Showing papers on "Frame bundle published in 2003"

Noncritical holomorphic functions on Stein manifolds

Abstract: We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle E in the tangent bundle TX with trivial quotient bundle TX/E is homotopic to the tangent bundle of a holomorphic foliation of X. If X is parallelizable, it admits a submersion to C^{n-1} and nonsingular holomorphic foliations of any dimension; the question whether such X also admits a submersion (=immersion) in C^n remains open. Our proof involves a blend of techniques (holomorphic automorphisms of Euclidean spaces, solvability of the di-bar equation with uniform estimates, Thom's jet transversality theorem, Gromov's convex integration method). A result of possible independent interest is a lemma on compositional splitting of biholomorphic mappings close to the identity (Theorem 4.1).

Noncommutative Riemannian and Spin Geometry of the Standard q-Sphere

Abstract: We study the quantum sphere $C_q[S^2]$ as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2-dimensional cotangent bundle as a direct sum $\Omega^{0,1}\oplus\Omega^{1,0}$ in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators. We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci curvature and q-Dirac operator $D$. We find the possibility of an antisymmetric volume form quantum correction to the Ricci curvature and Lichnerowicz-type formulae for $D^2$. We also remark on the geometric q-Borel-Weil-Bott construction.

Higgs Fields, Bundle Gerbes and String Structures

Abstract: We use bundle gerbes and their connections and curvings to obtain an explicit formula for a de Rham representative of the string class of a loop group bundle. This is related to earlier work on calorons.

Stable maps and Quot schemes

Abstract: In this paper we study the relationship between two different compactifications of the space of vector bundle quotients of an arbitrary vector bundle on a curve. One is Grothendieck's Quot scheme, while the other is a moduli space of stable maps to the relative Grassmannian. We establish an essentially optimal upper bound on the dimension of the two compactifications. Based on that, we prove that for an arbitrary vector bundle, the Quot schemes of quotients of large degree are irreducible and generically smooth. We precisely describe all the vector bundles for which the same thing holds in the case of the moduli spaces of stable maps. We show that there are in general no natural morphisms between the two compactifications. Finally, as an application, we obtain new cases of a conjecture on effective base point freeness for pluritheta linear series on moduli spaces of vector bundles.

Generalised connections over a vector bundle map

Abstract: A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in addition, is ‘parametrised’ in some specific way by a vector bundle map from a prescribed vector bundle over M into TM. Some basic properties of these generalised connections are investigated. Special attention is paid to the class of linear connections over a vector bundle map. It is pointed out that not only the more familiar types of connections encountered in the literature, but also the recently studied Lie algebroid connections, can be recovered as special cases within this more general framework.

On endomorphisms of projective bundles

Abstract: Let X be a projective bundle. We prove that X admits an endomorphism of degree >1 and commuting with the projection to the base, if and only if X trivializes after a finite covering. When X is the projectivization of a vector bundle E of rank 2, we prove that it has an endomorphism of degree >1 on a general fiber only if E splits after a finite base change.

Harmonic sections of homogeneous fibre bundles

Abstract: We show how the equations for harmonic maps into homogeneous spaces generalize to harmonic sections of homogeneous fibre bundles. Surprisingly, the generalization does not explicitly involve the curvature of the bundle. However, a number of special cases of the harmonic section equations (including the new condition of super-flatness) are studied in which the bundle curvature does appear. Some examples are given to illustrate these special cases in the non-flat environment. The bundle in question is the twistor bundle of an even-dimensional Riemannian manifold M whose sections are the almost-Hermitian structures of M.

Connections and Curvings on Lifting Bundle Gerbes

Abstract: We construct a connection and a curving on a bundle gerbe associated with lifting a structure group of a principal bundle to a central extension The construction is based on certain structures on the bundle, ie connections and splittings The Deligne cohomology class of the lifting bundle gerbe with the connection and with the curving coincides with the obstruction class of the lifting problem with these structures

Seshadri-exceptional foliations

Abstract: The maximal Seshadri number μ(L) of an ample line bundle L on a smooth projective variety X measures the local positivity of the line bundle L at a general point of X By refining the method of Ein-Kuchle-Lazarsfeld, lower bounds on μ(L) are obtained in terms of Ln, n=dim(X), for a class of varieties The main idea is to show that if a certain lower bound is violated, there exists a non-trivial foliation on the variety whose leaves are covered by special curves In a number of examples, one can show that such foliations must be trivial and obtain lower bounds for μ(L) The examples include the hyperplane line bundle on a smooth surface in ℙ3 and ample line bundles on smooth threefolds of Picard number 1

Noncommutative Ricci Curvature and Dirac Operator on Cq[SL2] at Roots of Unity

Abstract: We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group C q [ SL2], using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for q an odd rth root of unity that its eigenvalues are given by q-integers [m] q for m = 0,1...,r − 1 offset by the constant background curvature. We fully solve the Dirac equation for r = 3.

Gauge Natural Theories

Abstract: Gauge Natural field theories are introduced to generalize natural theories as well as pure gauge theories and to encompass many relevant physical situations. Conserved quantities and superpotentials are considered for an arbitrary gauge natural theory. Some relevant examples in Physics are considered in detail.

Geometric aspects of the maximum principle and lifts over a bundle map.

Abstract: A coordinate-free proof of the maximum principle is provided in the specific case of an optimal control problem with fixed time. Our treatment heavily relies on a special notion of variation of curves that consist of a concatenation of integral curves of time-dependent vector fields with unit time component, and on the use of a concept of lift over a bundle map. We further derive necessary and sufficient conditions for the existence of so-called abnormal extremals.

Integrable Systems Associated to a Hopf Surface

Abstract: A Hopf surface is the quotient of the complex surface C2 \ {0} by an infinite cyclic group of dilations of C 2 . In this paper, we study the moduli spaces M n of stable SL(2, C)-bundles on a Hopf surface H, from the point of view of symplectic geometry. An important point is that the surface H is an elliptic fibration, which implies that a vector bundle on H can be considered as a family of vector bundles over an elliptic curve. We define a map G: M n → P 2n+1 that associates to every bundle on H a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on M n . We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kit is an elliptic fibration that does not admit a section.

Finite rank vector bundles on inductive limits of grassmannians

Abstract: If P 1 is the projective ind-space, i.e. P 1 is the inductive limit of linear embeddings of complex projective spaces, the Barth-Van de Ven-Tyurin (BVT) Theorem claims that every finite rank vector bundle on P 1 is isomorphic to a direct sum of line bundles. We extend this theorem to general sequences of morphisms between projective spaces by proving that, if there are infinitely many morphisms of degree higher than one, every vector bundle of finite rank on the inductive limit is trivial. We then establish a relative version of these results, and apply it to the study of vector bundles on inductive limits of grassmannians. In particular we show that the BVT Theorem extends to the ind-grassmannian of subspaces commensurable with a fixed infinite dimensional and infinite codimensional subspace in C 1 . We also show that, for a class of twisted ind-grassmannians, every finite rank vector bundle is trivial. 2000 AMS Subject Classification: Primary 32L05, 14J60, Secondary 14M15.

Geodesics on the unit tangent bundle

Abstract: A geodesic γ on the unit tangent sphere bundle T1M of a Riemannian manifold (M, g), equipped with the Sasaki metric gS, can be considered as a curve x on M together with a unit vector field V along it. We study the curves x. In particular, we investigate for which manifolds (M, g) all these curves have constant first curvature κ1 or have vanishing curvature κi for some i = 1, 2 or 3.

Module derivations and cohomological splitting of adjoint bundles

Abstract: Let G be a nite loop space such that the mod p cohomology of the clas- sifying space BG is a polynomial algebra. We consider when the adjoint bundle associated with a G-bundle over M splits on mod p cohomology as an algebra. In the case p = 2, an obstruction for the adjoint bundle to admit such a splitting is found in the Hochschild homology concerning the mod 2 cohomologies of BG and M via a module derivation. Moreover the derivation tells us that the splitting is not compatible with the Steenrod operations in general. As a consequence, we can show that the isomorphism class of an SU(n)-adjoint bundle over a 4-dimensional CW complex coincides with the homotopy equivalence class of the bundle.

Gauge invariance and variational trivial problems on the bundle of connections

Abstract: Given a principal bundle P→M we classify all first order Lagrangian densities on the bundle of connections associated to P that are invariant under the Lie algebra of infinitesimal automorphisms. These are shown to be variationally trivial and to give constant actions that equal the characteristic numbers of P if dimM is even and zero if dimM is odd. In addition, we show that variationally trivial Lagrangians are characterized by the de Rham cohomology of the base manifold M and the characteristic classes of P of arbitrary degree

Restriction theorems for principal bundles

Abstract: Let E be a semistable (or stable) principal bundle over a smooth complex projective variety X, and let D⊂X be a complete intersection. We study the (semi)stability of the restriction E|D. Some of the results known for vector bundles, such as Grauert–Mulich, Flenner and Mehta–Ramanathan theorems, are generalized to principal bundles.

Principal bundle, parabolic bundle, and holomorphic connection

Abstract: Let E G be a principal G—bundle over a rationally connected variety, where G is a complex algebraic group. Then any holomorphic connection on E G is flat. We describe a necessary and sufficient condition for a parabolic vector bundle over a Riemann surface to admit a logarithmic connection compatible with the parabolic structure.

Sub-bundles of the complexified tangent bundle

Abstract: We study embeddings of complex vector bundles, especially line bundles, in the complexification of the tangent bundle of a manifold The aim is to understand implications of properties of interest in partial differential equations

Structure of Malicious Singularities

Abstract: In this paper, we investigate relativistic spacetimes, together with their singular boundaries (including the strongest singularities of the Big Bang type, called malicious singularities), as noncommutative spaces. Such a space is defined by a noncommutative algebra on the transformation groupoid Γ = Ē × G, where Ē is the total space of the frame bundle over spacetime with its singular boundary, and G is its structural group. We show that there exists the bijective correspondence between unitary representations of the groupoid Γ and the systems of imprimitivity of the group G. This allows us to apply the Mackey theorem to this case, and deduce from it some information concerning singular fibers of the groupoid Γ. At regular points the group representation, which is a part of the corresponding system of imprimitivity, does not have discrete components, whereas at the malicious singularity such a group representation can be a single representation (in particular, an irreducible one) or a direct sum of such representations. A subgroup K ⊂ G, from which—according to the Mackey theorem—the representation is induced to the whole of G, can be regarded as measuring the “richness” of the singularity structure. In this sense, the structure of malicious singularities is richer than those of milder ones.

Heisenberg groups?the fundamental ingredient to describe information, its transmission and quantization

Abstract: For a singularity free gradient field in an open set of an oriented Euclidean space of dimension three we define a natural principal bundle out of an immanent complex line bundle. The fibres of this bundle encode information. The elements of both bundles are called internal variables. Several other natural bundles are associated with the principal bundle and, in turn, determine the vector field. Two examples are given and it is shown that for a constant vector field circular polarized waves with values in the principal bundle are associated with the vector field. These waves transmit information encoded in internal variables and, moreover, determine a Schrodinger representation. On U(2) a relation between spin representations and Schrodinger representations is established. The link between the spin ½ model and the Schrodinger representations yields a connection between a microscopic and a macroscopic viewpoint. Quantization and its link to information is derived out of the Schrodinger representation.

On the fiber product preserving gauge bundle functors on vector bundles

Abstract: We present a complete description of all fiber product preserving gauge bundle functors F on the category VBm of vector bundles with m-dimensional bases and vector bundle maps with local diffeomorphisms as base maps. Some corollaries of this result are presented. Introduction. Modern differential geometry has clarified that product preserving bundle functors on the categoryMf of manifolds and maps play very important roles. To such bundle functors one can lift some geometric structures as vector fields, forms, connections, etc. To define such lifts the only important property is the product preservation. Such functors have been classified by means of Weil algebras [5]. Research quite similar to that on manifolds has been done on fibered manifolds. A wide class of bundle functors on the category FMm of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps is the class of fiber product preserving functors. Such functors have been classified in [6], and studied in [1]–[4], [8]. In turn research similar to that on fibered manifolds has been done on vector bundles. A wide class of (gauge) bundle functors on the category VBm of vector bundles with m-dimensional bases and vector bundle maps with local diffeomorphisms as base maps is the class of fiber product preserving functors. For example the r-jet prolongation functor plays an important role in the theory of higher order connections, Lagrangians, differential equations, etc. Below, we present many examples of such functors. Some of them are well known. It seems natural and useful to classify all such functors. The purpose of the present paper is to describe all fiber product preserving gauge bundle functors on VBm. 2000 Mathematics Subject Classification: 58A05, 58A20.

The Berwald-type linearization of generalized connections

Abstract: We study the existence of a natural 'linearization' process for generalized connections on an affine bundle. It is shown that this leads to an affine generalized connection over a prolonged bundle, which is the analogue of what is called a connection of Berwald type in the standard theory of connections. Various new insights are being obtained in the fine structure of affine bundles over an anchored vector bundle and affineness of generalized connections on such bundles.

Reduction theory for a rational function field

Abstract: Let $G$ be a split reductive group over a finite field $\Fq$. Let $F=\Fq(t)$ and let $\A$ denote the ad\`eles of $F$. We show that every double coset in $G(F)\bsl G(\A)/ K$ has a representative in a maximal split torus of $G$. Here $K$ is the set of integral ad\`elic points of $G$. When $G$ ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.