Showing papers on "Frame bundle published in 2009"

On the canonical line bundle and negative holomorphic sectional curvature

Abstract: We prove that a smooth complex projective threefold with a Kahler metric of negative holomorphic sectional curvature has ample canonical line bundle. In dimensions greater than three, we prove that, under equal assumptions, the nef dimension of the canonical line bundle is maximal. With certain additional assumptions, ampleness is again obtained. The methods used come from both complex differential geometry and complex algebraic geometry.

On semistable principal bundles over a complex projective manifold, II

Abstract: Let G be a simple linear algebraic group defined over the complex numbers. Fix a proper parabolic subgroup P of G and a nontrivial antidominant character \chi of P. We prove that a holomorphic principal G-bundle E over a connected complex projective manifold M is semistable and the second Chern class of its adjoint bundle vanishes in rational cohomology if and only if the line bundle over E/P defined by \chi is numerically effective. Similar results remain valid for principal bundles with a reductive linear algebraic group as the structure group. These generalize an earlier work of Y. Miyaoka where he gave a characterization of semistable vector bundles over a smooth projective curve. Using these characterizations one can also produce similar criteria for the semistability of parabolic principal bundles over a compact Riemann surface.

Regularizing Flows over Lie Groups

Abstract: In this paper we discuss regularization of images that take their value in matrix Lie groups. We describe an image as a section in a principal bundle which is a fibre bundle where the fiber (the feature space) is a Lie group. Via the scalar product on the Lie algebra, we define a bi-invariant metric on the Lie-group manifold. Thus, the fiber becomes a Riemannian manifold with respect to this metric. The induced metric from the principal bundle to the image manifold is obtained by means of the bi-invariant metric. A functional over the space of sections, i.e., the image manifolds, is defined. The resulting equations of motion generate a flow which evolves the sections in the spatial-Lie-group manifold. We suggest two different approaches to treat this functional and the corresponding PDEs. In the first approach we derive a set of coupled PDEs for the local coordinates of the Lie-group manifold. In the second approach a coordinate-free framework is proposed where the PDE is defined directly with respect to the Lie-group elements. This is a parameterization-free method. The differences between these two methods are discussed. We exemplify this framework on the well-known orientation diffusion problem, namely, the unit-circle S 1 which is identified with the group of rotations in two dimensions, SO(2). Regularization of the group of rotations in 3D and 4D, SO(3) and SO(4), respectively, is demonstrated as well.

The G2 sphere of a 4-manifold

Abstract: We bring to light a G2 structure existing on the unit sphere tangent bundle SM of any given orientable Riemannian 4-manifold M. The associated 3-form φ is co-calibrated if, and only if, M is an Einstein manifold—a result which leads to new examples of co-calibrated G2 spaces. We hope to be contributing both to the knowledge of special geometries and to the study of 4-manifolds.

Continuous trace c ∗ -algebras, gauge groups and rationalization

Abstract: Let ζ be an n-dimensional complex matrix bundle over a compact metric space X and let Aζ denote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UAζ, the group of unitaries of Aζ. The answer turns out to be independent of the bundle ζ and depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.

Stability of projective Poincare and Picard bundles

Abstract: Let X be an irreducible smooth projective curve of genus g ≥ 3 defined over the complex numbers, and let ℳ ξ denote the moduli space of stable vector bundles on X of rank n and determinant ξ, where ξ is a fixed line bundle of degree d. If n and d have a common divisor, then there is no universal vector bundle on X × ℳ ξ . We prove that there is a projective bundle on X × ℳ ξ with the property that its restriction to X × {E} is isomorphic to P(E) for all E ∈ ℳ ξ and that this bundle (called the projective Poincare bundle) is stable with respect to any polarization; moreover its restriction to {x} × ℳ ξ is also stable for any x ∈ X. We also prove stability results for bundles induced from the projective Poincare bundle by homomorphisms PGL(n) → H for any reductive H. We further show that there is a projective Picard bundle on a certain open subset ℳ′ of ℳ ξ for any d > n(g−1) and that this bundle is also stable. Also, we obtain new results on the stability of the Picard bundle even when n and d are coprime.

Algebraic independence of generalized Morita-Miller-Mumford classes

Abstract: The generalized Morita-Miller-Mumford classes of a smooth oriented manifold bundle are defined as the image of the characteristic classes of the vertical tangent bundle under the Gysin homomorphism. We show that if the dimension of the manifold is even, then all MMM-classes in rational cohomology are nonzero for some bundle. In odd dimensions, this is also true with one exception: the MMM-class associated with the Hirzebruch $\cL$-class is always zero. We also show a similar result for holomorphic fibre bundles.

Loop Groups, Higgs Fields and Generalised String Classes

Abstract: We consider various generalisations of the string class of a loop group bundle. The string class is the obstruction to lifting a bundle whose structure group is the loop group $LG$ to one whose structure group is the Kac-Moody central extension of the loop group. We develop a notion of higher string classes for bundles whose structure group is the group of based loops, $\Omega G$. In particular, we give a formula for characteristic classes in odd dimensions for such bundles which are associated to characteristic classes for $G$-bundles in the same way that the string class is related to the first Pontrjagyn class of a certain $G$-bundle associated to the loop group bundle in question. This provides us with a theory of characteristic classes for $\Omega G$-bundles analogous to Chern-Weil theory in finite dimensions. This also gives us a geometric interpretation of the well-known transgression map $H^{2k}(BG) \to H^{2k-1}(G).$ We also consider the obstruction to lifting a bundle whose structure group is not the loop group but the semi-direct product of the loop group with the circle, $LG \rtimes S^1$. We review the theory of bundle gerbes and their application to central extensions and lifting problems and use these methods to obtain an explicit expression for the de Rham representative of the obstruction to lifting such a bundle. We also relate this to a generalisation of the so-called `caloron correspondence' (which relates $LG$-bundles over $M$ to $G$-bundles over $M \times S^1$) to a correspondence which relates $LG \rtimes S^1$-bundles over $M$ to $G$-bundles over $S^1$-bundles over $M$.

Linear connections along the tangent bundle projection

Abstract: An intrinsic characterization is given of the concept of linear connection along the tangent bundle projection τ : TM → M . The main observation thereby is that every such connection D gives rise to a horizontal lift, which is needed to extend the action of the associated covariant derivative operator to tensor fields along τ in a meaningful way. The interplay is discussed between the given D and various related connections, such as the canonical non-linear connection of the geodesic equations and certain linear connections on the pullback bundle τ∗τ . This is particularly relevant to understand similarities and differences between various notions of torsion and curvature. I further discuss aspects of variationality and metrizability of a linear D along τ and let me guide for the selected topics by a very short, old paper of Krupka and Sattarov.

Learning bundle manifold by double neighborhood graphs

Abstract: In this paper, instead of the ordinary manifold assumption, we introduced the bundle manifold assumption that imagines data points lie on a bundle manifold Under this assumption, we proposed an unsupervised algorithm, named as Bundle Manifold Embedding (BME), to embed the bundle manifold into low dimensional space In BME, we construct two neighborhood graphs that one is used to model the global metric structure in local neighborhood and the other is used to provide the information of subtle structure, and then apply the spectral graph method to obtain the low-dimensional embedding Incorporating some prior information, it is possible to find the subtle structures on bundle manifold in an unsupervised manner Experiments conducted on benchmark datasets demonstrated the feasibility of the proposed BME algorithm, and the difference compared with ISOMAP, LLE and Laplacian Eigenmaps.

A survey on rankin-cohen deformations

Abstract: This is a survey about recent progress in Rankin-Cohen deformations. We explain a connection between Rankin-Cohen brackets and higher order Hankel forms. The famous Erlanger Programm of Klein says that geometry is about to study the transfor- mation groups of various spaces, or more precisely the properties invariant under the actions of such groups, i.e., the symmetries. Noncommutative geometry(NCG), which originated from Connes' study in operator alge- bras in 1970's, brought the landscape of geometry many new objects and some astonishing phenomena. Back to the early 1990s, Connes and Moscovici pointed out that in noncommutative geome- try(NCG) while noncommutative spaces are represented by the algebras (usually noncommuta- tive C ∗ -algebras) of "continuous functions" over noncommutative spaces, the local symmetries are reflected in some Hopf algebras. One of the first noncommutative spaces studied in NCG is the C ∗ -algebra of a foliated space. In the case of codimension n foliations, Connes and Moscovici discovered a Hopf algebra Hn, which governs the local symmetry of leaf spaces of foliations of codimension n. The Hopf algebra Hn is universal in the sense that it depends only on the codimension of a foliation. This family of Hopf algebras {Hn} is very useful in the study of transverse index theory, and later was found to have connections with various different areas of mathematics, c.f. (10), (13). In this paper, we review the application of the Hopf algebra H1 in Rankin-Cohen deformations, which was initiated by Connes and Moscovici (14). We start by recalling the general setting of transverse geometry. Let M be a smooth manifold and F be a foliation on M of codimension n. Let X be a complete flat transversal of F, and F + X be the oriented frame bundle of X. The holonomy pseudogroup acts on X and therefore F + X by transforming X parallelly along paths in leaves of F. The "transverse geometry" is to study the transversal X along with the action by the holonomy pseudogroup . In what follows we focus on the case when n = 1, and define Connes-Moscovici's Hopf algebra H1. Now the complete transversal X is a flat 1-dim manifold; and the oriented frame bundle F + X is diffeomorphic to X × R + , and is a discrete holonomy pseudogroup acting on X as local diffeomorphisms. We introduce coordinates x on X and y on R + . The lifted action of on F + X is

Brill–Noether theory and non-special scrolls

Abstract: In this paper we study the Brill-Noether theory of invertible subsheaves of a general, stable rank-two vector bundle on a curve C with general moduli. We relate this theory to the geometry of unisecant curves on smooth, non-special scrolls with hyperplane sections isomorphic to C. Most of our results are based on degeneration techniques.

Connections on a Parabolic Principal Bundle, II

Abstract: In Connections on a parabolic principal bundle over a curve, I we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in the above article that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalizesto the contextof parabolic principalbundles. For usualprincipalbundles, giving a splitting of this twisted Atiyah exact sequence is equivalent to giving a splitting of the Atiyah exact sequence. Connections on a parabolic principal bundle can be defined using the generalization of the twisted Atiyahexact sequence.

Exponents of a meromorphic connection on a compact Riemann surface

Abstract: We give a general definition of the exponents of a meromorphic connection ∇ on a holomorphic vector bundle E of rank n over a compact Riemann surface X. We prove that they can be computed as invariants of a vector bundle E L canonically attached to E, which we construct and call the Levelt bundle of E, and whose degree (equal to the sum of the exponents) we estimate by upper and lower bounds (Fuchs' relations). We use this definition to construct, for every linear differential equation on a compact Riemann surface (with regular or irregular singularities), the companion bundle of the equation, a vector bundle endowed with a meromorphic connection that is equivalent to the given equation and has precisely the same singularities and the same set of exponents.

Cohomology of line bundles on the cotangent bundle of a Grassmannian

Abstract: We show that certain line bundles on the cotangent bundle of a Grassmannian arising from an anti-dominant character λ have cohomology groups isomorphic to those of a line bundle on the cotangent bundle of the dual Grassmannian arising from the dominant character wo(λ), where ω 0 is the longest element of the Weyl group of SL l+1 (k).

A splitting theorem for holomorphic Banach bundles

Abstract: This paper is motivated by Grothendieck’s splitting theorem. In the 1960s, Gohberg generalized this to a class of Banach bundles. We consider a compact complex manifold X and a holomorphic Banach bundle E → X that is a compact perturbation of a trivial bundle in a sense recently introduced by Lempert. We prove that E splits into the sum of a finite rank bundle and a trivial bundle, provided \({H^{1}(X, \mathcal {O})=0}\) .

Heat kernel expansions in vector bundles

Abstract: Let M be a complete connected smooth (compact) Riemannian manifold of dimension n . Let Π : V → M be a smooth vector bundle over M . Let L = 1 2 Δ + b be a second order differential operator on M , where Δ is a Laplace-Type operator on the sections of the vector bundle V and b a smooth vector field on M . Let k t ( − , − ) be the heat kernel of V relative to L . In this paper we will derive an exact and an asymptotic expansion for k t ( x , y 0 ) where y 0 is the center of normal coordinates defined on M , x is a point in the normal neighborhood centered at y 0 . The leading coefficients of the expansion are then computed at x = y 0 in terms of the linear and quadratic Riemannian curvature invariants of the Riemannian manifold M , of the vector bundle V , and of the vector bundle section ϕ and its derivatives. We end by comparing our results with those of previous authors (I. Avramidi, P. Gilkey, and McKean–Singer).

On the Picard bundle

Abstract: Fix a holomorphic line bundle ξ over a compact connected Riemann surface X of genus g, with g ⩾ 2 , and also fix an integer r such that degree ( ξ ) > r ( 2 g − 1 ) . Let M ξ ( r ) denote the moduli space of stable vector bundles over X of rank r and determinant ξ. The Fourier–Mukai transform, with respect to a Poincare line bundle on X × J ( X ) , of any F ∈ M ξ ( r ) is a stable vector bundle on J ( X ) . This gives an injective map of M ξ ( r ) in a moduli space associated to J ( X ) . If g = 2 , then M ξ ( r ) becomes a Lagrangian subscheme.

Representations attached to vector bundles on curves over finite and p-adic fields, a comparison

Abstract: For a vector bundle E on a model of a smooth projective curve over a p-adic number field a p-adic representation of the geometric fundamental group of X has been defined in work with Annette Werner if the reduction of E is strongly semistable of degree zero. In the present note we calculate the reduction of this representation using the theory of Nori's fundamental group scheme.

On synthetic interpretation of quantum principal bundles

Abstract: Quantum principal bundles or principal comodule algebras are re-interpreted as principal bundles within a framework of Synthetic Noncommutative Differential Geometry. More specifically, the notion of a noncommutative principal bundle within a braided monoidal category is introduced and it is shown that a noncommutative principal bundle in the category opposite to the category of vector spaces is the same as a faithfully flat Hopf-Galois extension.

Spin-structures and 2-fold coverings

Abstract: We prove that the existence of a Spin-structure on an oriented real vector bundle and the number of them can be obtained in terms of 2-fold coverings of the total space of the SO(n)-principal bundle associated to the vector bundle. Basically we use theory of covering spaces. We give a few elementary applications making clear that the Spin-bundle associated to a Spin-structure is not suffcient to classify such structure, as pointed out by [6].

Connections on higher order frame bundles and their gauge analogies

Abstract: We present a survey of the basic properties of connections on the r-th order frame bundle of a manifold. Special attention is paid to the torsion and torsion-free connections. Connections on the r-th principal prolongation of a principal bundle are treated from similar points of view.

Noncommutative Field Theories from a Deformation Point of View

Abstract: In this review we discuss the global geometry of noncommutative field theories from a deformation point of view: The space-times under consideration are deformations of classical space-time manifolds using star products. Then matter fields are encoded in deformation quantizations of vector bundles over the classical space-time. For gauge theories we establish a notion of deformation quantization of a principal fibre bundle and show how the deformation of associated vector bundles can be obtained.

Generalization of hasimoto's transformation

Abstract: In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold M of constant curvature, gives rise under Hasimoto's transformation to the nonlinear Schrodinger equation. We also give a natural interpretation of the function. introduced by Hasimoto in terms of moving frames associated to a natural complex bundle over the filament.

The Moduli Space of Compact Fiber Bundle Structures on a Fibration

Abstract: by Stacy L. Hoehn Suppose that p is a fibration with total space E over a connected, finite CW complex B whose fibers are homotopy equivalent to a finite CW complex F . Then there is a fibration associated to p whose fibers are all homotopy equivalent to A(F ), the algebraic K-theory space of F . Dwyer, Weiss, and Williams have constructed a section of this fibration, called the parametrized Euler characteristic of p, which has a lift to a parametrized excisive Euler characteristic if and only if p is fiber homotopy equivalent to a topological fiber bundle whose fibers are homeomorphic to a compact topological manifold, possibly with boundary. Assuming that p does admit at least one compact topological fiber bundle structure, we can also try to classify the space of all such structures on p. We show that this space of structures on p is homotopy equivalent to the product of the space of all lifts of the parametrized Euler characteristic of p to a parametrized excisive Euler characteristic with the space of stable Euclidean bundles over E.

Geometric Formulation of the Averaging Operation

Abstract: A general framework for geometric averaging procedure is introduced. We motivate the existence of this framework through three examples: 1. The averaging principle appearing in classical mechanics and constitutes the basis of perturbation theory, 2. The integration along the fiber operation, leading to Thom's isomorphism theorem in algebraic topology and 3. The averaging of linear connections in pull-back bundles. A second motivation is suggested by the problem of given a vector bundle automorphism define a natural {\it push-forward} vector bundle automorphism. We will see that a definition of averaging exists such that it provides a solution to the problem and contains the three examples of averaging procedure described before. We follow discussing the notion of {\it convex invariance} and a new approach to investigate Finsler geometry.

Degeneracy of triality-symmetric morphisms

Abstract: We define a new symmetry for morphisms of vector bundles, called triality symmetry, and compute Chern class formulas for the degeneracy loci of such morphisms. In an appendix, we show how to canonically associate an octonion algebra bundle to any rank 2 vector bundle.

Lectures on normal Coulomb frames in the normal bundle of twodimensional immersionen of higher codimension

Abstract: We establish existence and regularity results for normal Coulomb frames in the normal bundle of two-dimensional surfaces of disc-type embedded in Euclidean spaces of higher dimensions.

On the Non-Splitting of the Normal Bundle Sequence

Abstract: In a recent article, Paltin Ionescu and Flavia Repetto proved that if X ⊂ ℙ n is a smooth projective variety over ℂ such that its normal bundle sequence splits over some curve C ⊂ X, then X a linear subspace in ℙ n . In this note, we give a purely geometric proof of this result that is valid in arbitrary characteristic.