about the energy shaping method. In particular, a geometric proof is provided which shows that linear controllability is sufficient for energy shaping of linear simple mechanical systems. Furthermore, it is shown that all linearly controllable simple mechanical control