Showing papers on "Distribution (differential geometry) published in 2013"

Spine Segmentation in Medical Images Using Manifold Embeddings and Higher-Order MRFs

Abstract: We introduce a novel approach for segmenting articulated spine shape models from medical images. A nonlinear low-dimensional manifold is created from a training set of 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 higher-order Markov random field (HOMRF). Singleton and pairwise potentials measure the support from the global data and shape coherence in manifold space respectively, while higher-order cliques encode geometrical modes of variation to segment each localized vertebra models. Generic feature functions learned from ground-truth data assigns costs to the higher-order terms. Optimization of the 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. Clinical experiments demonstrated promising results in terms of spine segmentation. Quantitative comparison to expert identification yields an accuracy of 1.6 ± 0.6 mm for CT imaging and of 2.0 ± 0.8 mm for MR imaging, based on the localization of anatomical landmarks.

A partial analog of the integrability theorem for distributions on p-adic spaces and applications

Abstract: Let X be a smooth real algebraic variety. Let ξ be a distribution on it. One can define the singular support of ξ to be the singular support of the DX-module generated by ξ (sometimes it is also called the characteristic variety). A powerful property of the singular support is that it is a coisotropic subvariety of T*X. This is the integrability theorem (see [KKS, Mal, Gab]). This theorem turned out to be useful in representation theory of real reductive groups (see, e.g., [AG4, AS, Say]).

Normal forms and invariants for 2-dimensional almost-Riemannian structures

Abstract: 2-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. Generically, there are three types of points: Riemannian points where the two vector fields are linearly independent, Grushin points where the two vector fields are collinear but their Lie bracket is not, and tangency points where the two vector fields and their Lie bracket are collinear and the missing direction is obtained with one more bracket. In this paper we consider the problem of finding normal forms and functional invariants at each type of point. We also require that functional invariants are “complete” in the sense that they permit to recognize locally isometric structures. The problem happens to be equivalent to the one of finding a smooth canonical parameterized curve passing through the point and being transversal to the distribution. For Riemannian points such that the gradient of the Gaussian curvature K is different from zero, we use the level set of K as support of the parameterized curve. For Riemannian points such that the gradient of the curvature vanishes (and under additional generic conditions), we use a curve which is found by looking for crests and valleys of the curvature. For Grushin points we use the set where the vector fields are parallel. Tangency points are the most complicated to deal with. The cut locus from the tangency point is not a good candidate as canonical parameterized curve since it is known to be non-smooth. Thus, we analyse the cut locus from the singular set and we prove that it is not smooth either. A good candidate appears to be a curve which is found by looking for crests and valleys of the Gaussian curvature. We prove that the support of such a curve is uniquely determined and has a canonical parametrization.

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.

A further improvement of the Quantitative Subspace Theorem

Abstract: In 2002, Evertse and Schlickewei [11] obtained a quantitative version of the so-called Absolute Parametric Subspace Theorem. This result deals with a parametrized class of twisted heights. One of the consequences of this result is a quantitative version of the Absolute Subspace Theorem, giving an explicit upper bound for the number of subspaces containing the solutions of the Diophantine inequality under consideration. In the present paper, we further improve Evertse’s and Schlickewei’s quantitative version of the Absolute Parametric Subspace Theorem, and deduce an improved quantitative version of the Absolute Subspace Theorem. We combine ideas from the proof of Evertse and Schlickewei (which is basically a substantial refinement of Schmidt’s proof of his Subspace Theorem from 1972 [22]), with ideas from Faltings’ and Wustholz’ proof of the Subspace Theorem [14]. A new feature is an “interval result,” which gives more precise information on the distribution of the heights of the solutions of the system of inequalities considered in the Subspace Theorem.

Deforming metrics of foliations

Abstract: Let M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.

L-infinity Algebras From Multicontact Geometry

Abstract: I define higher codimensional versions of contact structures on manifolds as maximally non-integrable distributions. I call them multicontact structures. Cartan distributions on jet spaces provide canonical examples. More generally, I define higher codimensional versions of pre-contact structures as distributions on manifolds whose characteristic symmetries span a constant dimensional distribution. I call them pre-multicontact structures. Every distribution is almost everywhere, locally, a pre-multicontact structure. After showing that the standard symplectization of contact manifolds generalizes naturally to a (pre-)multisymplectization of (pre-)multicontact manifolds, I make use of results by C. Rogers and M. Zambon to associate a canonical $L_{\infty}$-algebra to any (pre-)multicontact structure. Such $L_{\infty}$-algebra is a multicontact version of the Jacobi bracket on a contact manifold. However, unlike the multisymplectic $L_\infty$-algebra of Rogers and Zambon, the multicontact $L_\infty$-algebra is always a homological resolution of a Lie algebra. Finally, I describe in local coordinates the $L_{\infty}$-algebra associated to the Cartan distribution on jet spaces.

Vector fields with distributions and invariants of ODEs

Abstract: We study dynamic pairs $(X,V)$ where $X$ is a vector field on a smooth manifold $M$ and $V\subset TM$ is a vector distribution, both satisfying certain regularity conditions. We construct basic invariants of such objects and solve the equivalence problem. In particular, we assign to $(X,V)$ a canonical connection and a canonical frame on a certain frame bundle. We compute the curvature and torsion. The results are applied to the problem of time scale preserving equivalence of ordinary differential equations and of Veronese webs. The framework of dynamic pairs $(X,V)$ is shown to include sprays, control-affine systems, mechanical control systems, Veronese webs and other structures.

Constrained object tracking on compact one-dimensional manifolds based on directional statistics

Abstract: In this paper, we present a novel approach for tracking objects whose movement is constrained to a compact one-dimensional manifold, for example a conveyer belt or a mobile robot whose movement is restricted to tracks. Standard approaches either ignore the constraint at first and retroactively move the estimate to lie on the manifold, or consider the tracking problem on a manifold but falsely assume a Gaussian distribution. Our method explicitly takes the actual topology into account from the beginning and relies on special types of probability distributions defined on the proper manifold. In particular, we consider objects moving along a closed one-dimensional track, for example an ellipse, a polygon, or similar closed shapes. This shape is transformed to a circle with a homeomorphism. Thus, we can apply a recursive circular filtering algorithm to the constrained tracking problem. Finally, the estimate is transformed back to the original manifold. We evaluate the proposed method in an experiment by tracking a toy train moving along a track and comparing the results to those of traditional approaches for this problem.

Families of distributions and Pfaff systems under duality

Abstract: A singular distribution on a non-singular variety $X$ can be defined either by a subsheaf $D$ of the tangent sheaf, or by the zeros of a subsheaf $D^0$ of the sheaf of 1-forms. Although both definitions are equivalent under mild conditions on $D$, they give rise, in general, to non-equivalent notions of flat families. In this work we investigate conditions under which both notions of flat families are equivalent. In the last sections we focus on the case where the distribution is integrable, and we use our results to generalize a theorem of Cukierman and Pereira.

On the Weyl Tensor Classification in All Dimensions and its Relation with Integrability Properties

Abstract: In this paper the Weyl tensor is used to define operators that act on the space of forms. These operators are shown to have interesting properties and are used to classify the Weyl tensor, the well known Petrov classification emerging as a special case. Particularly, in the Euclidean signature this classification turns out be really simple. Then it is shown that the integrability condition of maximally isotropic distributions can be described in terms of the invariance of certain subbundles under the action of these operators. Here it is also proved a new generalization of the Goldberg-Sachs theorem, valid in all even dimensions, stating that the existence of an integrable maximally isotropic distribution imposes restrictions on the optical matrix. Also the higher-dimensional versions of the self-dual manifolds are investigated. These topics can shed light on the integrability of Einstein's equation in higher dimensions.

Anti-invariant Riemannian Submersions from Nearly Kaehler Manifolds

Abstract: We extend the notion of anti-invariant and Langrangian Riemannian submersion to the case when the total manifold is nearly Kaehler. We obtain the integrability conditions for the horizontal distribution while it is noted that the vertical distribution is always integrable. We also investigate the geometry of the foliations of the two distributions and obtain the necessary and sufficient condition for a Langrangian submersion to be totally geodesic. The decomposition theorems for the total manifold of the submersion are obtained.

Integrability conditions for the Grushin and Martinet distributions

Abstract: OVIDIU CALIN, DER-CHEN CHANG, AND MICHAEL EASTWOODAbstract. We realise the first and second Grushin distributionsas symmetry reductions of the 3-dimensional Heisenberg distribu-tion and 4-dimensional Engel distribution respectively. Similarly,we realise the Martinet distribution as an alternative symmetryreduction of the Engel distribution. These reductions allow us toderive the integrability conditions for the Grushin and Martinetdistributions and build certain complexes of differential operators.These complexes are well-behaved despite the distributions theyresolve being non-regular.

A classification of half lightlike submanifolds of a semi-riemannian manifold with a semi-symmetric non-metric connection

Abstract: In this paper, we study the geometry of half lightlike sub- manifolds M of a semi-Riemannian manifold f M with a semi-symmetric non-metric connection subject to the conditions; (1) the characteristic vector field of f M is tangent to M, the screen distribution on M is totally umbilical in M and the co-screen distribution on M is conformal Killing, or (2) the screen distribution is integrable and the local lightlike second fundamental form of M is parallel.

Pseudodifferential operators of mixed type adapted to distributions of $k$-planes

Abstract: We study the phenomena that arise when we combine the standard pseudodifferential operators with those operators that appear in the study of some sub-elliptic estimates, and on strongly pseudoconvex domains. The algebra of operators we introduce is geometrically invariant, and is adapted to a smooth distribution of tangent subspaces of constant rank. We isolate certain ideals in the algebra whose analysis is of particular interest.

On the Weyl tensor classification in all dimensions and its relation with integrability properties

Abstract: In this paper the Weyl tensor is used to define operators that act on the space of forms. These operators are shown to have interesting properties and are used to classify the Weyl tensor, the well-known Petrov classification emerging as a special case. Particularly, in the Euclidean signature this classification turns out to be really simple. Then it is shown that the integrability condition of maximally isotropic distributions can be described in terms of the invariance of certain subbundles under the action of these operators. Here it is also proved a new generalization of the Goldberg-Sachs theorem, valid in all even dimensions, stating that the existence of an integrable maximally isotropic distribution imposes restrictions on the optical matrix. Also the higher-dimensional versions of the self-dual manifolds are investigated. These topics can shed light on the integrability of Einstein's equation in higher dimensions.

Definition and Properties of m-Dimensional n-Principal Points

Abstract: In this article, we introduce the notion of “m-dimensional n-principal points,” which is a generalization of the notion of n-principal points. A set of m-dimensional n-principal points of a distribution is defined as a set of n points that optimally represents the distribution in terms of mean squared distance subject to the condition that the dimension of the linear subspace spanned by the n points is at most m. Its properties and connections to principal components are investigated for elliptically symmetric distributions and a location mixture of spherically symmetric distributions.

Towards a Classification of pseudo-Riemannian Geometries Admitting Twistor Spinors

Abstract: We show that given a conformal structure whose holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is always a local metric in the conformal class off a singular set which is Ricci-isotropic and gives rise to a parallel, totally lightlike distribution on the tangent bundle. This naturally applies to parallel spin tractors resp. twistor spinors on conformal spin manifolds and clarifies which twistor spinors are locally equivalent to parallel spinors. Moreover, we study the zero set of a twistor spinor using the curved orbit decomposition for parabolic geometries. We can completely describe its local structure, construct a natural projective structure on it, and show that locally every twistor spinor with zero is equivalent to a parallel spinor off the zero set. An application of these results in low-dimensional split-signatures leads to a complete geometric description of local geometries admitting non-generic twistor spinors in signatures (3,2) and (3,3) which complements the well-known description of the generic case.

Geometry of rank 2 distributions with nonzero Wilczynski invariants and affine control systems with one input

Abstract: We demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works in the following two situations: rank 2 distributions of maximal class in R^n with non-zero generalized Wilczynski invariants and rank 2 distributions of maximal class in R^n with additional structures such as affine control system with one input spanning these distributions, sub-(pseudo)Riemannian structures etc. The common feature of these two situations is that each abnormal extremal (of the underlying rank 2 distribution) possesses a distinguished parametrization. This fact allows one to construct the canonical frame on a (2n-3)-dimensional bundle in both situations for arbitrary n greater than 4. The moduli spaces of the most symmetric models for both situations are described as well. The relation of our results to the divergence equivalence of Lagrangians of higher order is given

Lightlike Hypersurfaces in Indefinite Trans-Sasakian Manifolds

Abstract: This paper deals with lightlike hypersurfaces of indefinite trans-Sasakian manifolds of type (α, β), tangent to the structure vector field. Characterization Theorems on parallel vector fields, integrable distributions, minimal distributions, Ricci-semi symmetric, geodesibility of lightlike hypersurfaces are obtained. The geometric configuration of lightlike hypersurfaces is established. We prove, under some conditions, that there are no parallel and totally contact umbilical lightlike hypersurfaces of trans-Sasakian space forms, tangent to the structure vector field. We show that there exists a totally umbilical distribution in an Einstein parallel lightlike hypersurface which does not contain the structure vector field. We characterize the normal bundle along any totally contact umbilical leaf of an integrable screen distribution. We finally prove that the geometry of any leaf of an integrable distribution is closely related to the geometry of a normal bundle and its image under \({\overline{\phi}}\) .

Testing the Manifold Hypothesis

Abstract: The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for fitting a manifold to an unknown probability distribution supported in a separable Hilbert space, only using iid samples from that distribution More precisely, our setting is the following Suppose that data are drawn independently at random from a probability distribution $P$ supported on the unit ball of a separable Hilbert space $H$ Let $G(d, V, \tau)$ be the set of submanifolds of the unit ball of $H$ whose volume is at most $V$ and reach (which is the supremum of all $r$ such that any point at a distance less than $r$ has a unique nearest point on the manifold) is at least $\tau$ Let $L(M, P)$ denote mean-squared distance of a random point from the probability distribution $P$ to $M$ We obtain an algorithm that tests the manifold hypothesis in the following sense The algorithm takes iid random samples from $P$ as input, and determines which of the following two is true (at least one must be): (a) There exists $M \in G(d, CV, \frac{\tau}{C})$ such that $L(M, P) \leq C \epsilon$ (b) There exists no $M \in G(d, V/C, C\tau)$ such that $L(M, P) \leq \frac{\epsilon}{C}$ The answer is correct with probability at least $1-\delta$

A vertical Liouville subfoliation on the cotangent bundle of a Cartan space and some related structures

Abstract: In this paper we study some problems related to a vertical Liouville distribution (called vertical Liouville-Hamilton distribution) on the cotangent bundle of a Cartan space. We study the existence of some linear connections of Vranceanu type on Cartan spaces related to some foliated structures. Also, we identify a certain $(n,2n-1)$--codimensional subfoliation $(\mathcal{F}_V,\mathcal{F}_{C^*})$ on $T^*M_0$ given by vertical foliation $\mathcal{F}_V$ and the line foliation $\mathcal{F}_{C^*}$ spanned by the vertical Liouville-Hamilton vector field $C^*$ and we give a triplet of basic connections adapted to this subfoliation. Finally, using the vertical Liouville foliation $\mathcal{F}_{V_{C^*}}$ and the natural almost complex structure on $T^*M_0$ we study some aspects concerning the cohomology of $c$--indicatrix cotangent bundle.

Co-screen conformal half lightlike submanifolds

Abstract: In this paper, we introduce and study the geometry of half lightlike submanifold $M$ of a semi-Riemannian manifold $\overline{M}$ satisfying that the shape operator of screen transversal bundle is conformal to the shape operator of lightlike transversal bundle of $M$. Using this geometric condition we obtain some results to characterize the unique existence of screen distribution of $M$, also, we present some sufficient conditions for the induced Ricci curvature tensor of $M$ to be symmetric.

A note on lightlike hypersurfaces of semi-riemannian space form

Abstract: In this paper, we mainly study lightlike hypersurfaces of semi-Riemannian space form. Our main result is a classication theo- rem of screen conformal lightlike hypersurfaces. Also, we obtain some geometric properties of lightlike hypersurfaces with a conformal Killing distribution.

Special solutions of the first and second Painlevé equations and singularities of the monodromy data manifold

Abstract: A classification of solutions of the first and second Painleve equations corresponding to a special distribution of poles at infinity is considered. The relation between this distribution and singularities of the two-dimensional complex monodromy data manifold used for the pa- rameterization of the solutions is analyzed. It turns out that solutions of the Painlevee quations have no poles in a certain critical sector of the complex plane if and only if their monodromy data lie in the singularity submanifold. Such solutions belong to the so-called class of "truncated" solutions (integrales tronquee) according to P. Boutroux's classification. It is shown that all known special solutions of the first and second Painleve equations belong to this class.

Contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold

Abstract: In this paper we prove some properties of the indefinite Lorentzian para-Sasakian manifolds. Section 1 is introductory. In Sec- tion 2 we define D-totally geodesic and D ? -totally geodesic contact CR- submanifolds of an indefinite Lorentzian para-Sasakian manifold and de- duce some results concerning such a manifold. In Section 3 we state and prove some results on mixed totally geodesic contact CR-submanifolds of an indefinite Lorentzian para-Sasakian manifold. Finally, in Section 4 we obtain a result on the anti-invariant distribution of totally umbilic contact CR-submanifolds of an indefinite Lorentzian para-Sasakian man- ifold.