Showing papers on "Frame bundle published in 2015"

A Categorical Equivalence between Generalized Holonomy Maps on a Connected Manifold and Principal Connections on Bundles over that Manifold

Abstract: A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30(9), (1991)], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation.

A remark on bigness of the tangent bundle of a smooth projective variety and D-simplicity of its section rings

Abstract: We point out a connection between bigness of the tangent bundle of a smooth projective variety X over ℂ and simplicity of the section rings of X as modules over their rings of differential operators. As a consequence, we see that the tangent bundle of a smooth projective toric variety or a (partial) flag variety is big. Some other applications and related questions are discussed.

Nonnegative curvature on stable bundles over compact rank one symmetric spaces

Abstract: In this note we show that every (real or complex) vector bundle over a compact rank one symmetric space carries, after taking the Whitney sum with a trivial bundle of sufficiently large rank, a metric with nonnegative sectional curvature We also examine the case of complex vector bundles over other manifolds, and give upper bounds for the rank of the trivial bundle that is necessary to add when the base is a sphere

A Note on Vanishing Theorems

Abstract: On a Riemannian manifold we define a one-parameter family of Laplacians acting on sections of any bundle associated to the principal frame bundle via a representation, and show how various examples fit into this framework.

Evolution Equations with Anisotropic Distributions and Diffusion PCA

Abstract: This paper presents derivations of evolution equations for the family of paths that in the Diffusion PCA framework are used for approximating data likelihood. The paths that are formally interpreted as most probable paths generalize geodesics in extremizing an energy functional on the space of differentiable curves on a manifold with connection. We discuss how the paths arise as projections of geodesics for a (non bracket-generating) sub-Riemannian metric on the frame bundle. Evolution equations in coordinates for both metric and cometric formulations of the sub-Riemannian geometry are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we use the construction to show how the evolution equations can be implemented on finite dimensional LDDMM landmark manifolds.

Jumping pairs of Steiner bundles

Abstract: In this work we introduce the definition of Schwarzenberger bundle on a Grassmannian. Recalling the notion of Steiner bundle, we generalize the concept of jumping pair for a Steiner bundle on a Grassmannian. After studying the jumping locus variety and bounding its dimension, we give a complete classification of Steiner bundles with jumping locus of maximal dimension, which all are Schwarzenberger bundles

Prolongations of Golden Structure to Tangent Bundle of Order 2

Abstract: In this paper, we study 2 nd lift of golden structure to tangent bundle of order 2. We investigate integrability and parallelism of Gloden structures in $T_2(M)$. Moreover, we define golden semi-Riemannian metric in $T_2(M)$.

Holomorphic sectional curvature and the canonical bundle

Abstract: We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture of Yau. A key ingredient is the recent solution of this conjecture in the projective case by Wu-Yau.

A Stabilization Theorem for Fell Bundles over groupoids

Abstract: We study the $C^*$-algebras associated to upper-semicontinuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer--Raeburn "Stabilization Trick," we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced $C^*$-algebras of any saturated upper-semicontinuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the $C^*$-algebra of a continuous Fell-bundle by applying Renault's results about the ideals of the $C^*$-algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle $C^*$-algebra of a bundle over $G$ in terms of an action, described by the first and last named authors, of $G$ on the primitive-ideal space of the $C^*$-algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted $k$-graph algebras, where the components of our results become more concrete.

Segre forms and Kobayashi-L\"ubke inequality

Abstract: Starting from the description of Segre forms as direct images of (powers of) the first Chern form of the (anti)tautological line bundle on the projectivized bundle of a holomorphic hermitian vector bundle, we derive a version of the pointwise Kobayashi-Lubke inequality.

An algorithm for the normal bundle of rational monomial curves

Abstract: We give an algorithm for calculating the splitting type of the normal bundle of any rational monomial curve. The algorithm is obtained by reducing the calculus to a combinatorial problem and then by solving this problem.

On the (1, 1)-tensor bundle with Cheeger–Gromoll type metric

Abstract: The main purpose of the present paper is to construct Riemannian almost product structures on the (1, 1)-tensor bundle equipped with Cheeger–Gromoll type metric over a Riemannian manifold and present some results concerning these structures.

Geometry, dynamics and spectral analysis on manifolds : the Pestov Identity on frame bundles and eigenvalue asymptotics on graph-like manifolds

Abstract: This dissertation is made up of two independent parts. In Part I we consider the Pestov Identity, an identity stated for smooth functions on the tangent bundle of a manifold and linking the Riemannian curvature tensor to the generators of the geodesic flow, and we lift it to the bundle of k-tuples of tangent vectors over a compact manifold M of dimension n. We also derive an integrated version over the bundle of orthonormal k-frames of M as well as a restriction to smooth functions on such a bundle. Finally, we present a dynamical application for the parallel transport of the Grassmannian of oriented k-planes of M. In Part II we consider a family of compact and connected n-dimensional manifolds, called graph-like manifold, shrinking to a metric graph in the appropriate limit. We describe the asymptotic behaviour of the eigenvalues of the Hodge Laplacian acting on differential forms on those manifolds in the appropriate limit. As an application, we produce manifolds and families of manifolds with arbitrarily large spectral gaps in the spectrum of the Hodge Laplacian.

Fiber Bundles and Parseval Frames

Abstract: Continuous frames over a Hilbert space have a rich and sophisticated structure that can be represented in the form of a fiber bundle. The fiber bundle structure reveals the central importance of Parseval frames and the extent to which Parseval frames generalize the notion of an orthonormal basis.

Geometry and general relativity in the groupoid model with a finite structure group

Abstract: In a series of papers (M. Heller et al. J. Math. Phys. 38, 5840 (1997). doi:10.1063/1.532186; M. Heller and W. Sasin. Int. J. Theor. Phys. 38, 1619 (1999). doi:10.1023/A:1026617913754; M. Heller et al. Int. J. Theor. Phys. 44, 619 (2005). doi:10.1007/s10773-005-3992-7) we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra, AΓ , defined on a transformation groupoid Γ determined by the action of the Lorentz group on the frame bundle (E, πM, M) over space–time M. In the present work, we construct a simplified version of the gravitational sector of this model in which the Lorentz group is replaced by a finite group, G, and the frame bundle is trivial E = M × G. The model is fully computable. We define the Einstein–Hilbert action, with the help of which we derive the generalized vacuum Einstein equations. When the equations are projected to space–time (giving the “general relativistic limit”), th...

Hopf algebra $\mathcal{K}_n$ and universal Chern classes

Abstract: We construct a variant $\mathcal{K}_n$ of the Hopf algebra $\mathcal{H}_n$, which acts directly on the noncommutative model for the generic space of leaves rather than on its frame bundle. We prove that the Hopf cyclic cohomology of $\mathcal{K}_n$ is isomorphic to that of the pair $(\mathcal{H}_n, {\mathop{\rm GL}_n})$ and thus consists of the universal Hopf cyclic classes. We then realize these classes in terms of geometric cocycles.

Paracontact Tangent Bundles with Cheeger-Gromoll Metric

Abstract: In this paper, we define the almost paracontact metric structure on a tangent bundle TM with Cheeger–Gromoll (C–G) metric and obtain the normality condition for it. We define the paracontact C–G metric tangent bundle, K-paracontact C–G metric tangent bundle and C–G para-Sasakian tangent bundle and give some characterizations about them. Also, we give the Riemannian curvature tensor R and the sectional curvature K of the almost paracontact C–G metric tangent bundle TM. Finally, we obtain the Ricci curvature S and the scalar curvature $${\tilde{\sigma}}$$ of the almost paracontact C–G metric tangent bundle TM with the aid of the orthonormal basis of TM.

Modelling Anisotropic Covariance using Stochastic Development and Sub-Riemannian Frame Bundle Geometry

Abstract: We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal distributions. We discuss a factorization of the frame bundle projection map through this bundle, the natural sub-Riemannian structure of the frame bundle, the effect of holonomy, and the existence of subbundles where the Hormander condition is satisfied such that the Brownian motions have smooth transition densities. We identify the most probable paths for the underlying Euclidean Brownian motion and discuss small time asymptotics of the transition densities on the manifold. The geometric setup yields an intrinsic approach to the estimation of mean and covariance in non-linear spaces.

p-Buchsbaum rank 2 bundles on the projective space

Abstract: It has been proved by various authors that a normalized, 1-Buchsbaum rank 2 vector bundle on P 3 is a nullcorrelation bundle, while a normalized, 2-Buchsbaum rank 2 vector bundle on P 3 is an instanton bundle of charge 2. We find that the same is not true for 3-Buchsbaum rank 2 vector bundles on P 3 , and propose a conjecture regarding the classification of such objects.

Hörmander's condition for normal bundles on spaces of immersions

Abstract: Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and vertical bundle. We investigate in these notes the sub-Riemannian geometries of these bundles. In particular, we show for a selection of bundles which naturally occur in applications that they are either bracket generating or integrable.

Equivariant bundle gerbes

Abstract: We develop the theory of simplicial extensions for bundle gerbes and their characteristic classes with a view towards studying descent problems and equivariance for bundle gerbes. Equivariant bundle gerbes are important in the study of orbifold sigma models. We consider in detail two examples: the basic bundle gerbe on a unitary group and a string structure for a principal bundle. We show that the basic bundle gerbe is equivariant for the conjugation action and calculate its characteristic class; we show also that a string structure gives rise to a bundle gerbe which is equivariant for a natural action of the String 2-group.

On the space of connections having non-trivial twisted harmonic spinors

Abstract: We consider Dirac operators on odd-dimensional compact spin manifolds which are twisted by a product bundle. We show that the space of connections on the twisting bundle which yield an invertible operator has infinitely many connected components if the untwisted Dirac operator is invertible and the dimension of the twisting bundle is sufficiently large.

Vector bundles on real algebraic curves

Abstract: We prove that any topological real line bundle on a compact real algebraic curve X is isomorphic to an algebraic line bundle. The result is then generalized to vector bundles of an arbitrary constant rank. As a consequence we prove that any continuous map from X into a real Grassmannian can be approximated by regular maps.

Foliation by G-orbits ∗

Abstract: We study the properties of the normal bundle defined by the bundle of the G-orbits of the action of a semisimple Lie group G on a pseudo-Riemannian manifold M, as a consequence we obtain that the foliation induced by the normal bundle is integrable and totally geodesic.

Nambu Structures And Associated Bialgebroids

Abstract: This paper investigates higher order generalizations of well known results for Lie algebroids and bialgebroids. It is proved that $n$-Lie algebroid structures correspond to $n$-ary generalization of Gerstenhaber algebras and are implied by $n$-ary generalization of linear Poisson structures on the dual bundle. A Nambu-Poisson manifold (of order $n>2$) gives rise to a special bialgebroid structure which is referred to as a weak Lie-Filippov bialgebroid (of order $n$). It is further demonstrated that such bialgebroids canonically induce a Nambu-Poisson structure on the base manifold. Finally, the tangent space of a Nambu Lie group gives an example of a weak Lie-Filippov bialgebroid over a point.

A Universal Spinor Bundle and the Einstein-Dirac-Maxwell Equation as a Variational Theory

Abstract: Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein-Dirac-Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.

On the Topology of Real Bundle Pairs over Nodal Symmetric Surfaces

Abstract: We give an alternative argument for the classification of real bundle pairs over smooth symmetric surfaces and extend this classification to nodal symmetric surfaces. We also classify the homotopy classes of automorphisms of real bundle pairs over symmetric surfaces. The two statements together describe the isomorphisms between real bundle pairs over symmetric surfaces up to deformation.