Showing papers on "Frame bundle published in 2004"

A hyperkahler structure on the cotangent bundle of a complex Lie group

Abstract: Let G be compact Lie group. It is shown that the cotangent bundle of the complexification of G admits a hyperkahler structure which is invariant under left and right translations by elements of G. The proof is to realize the cotangent bundle of the complex group as a moduli space of solutions to Nahm's equations on the closed interval.

Harder-Narasimhan reduction of a principal bundle

Abstract: Let $E$ be a principal $G$--bundle over a smooth projective curve over an algebraically closed field $k$, where $G$ is a reductive linear algebraic group over $k$. We construct a canonical reduction of $E$. The uniqueness of canonical reduction is proved under the assumption that the characteristic of $k$ is zero. Under a mild assumption on the characteristic, the uniqueness is also proved when the characteristic of $k$ is positive.

Bundle 2-gerbes

Abstract: We make the category $\textbf{BGrb}_M$ of bundle gerbes on a manifold $M$ into a $2$-category by providing $2$-cells in the form of transformations of bundle gerbe morphisms. This description of $\textbf{BGrb}_M$ as a $2$-category is used to define the notion of a bundle $2$-gerbe. To every bundle $2$-gerbe on $M$ is associated a class in $H^4(M ; \mathbb{Z})$. We define the notion of a bundle $2$-gerbe connection and show how this leads to a closed, integral, differential $4$-form on $M$ which represents the image in real cohomology of the class in $H^4(M ; \mathbb{Z})$. Some examples of bundle $2$-gerbes are discussed, including the bundle $2$-gerbe associated to a principal $G$ bundle $P \to M$. It is shown that the class in $H^4(M ; \mathbb{Z})$ associated to this bundle $2$-gerbe coincides with the first Pontryagin class of $P$: this example was previously considered from the point of view of $2$-gerbes by Brylinski and McLaughlin.

Geometry of Multiplicity-Free Representations of GL(n), Visible Actions on Flag Varieties, and Triunity

Abstract: We analyze the criterion of the multiplicity-free theorem of representations [5, 6] and explain its generalization. The criterion is given by means of geometric conditions on an equivariant holomorphic vector bundle, namely, the ‘visibility’ of the action on a base space and the multiplicity-free property on a fiber.

On Manifolds Whose Tangent Bundle is Big and 1-Ample

Abstract: A line bundle over a complex projective variety is called big and 1-ample if a large multiple of it is generated by global sections and a morphism induced by the evaluation of the spanning sections is generically finite and has at most 1-dimensional fibers. A vector bundle is called big and 1-ample if the relative hyperplane line bundle over its projectivisation is big and 1-ample. The main theorem of the present paper asserts that any complex projective manifold of dimension 4 or more, whose tangent bundle is big and 1-ample, is equal either to a projective space or to a smooth quadric. Since big and 1-ample bundles are ?almost? ample, the present result is yet another extension of the celebrated Mori paper ?Projective manifolds with ample tangent bundles? (Ann. of Math. 110 (1979) 593?606). The proof of the theorem applies results about contractions of complex symplectic manifolds and of manifolds whose tangent bundles are numerically effective. In the appendix we re-prove these results.

Hypercomplex manifolds with trivial canonical bundle and their holonomy

Abstract: Let (M,I,J,K) be a compact hypercomplex manifold admitting an HKT-metric Assume that the canonical bundle of (M,I) is trivial as a holomorphic line bundle We show that the holonomy of Obata connection on M is contained in SL(n,H) In Appendix we apply these arguments to compact nilmanifolds equipped with abelian hypercomplex structures, showing that such manifolds have holonomy in SL(n,H)

On manifolds with trivial logarithmic tangent bundle

Abstract: By a classical result of Wang [14] a connected compact complex manifold X has holomorphically trivial tangent bundle if and only if there is a connected complex Lie group G and a discrete subgroup Γ such that X is biholomorphic to the quotient manifold G/Γ. In particular X is homogeneous. If X is Kähler, G must be commutative and the quotient manifold G/Γ is a compact complex torus. The purpose of this note is to generalize this result to the noncompact Kähler case. Evidently, for arbitrary non-compact complex manifold such a result can not hold. For instance, every domain over C has trivial tangent bundle, but many domains have no automorphisms. So we consider the “open case” in the sense of Iitaka ([7]), i.e. we consider manifolds which can be compactified by adding a divisor. Following a suggestion of the referee, instead of only considering Kähler manifolds we consider manifolds in class C as introduced in [5]. A compact complex manifold X is said to be class in C if there is a surjective holomorphic map from a compact Kähler manifold onto X. Equivalently, X is bimeromorphic to a Kähler manifold ([13]). For example, every Moishezon manifold is in class C. We obtain the following characterization:

The Moduli of reducible vector bundles

Abstract: A procedure for computing the dimensions of the moduli spaces of reducible, holomorphic vector bundles on elliptically fibered Calabi-Yau threefolds X is presented. This procedure is applied to poly-stable rank n+m bundles of the form V + pi* M, where V is a stable vector bundle with structure group SU(n) on X and M is a stable vector bundle with structure group SU(m) on the base surface B of X. Such bundles arise from small instanton transitions involving five-branes wrapped on fibers of the elliptic fibration. The structure and physical meaning of these transitions are discussed.

Computations of Bott-Chern classes on ℙ(E)

Abstract: We compute the Bott-Chern classes of the metrized relative Euler sequence describing the relative tangent bundle of the variety ℙ(E) of the hyperplanes in a holomorphic Hermitian vector bundle (E,h) on a complex manifold. We give applications to the construction of the arithmetic characteristic classes of an arithmetic vector bundle $\overline{\mathscr{E}}$ and to the computation of the height of ℙ($\overline{\mathscr{E}}$) with respect to the tautological quotient bundle $\mathscr{O}_\overline{\mathscr{E}}$(1).

Differential and Geometric Structure for the Tangent Bundle of a Projective Limit Manifold.

Abstract: The tangent bundle of a wide class of Frechet manifolds is studied he- re. A vector bundle structure is obtained with structural group a topological subgroup of the general linear group of the fiber type. Moreover, basic geo- metric results, known form the classical case of finite dimensional manifolds, are recovered here: Connections can be defined and are characterized by a generalized type of Christoffel symbols while, at the same time, parallel di- splacements of curves are possible despite the problems concerning differen- tial equations in Frechet spaces.

From E(8) to F via T

Abstract: We argue that T-duality and F-theory appear automatically in the E 8 ``gauge'' bundle perspective of M-theory. The 11-dimensional supergravity four-form determines an E 8 bundle over the 11-dimensional bulk. If we compactify on a two-torus, this data specifies an LLE 8 bundle over the remaining 9-dimensions where LG is a centrally-extended loopgroup of G. If one of the circles of the torus is smaller than (α')1/2 then it is also smaller than a nontrivial circle S in the LLE 8 fiber and so a dimensional reduction on the total space of the LLE 8 bundle is not valid. We conjecture that S is the circle on which the T-dual type IIB supergravity is compactified, with the aforementioned torus playing the role of the F-theory torus. As tests we reproduce the known T-dualities between NS5-branes and KK-monopoles, as well as D6 and D7-branes where we find the desired F-theory monodromy. Using Hull's proposal for massive IIA, this realization of T-duality allows us to confirm that the Romans mass is the central extension of our LE 8 . In addition this construction immediately reproduces the conjectured formula for global topology change from T-duality with H-flux.

Image processing via shift-twist invariant operations on orientation bundle functions

Abstract: Inspired by our own visual system we consider the construction of-and reconstruction from- an orientation bundle function (OBF) Uf : R2 ? S1 ?C as a local orientation score of an image, f : R2 ? R, via a wavelet transform W? corresponding to a representation of the Euclidean motion group onto L2(R2) and oriented wavelet ? ?L2(R2). This wavelet transform is a unitary mapping with stable inverse, which allows us to directly relate each operation ? on OBF’s to an operation ? on images in a robust manner. We examine the geometry of the domain of an OBF and show that the only sensible operations on OBF’s are non-linear and shift-twist invariant. As an example we consider all linear 2nd order shift-twist invariant evolution equations on OBF’s corresponding to stochastic processes on the Euclidean motion group in order to construct nonlinear shift-twist invariant operations on OBF’s. Given two such stochastic processes we derive the probability density that particles of the different processes collide. As an application we detect elongated structures in images and automatically close the gaps between them.

Classification of extensions of principal bundles and transitive Lie groupoids with prescribed kernel and cokernel

Abstract: The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well-known result. A remarkable generalization of this equivalence, given by Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admit an action of the structure group by automorphisms. In this paper the existence of suitably equivariant transition functions is proved for such groupoids, generalizing consequently the classification of principal bundles by means of their transition functions, to extensions of principal bundles by an equivariant form of Cech cohomology.

Rigid and Finite Type Geometric Structures

Abstract: Rigid geometric structures on manifolds, introduced by Gromov, are characterized by the fact that their infinitesimal automorphisms are determined by their jets of a fixed order. Important examples of such structures are those given by an H-reduction of the first order frame bundle of a manifold, where the Lie algebra of H is of finite type; in fact, for structures given by reductions to closed subgroups of first order frame bundles, finite type implies rigidity. The goal of this paper is to generalize this to geometric structures defined by reductions of frame bundles of arbitrary order, and to give an algebraic characterization of the property of being rigid in terms of a suitable notion of finite type.

On higher order geometry on anchored vector bundles

Abstract: Some geometric objects of higher order concerning extensions, semi-sprays, connections and Lagrange metrics are constructed using an anchored vector bundle.

Existence of holomorphic sections and perturbation of positive line bundles over $q$--concave manifolds

Abstract: By using asymptotic Morse inequalities we give a lower bound for the space of holomorphic sections of high tensor powers in a positive line bundle over a q-concave domain The curvature of the positive bundle induces a hermitian metric on the manifold The bound is given explicitely in terms of the volume of the domain in this metric and a certain integral on the boundary involving the defining function and its Levi form As application we study the perturbattion of the complex structure of a q-concave manifold

Stratified fibre bundles

Abstract: A stratified bundle is a fibered space in which strata are classical bundles and in which attachment of strata is controlled by a structure category F of fibers. Well known results on fibre bundles are shown to be true for stratified bundles; namely the pull back theorem, the bundle theorem and the principal bundle theorem. AMS SC : 55R55 (Fiberings with singularities); 55R65 (Generalizations of fiber spaces and bundles); 55R70 (Fiberwise topology); 55R10 (Fibre bundles); 18F15 (Abstract manifolds and fibre bundles); 54H15 (Transformation groups and semigroups); 57S05 (Topological properties of groups of homeomorphisms or diffeomorphisms).

Holomorphic Vector Bundle on Hopf Manifolds with Abelian Fundamental Groups

Abstract: Let X be a Hopf manifolds with an Abelian fundamental group. E is a holomorphic vector bundle of rank r with trivial pull-back to W = ℂn–{0}. We prove the existence of a non-vanishing section of L⊗E for some line bundle on X and study the vector bundles filtration structure of E. These generalize the results of D. Mall about structure theorem of such a vector bundle E.

Twistor bundle theory and its application

Abstract: Over an oriented even dimensional Riemannian manifold(M 2m ,ds2 ), in terms of the Levi-Civita connection form Ω and the canonical form Θ on the bundle of positive orthonormal frames, we give a detailed description of the twistor bundle Гm = SO(2m)/U(m)↪ J +(@#@ M,ds2 ) →M. The integrability on an almost complex structureJ compatible with the metric and the orientation, is shown to be equivalent to the fact that the corresponding cross section of the twistor bundle is holomorphic with respect toJ and the canonical almost complex structureJ 1 onJ +(M,ds2 ), by using moving frame theory. Moreover, for various metrics and a fixed orientation onM, a canonical bundle isomorphism is established. As a consequence, we generalize a celebrated theorem of LeBrun.

A class of Poisson–Nijenhuis structures on a tangent bundle

Abstract: Equipping the tangent bundle TQ of a manifold with a symplectic form coming from a regular Lagrangian L, we explore how to obtain a Poisson–Nijenhuis structure from a given type (1, 1) tensor field J on Q. It is argued that the complete lift Jc of J is not the natural candidate for a Nijenhuis tensor on TQ, but plays a crucial role in the construction of a different tensor R, which appears to be the pullback under the Legendre transform of the lift of J to T*Q. We show how this tangent bundle view brings new insights and is capable also of producing all important results which are known from previous studies on the cotangent bundle, in the case when Q is equipped with a Riemannian metric. The present approach further paves the way for future generalizations.

Some geometric structures on the tangent bundle of a Riemannian manifold

Abstract: It is defined a new almost complex structure with Norden metric (hyperbolic metric) on the tangent bundle TM of an n—dimensional Riemannian manifold M. Next, the conditions under which the considered almost complex structure with Norden metric belongs to one of the eight classes of almost complex manifolds with Norden metric obtained by G. T. Ganchev and D. V. Borisov in the classification from [2] there are studied.

Natural T-Functions on the Cotangent Bundle of a Weil Bundle

Abstract: A natural T-function on a natural bundle F is a natural operator transforming vector fields on a manifold M into functions on FM. For any Weil algebra A satisfying dim M ⩾ width(A) + 1 we determine all natural T-functions on T * T A M, the cotangent bundle to a Weil bundle T A M.

Hitchin's connection and differential operators with values in the determinant bundle

Abstract: For a family of smooth curves, we have the associated family of moduli spaces of stable bundles with fixed determinant on the curves. There exists a so called theta line bundle on the family of moduli spaces. When the Kodaira–Spencer map of the family of curves is an isomorphism, we prove in this paper an identification theorem between sheaves of differential operators on the theta line bundle and higher direct images of vector bundles on curves. As an application, the so called Hitchin's connection on the direct image of (powers of) theta line bundle is derived naturally from the identification theorem. A logarithmic extension of Hitchin's connection to certain singular stable curves is also presented in this paper.

A new surface model based on a fibre bundle of 1-parameter groups

Abstract: A new fibre bundle surface model is proposed for 3D object modeling. This fibre bundle model, consisting of local direct product of a base curve and a fibre curve, is able to represent arbitrary surfaces. In particular, the fibre bundle of 1-parameter groups, i.e. with fibres as 1-parameter groups are efficient in both synthesis and recognition. Indeed, the 1-parameter groups can be uniquely determined by finite, e.g. six invariants of their Lie algebras. Besides, the surfaces can be quickly generated by elementary functions without numerical integration errors. This model is also expected to be useful in object-based 3D image coding.

Holomorphic line bundles on a domain of a two-dimensional Stein manifold

Abstract: Let D be an open subset of a two-dimensional Stein manifold S. Then D is Stein if and only if every holomorphic line bundle L on D is the line bundle associated to some (not necessarily effective) Cartier divisor d on D.

The natural affinors on the r-jet prolongations of a vector bundle

Abstract: It is known that for integers m > 2, n > 1 and r > 3 there are only three r-jet prolongations of a vector bundle E with m-dimensional bases and n-dimensional fibers. The first one is the usual r-jet prolongation JrE, the second one is the vertical r-jet prolongation J%E and the third one is the [r]-jet prolongation In this paper for integers m > 2, n > 1 and r > 1 we classify all natural affinors on FrE, where FrE denotes JT E or JyE or As corollaries we obtain similar results for FrE*, (FrE)* and ( F r E * y instead of FrE. Introduction One can prove (a paper in preparation) that for integers r > 3 and m > 2 there are only three r-jet prolongations of a vector bundle E with m-dimensional basis. Namely, we have the usual r-jet prolongation JrE of E, the vertical r-jet prolongation J^E of E and the [r]-jet prolongation jMtfof E. In [15] for integers m > 2, n > 1 and r > 1 we classified all natural linear operators A lifting a linear vector field X from a vector bundle E with m-dimensional basis and n-dimensional fibers into a vector field A(X) on FrE, where FrE denotes JrE or JTVE or J^E. In the case FrE = JrE we proved that A{X) is a constant multiple of the flow operator JTX. In the case FrE = J£E we proved that A{X) is a linear combination of the flow operator J^X and some explicitely constructed linear natural operator V < l > ( X ) . In the case FrE = J^E we proved that A(X) is a linear combination of the flow operator J ^ X and some explicitely constructed linear natural operator U ^ ( X ) . An afiinor B on a manifold M is a tensor field of type (1,1) on M.

A geometric construction of Tango bundle on P 5

Abstract: The Tango bundle T over P 5 is proved to be the pull-back of the twisted Cayley bundle C(1) via a map f : P 5 ! Q5 existing only in character- istic 2. The Frobenius morphism ' factorizes via such f.

On adapted bi-conformal metrics

Abstract: The geometry of the tangent bundle is used to define a particular class of metrics called adapted bi-conformal. These metrics are defined and studied on the holomorphic cotangent bundle of a Kahler manifold. Kahlerian adapted bi-conformal metrics are totally classified and their curvature expressed. The Eguchi-Hanson metric appears as a particular example.