Fiber Bundle Theory Research Articles

Rigidity of fibrations over nonpositively curved manifolds

THE THEORY of manifold approximate fibrations is the correct bundle theory for topological manifolds and singular spaces. This theory plays the same role in the topological category as fiber bundle theory plays in the differentiable category and as block bundle theory plays in the piecewise linear category. For example, neighborhoods in topologically stratified spaces can be characterized and classified within the theory of manifold approximate fibrations (see [l 1,3]). This paper is concerned with manifold approximate fibrations over closed manifolds of nonpositive curvature. The main result is that this curvature assumption on the base space implies that such manifold approximate fibrations are rigid in the sense that if two manifold approximate fibrations over a manifold of nonpositive curvature are homotopy-theoretically equivalent, then they are equivalent in a much stronger, geometric way appropriate in this theory (they are controlled homeomorphic). Although we work within the setting of manifold approximate fibrations, our results also hold, and are new, for the more classical theories of fibrations and fiber bundle projections between manifolds. These results are established by working with the full moduli space of all manifold approximate fibrations over a closed manifold B of nonpositive curvature. We study a forgetful map from this moduli space which assigns to a manifold approximate fibratidn over B, the underlying map of a closed manifold to B. We call this map the forget control map for manifold approximate fibrations and show that it is homotopy-split injective. This paper is the second in a series dealing with controlled topology over manifolds of nonpositive curvature. In the first paper [8] we establish that the theory of controlled homeomorphisms coincides with the theory of bounded homeomorphisms over Hadamard manifolds. This point of view is extended in the present paper by showing that the theory of manifold approximate fibrations coincides with the theory of manifold bounded fibrations over Hadamard manifolds. We combine this finding with our classifying differential [6] to prove the homotopy injectivity of the forget control map mentioned above. In the future papers in this series [9, lo], we will show that the splitting of the forget control map for manifold approximate fibrations is compatible with various other constructions, such as taking the underlying Hurewicz fibration or taking the surgery-theoretic normal invariant. This will allow us to prove homotopy split injectivity for the forgetful

Quantum groups on fibre bundles

It is shown that the principle of locality and noncommutative geometry can be connected by a sheaf theoretical method. In this framework quantum spaces are introduced and examples in mathematical physics are given. Within the language of quantum spaces noncommutative principal and vector bundles are defined and their properties are studied. Important constructions in the classical theory of principal fibre bundles like associated bundles and differential calculi are carried over to the quantum case. At the endq-deformed instanton models are introduced for every integral index.

Quantum Einstein Gravity as a Differential Geometry: General Derivation of Gauge-Fixing Term and FP-Ghost Term Based on Fibre Bundle Theory

Intrinsic OSp(2D;2D)-symmetry (D≥3) is formally regarded as the intrinsic part of OSp(2D;2D)-symmetry which is derived from the quantum Einstein gravity theory. On the basis of the intrinsic OSp(2D;2D)-symmetry as the first principle, the geometrical structure of the theory is clarified and the gauge-fixing term plus the FP-ghost one in the quantum Einstein gravity is characterized as an orthosymplectic super-form on the totality of the cross-section of a tensor product bundle

A thin film model

Using the theory of fiber bundles we construct a new model of a thin polycrystal film. The quantum equations of motion thereby obtained are used to determine the regularity of the influence of substructural ordering on the physical characteristics of the film.

An application of the fibration theorem of Ehresmann

The main purpose of the paper is to prove that the map (7) and also its restriction to GL+ (n, IR) is a locaily trivial fibration. From the general theory of fiber bundles we know th a t a bu.ndle map between two C “ -differentiable manifolds is a surjeetive submersion. Here arise a natural problem: given M and N two C00— differentiable manifolds and f: M -> N a smooth surjeetive submersion, find suffi- cient conditions in order th a t f be a locaily trivial fibration. A such condition is given by:

Nonlinear gauge theory of Poincar� gravity

A Poincare affine frame bundle\(\hat P\)(M) and its associated bundleE are established. Using the connection theory of fiber bundles, nonlinear connections on the bundleE are introduced as nonlinear gauge fields. An action and two sets of gauge field equations are presented.

The geometry of compact Lie groups

We explain the sense in which the compact forms of simple Lie groups behave as products of odd-dimensional spheres; the results are obtained by extensive use of homotopy and fibre bundle theory. Applications include topological properties, the study of Casimir operators and some results in representation theory.

Gauge transformations and fiber bundle theory

The relationship between the group Γ of pure gauge transformations of Atiyah, Hitchin, and Singer [Proc. R. Soc. London Ser. A 362, 425 (1978)] of a principal fiber bundle and the group 𝒢 of gauge transformations consisting of the direct product of the local gauge groups on the base space is studied. Γ is an invariant subgroup of 𝒢 and the quotient 𝒢/Γ is identified with the group of inequivalent gauge transformations. In the framework of the category of principal fiber bundles with connections, a natural explanation for the relevance of the group Γ in the classical and quantum theories of gauge fields is presented. The paper is made self-contained by an introductory discussion of the concepts of principal coordinate fiber bundle (gauge fixed principal fiber bundle) and principal fiber bundle, and of the equivalence between the three different versions of the group of vertical automorphisms of the bundle.

The diffeomorphism group and flat principal bundles

Representations of the semidirect product group D■Diff(M) in the context of fiber bundle theory are studied. Here, D is the group of compactly supported functions and Diff(M) is the group of compactly supported diffeomorphisms of a manifold M. The carrier space is taken as the space of equivariant functions on a flat principal bundle over M where M is multiply connected. Two principal bundles are taken as equivalent if they are related by a gauge transformation. For the U(1) case it is found that the representations are irreducible, and the equivalence classes of representations are in one to one correspondence with the equivalence classes of bundles. Simple examples are discussed.

The Fibre Bundle Formulation of σ-Model and SUSY σ-Model

In this paper, we present an analysis of the general symmetry properties of some physical systems, where especially the method of decomposition of connection is used. Then we reformulate the σ-model and the SUSY σ-model in the framework of fibre bundle theory. By comparisons, we obtain the SUSY partner of “linearized” equation or Lax pair of σ-model.

Differential Geometric Methods in Hybrid Parametrization of Linear Dynamical Systems

This paper introduces a non-Euclidean parametrization of linear dynamical systems both in the open loop and in the closed loop. In particular, the various parametrizations introduced have a bundle structure which appears to be of relevance in system identification. Furthermore a quotient topology is obtained on the space of all systems and compared with the well-known graph topology. The article describes many desirable features of the quotient topology. Applications of vector bundle and fiber bundle theory in parametrization of control systems in the closed loop introduced in this paper are new.

Application of topological ideas to the Aharonov-Bohm effect

The Aharonov-Bohm effect is studied to elucidate the existence of global effects in field theory, originating from the topology of the field configuration. The problem is formulated in the language of Maxwell's equations, cohomology and principal fibre bundle theory. The authors offer a motivation for the use of these concepts. In particular, the idealised and realistic descriptions of the Aharonov-Bohm effect are distinguished.

Global bilinearization of non-linear systems and the existence of Volterra series

A global Volterra series is developed for an analytic system defined on an analytic manifold M. Local bilinearizations are pieced together using the theory of fibre bundles, giving rise to ‘twisted’ bilinear systems denned globally on M.

Method of Construction of Gauge-Invariant Lagrangians for Arbitrary Lie Groups

The general method of construction of gauge-invariant Lagrangians that is based on the fibre bundle theory and group transformations, is proposed. The method enables one to obtain Lagrangians with the exact and spontaneously broken symmetry. To illustrate, the Lagrangian of the inhomogeneous gauge group £3= 0(3) x) T3 is constructed. The possibility of constructing a gauge-invariant action for the Poincare group, which satisfies physical requirements is analyzed. It is shown that the action cannot be obtained for the exact symmetry and it can be constructed for the spontaneously broken symmetry. In the latter case the action corresponds to the generalized gravitation theory containing, as a particular case, Einstein's relativity and also the theorieacinvolving torsion and squared curvature. Moreover, it becomes possible that the massive vector particles corresponding to the gauge fields of the translation subgroup would exist.

The geometrical setting of Utiyama’s interaction theory

A mathematical deduction of the fiber bundle theory of interaction from Utiyama’s theory is drawn, in order to get some deeper insights into the geometry of classical gauge theories.

Geometrical approach to the dimensional reduction of symmetric gauge fields

The authors show that for the classification of G-invariant connections it is convenient to use the theory of reduction of fiber bundles, thus making it possible to reduce in a natural manner the action for symmetric gauge fields in a more general case. The classification of symmetric gauge fields and the reduction of their action in a pseudo-Riemannian space with arbitrary invariant metric are considered in the framework of fiber bundle theory.

Mathematical theory of super fibre bundles

A mathematically consistent global theory of super fibre bundles is laid down. In this framework the authors perform, in a rigorous way, the analysis of the construction of super fibre bundles, connections, reduced bundles and reduced connections. Some bundles of physical interest, constructed over a supermanifold (the Lorentz bundle and the graded Poincare bundle), are analysed in detail. A criterion for the existence of a Lorentz bundle over a supermanifold is established. Some results about subgroups of super Lie groups are also given.

A constructive approach to bundles of geometric objects on a differentiable manifold

A constructive approach to bundles of geometric objects of finite rank on a differentiable manifold is proposed, whereby the standard techniques of fiber bundle theory are extensively used. Both the point of view of transition functions (here directly constructed from the jets of local diffeomorphisms of the basis manifold) and that of principal fiber bundles are developed in detail. These, together with the absence of any reference to the current functorial approach, provide a natural clue from the point of view of physical applications. Several examples are discussed. In the last section the functorial approach is also presented in a constructive way, and the Lie derivative of a field of geometric objects is defined.

Connection Theory of Fibre Bundle and Prolongation Structures of Nonlinear Evolution Equations

It is shown that the proper geometrical framework for the nonlinear evolution equations (NEEs) and the soliton equations should be the fibre bundle theory, the principal bundle and its associated bundle and their connection theory. Based upon the requirement of covariance of the geometrical quantities, a covariant generic geometry theory for the prolongation structures of the NEEs is proposed and the fundamental equations for the prolongation structures are presented. From the fundamental equations it immediately follows that the connections corresponding to these NEEs always flat but with torsion and the covariant formulae satisfied by the conservation quantities associated with these NEEs are obtained. The prolongation structure of the MKdV equation, as an example, is concretely worked out by means of the covariant theory of the prolongation structure presented in this paper.

Dyon spin and statistics: A fiber-bundle theory of interacting magnetic and electric charges

The quantum mechanics of a system of nonrelativistic electrically and magnetically charged particles is formulated by introducing as the classical configuration space a certain U(1) bundle over the set of physical particle configurations. In this way one obtains a well-defined and single-valued classical action; and in the corresponding quantum theory, the origin of half-integral spin and of asymptotic Fermi-Dirac statistics in the description of composite systems of bosons finds its expression in terms of the bundle's structure.