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Showing papers on "Frame bundle published in 1992"


Book ChapterDOI
01 Jan 1992
TL;DR: In this paper, a singular hermitian metric on a holomorphic line bundle is introduced as a tool for the study of various algebraic questions, and the coefficients of isolated logarithmic poles of a plurisubharmonic singular metric are shown to have a simple interpretation in terms of the constant e of Seshadri's ampleness criterion.
Abstract: The notion of a singular hermitian metric on a holomorphic line bundle is introduced as a tool for the study of various algebraic questions. One of the main interests of such metrics is the corresponding L2 vanishing theorem for Open image in new window cohomology, which gives a useful criterion for the existence of sections. In this context, numerically effective line bundles and line bundles with maximum Kodaira dimension are characterized by means of positivity properties of the curvature in the sense of currents. The coefficients of isolated logarithmic poles of a plurisubharmonic singular metric are shown to have a simple interpretation in terms of the constant e of Seshadri’s ampleness criterion. Finally, we use singular metrics and approximations of the curvature current to prove a new asymptotic estimate for the dimension of cohomology groups with values in high multiples O(kL) of a line bundle L with maximum Kodaira dimension.

359 citations


Journal ArticleDOI
TL;DR: In this article, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined, where the bundle E fails to be associated with a principal bundle and the canonical jet bundle morphism J1EH×J1Σ→J1E is used.
Abstract: Given a principal bundle P→X with a structure group G and an associated Higgs bundle Σ with a standard fiber G/H, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined. In the presence of a fixed Higgs field σ: X→Σ, matter fields are represented by sections of a matter bundle Eh associated with the corresponding reduced subbundle Ph of P. The totality of matter fields and Higgs fields is described by sections of the bundle E which is the composite bundle EH→Σ→X where EH→Σ is the bundle associated with the principal H‐bundle P→Σ. The bundle E fails to be associated with a principal bundle. To construct a connection Γ: E→J1E on E, the canonical jet bundle morphism J1EH×J1Σ→J1E is used.

78 citations


Journal ArticleDOI
TL;DR: For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold M, a family of diffential operators is given, which acts on the space of smooth sections of a vector bundle on M as discussed by the authors.
Abstract: For every pointwise polynomial function on each fiber of the cotangent bundle of a Riemannian manifold M, a family of diffential operators is given, which acts on the space of smooth sections of a vector bundle on M. Such a correspondence may be considered as a rule to quantize classical systems moving in a Riemannian manifold or in a gauge field. Some applications of our construction are also given in this paper

35 citations


Journal ArticleDOI
TL;DR: In this paper, conditions for the Levi-Civita connection of the spacetime tangent bundle of a Finsler manifold to be almost complex or Kahlerian were investigated.
Abstract: Conditions are investigated under which the Levi-Civita connection of the spacetime tangent bundle corresponds to that of a generic tangent bundle of a Finsler manifold. Also, requirements are specified for the spacetime tangent bundle to be almost complex or Kahlerian.

25 citations


Journal ArticleDOI
TL;DR: The natural vector valued l-forms Q on the natural bundles associated with product-preserving functors (including the tangent bundle, the bundle of first order k-velocities, the bundles of second order l-vectors and the bundling of linear frames) and on the cotangent bundle are classified in this paper.
Abstract: The natural vector valued l-forms Q on the natural bundles associated with product preserving functors (including the tangent bundle, the bundle of first order k-velocities, the bundle of second order l-velocities and the bundle of linear frames) and on the cotangent bundle are classified. Then, these forms Q are used to study the torsion r = (I', Q) of connections P on the above bundles, where ( 1, -1 is the Frolicher-Nijenhuis bracket.

21 citations



Journal ArticleDOI
TL;DR: On the tangent bundle of a nonholonomic manifold one defines a canonical structure of bundle of homogeneous nilpotent Lie algebras, and on the local ring of a nonsmooth manifold one determines a quasijet structure with the help of these constructions one uniformizes and simplifies a number of results as mentioned in this paper.
Abstract: On the tangent bundle of a nonholonomic manifold one defines a canonical structure of bundle of homogeneous nilpotent Lie algebras. On the local ring of a nonholonomic manifold one defines a quasijet structure. With the help of these constructions one uniformizes and simplifies a number of results of nonholonomic geometry.

20 citations


Journal ArticleDOI
TL;DR: In this paper, the Levi-Civita connection coefficients of the spacetime tangent bundle, for the case of a Finsler spacetime, are reduced to the form given by Yano and Davies.
Abstract: The Levi-Civita connection coefficients of the spacetime tangent bundle, for the case of a Finsler spacetime, are reduced to the form given by Yano and Davies for a generic tangent bundle of a Finsler manifold. A useful expression is also obtained for the Riemann curvature scalar of a Finsler-spacetime tangent bundle.

19 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived a useful expression for the Riemann curvature scalar of the space-time tangent bundle manifold by working in an anholonomic basis adapted to the spacetime affine connection.
Abstract: The maximum possible proper acceleration relative to the vacuum determines much of the differential geometric structure of the space-time tangent bundle. By working in an anholonomic basis adapted to the spacetime affine connection, one derives a useful expression for the Riemann curvature scalar of the bundle manifold. The explicit documentation of the proof is important because of the central role of the curvature scalar in the formulation of an action with resulting field equations and associated solutions to physical problems.

19 citations


Journal ArticleDOI
TL;DR: In this paper, an analysis of the holomorphic Pfaffian line bundle defined over an infinite dimensional isotropic Grassmannian manifold is presented, and a Fock space structure on the space of holomorphic sections of the dual of this bundle is given.
Abstract: We analyze the holomorphic Pfaffian line bundle defined over an infinite dimensional isotropic Grassmannian manifold. Using the infinite dimensional relative Pfaffian, we produce a Fock space structure on the space of holomorphic sections of the dual of this bundle. On this Fock space, an explicit and rigorous construction of the spin representations of the loop groupsLOn is given. We also discuss and prove some facts about the connection between the Pfaffian line bundle over the Grassmannian and the Pfaffian line bundle of a Dirac operator.

16 citations


Journal ArticleDOI
TL;DR: In this paper, a family of transition paths which reproduce all the allowed combinations of genus g ⪖ 2 spaces was constructed, where the amplitude is non-vanishing only if the combination of the space topologies satisfies a certain selection rule.

Journal ArticleDOI
TL;DR: In this paper, a geometro-stochastic quantization of a gauge theory for extended objects based on the (4, 1)-de Sitter group is used for the description of quantized matter in interaction with gravitation.
Abstract: The geometro-stochastic quantization of a gauge theory for extended objects based on the (4, 1)-de Sitter group is used for the description of quantized matter in interaction with gravitation. In this context a Hilbert bundle ℋ over curved space-time B is introduced, possessing the standard fiber ℋ\(_{\bar \eta }^{(\rho )} \), being a resolution kernel Hilbert space (with resolution generator\(\tilde \eta \)and generalized coherent state basis) carrying a spin-zero phase space representation of G=SO(4, 1) belonging to the principal series of unitary irreducible representations determined by the parameter ρ. The bundle ℋ, associated to the de Sitter frame bundle P(B, G), provides a geometric arena with built-in fundamental length parameter R (taken to be of the order of 10−13 cm characterizing hadron physics) yielding, in the presence of gravitation, a quantum kinematical framework for the geometro-stochastic description of spinless matter described in terms of generalized quantum mechanical wave functions, Ψxρ(ξ, ζ), defined on #x210B;. By going over to a nonlinear realization of the de Sitter group with the help of a section ξ(x) on the soldered bundle E, associated to P, with homogeneous fiber V′4⋍G/H, one is able to recover gravitation in a de Sitter gauge invariant manner as a gauge theory related to the Lorentz subgroup H of G. ξ(x) plays the dual role of a symmetry-reducing and an extension field. After introducing covariant bilinear source currents in the fields Ψxρ(ξ, ζ) and their adjoints determined by G-invariant integration over the local fibers in ℋ, a quantum fiber dynamical (QFD) framework is set up for the dynamics at small distances in B determining the geometric quantities beyond the classical metric of Einstein's theory through a set of current-curvature field equations representing the source equations for axial vector torsion and the de Sitter boost contributions to the bundle connection (the latter defining the soldering forms of the Cartan connection in P(B, G) in the nonlinear gauge). The presented bundle framework yields a theory for quantized material objects in interaction with gravitation, the long-range metrical part of which remains classical.

Book
01 Apr 1992
TL;DR: In this article, the authors describe the evolution of dynamical space-time theories, including the manifold model of space time, general relativity, and the cosmological problem geometry.
Abstract: Manifold model of space-time relativistic model of space-time method of fibre bundles the frame bundle and general relativity the cosmological problem geometry of the standard cosmological model bibliographical essay evolution of dynamical space-time theories.

Journal ArticleDOI
TL;DR: A fiber bundle is a fiber bundle that is connected, totally geodesic and geodesically complete submanifolds of a pseudoriemannian manifold if its fibres are connected.
Abstract: A surjective submersion π: M→B from a pseudoriemannian manifold M is a fibre bundle, if its fibres are connected, totally geodesic and geodesically complete submanifolds of M.

Journal ArticleDOI
TL;DR: In this article, the concept of bundle realization of a Lie group on an Abelian principal bundle is defined, based on the theory of locally operating realizations of Lie groups, and the bundle realizations are studied and characterized into pseudoequivalence classes.
Abstract: In this paper the concept of bundle realization of a Lie group on an Abelian principal bundle is defined. This definition is based on the theory of locally operating realizations of Lie groups. Afterward the bundle realizations are studied and characterized into pseudoequivalence classes. This theory is applied to a systematic study and classification of invariant connections under a Lie group. In particular, some examples of gauge invariant potentials under some subgroups of the Poincare group are worked out.

Journal ArticleDOI
TL;DR: It was shown in this paper that any Lagrangian distribution on a 2n-dimensional symplectic manifold is equivalent to a principal O(n) bundle obtained as a reduction of the frame bundle.
Abstract: It is shown that any Lagrangian distribution on a 2n‐dimensional symplectic manifold Γ is equivalent to a principal O(n) bundle obtained as a reduction of the frame bundle. Bisymplectic manifolds with Nijenhuis recursion operators are studied and it is shown that the set of all bi‐Lagrangian distributions is a trivial bundle the structure group being the homogeneous space U(1)n/O(1)n. Various formulas for Maslov indices of closed curves are given including one using only data from the recursion operator.

Journal ArticleDOI
TL;DR: In this paper, the derivative strings of Barndorff-Nielsen and the differential strings of Blaesild & Mora are considered from the coordinate-free viewpoint. And it is shown that the derivative string of given length and degree over a differentiable manifold form a vector bundle associated to a principal bundle of higher-order frames and that there is an analogous result for differential strings.
Abstract: The derivative strings of Barndorff-Nielsen and the differential strings of Blaesild & Mora are considered here from the coordinate-free viewpoint. It is shown that the derivative strings of given length and degree over a differentiable manifold form a vector bundle associated to a principal bundle of higher-order frames and that there is an analogous result for differential strings. Bundles of derivative strings are identified with vector bundles obtained from 0-truncated versions of Ehresmann's semi-holonomic jets by dualization and by taking tensor products. Similarly, bundles of differential strings are identified with vector bundles obtained from semiholonomic jets of certain tensor fields.

Journal ArticleDOI
TL;DR: In this paper, it was shown that h 0(X, L) ≥ 2k for every k-spanned line bundle on a smooth complete complex surface X, L.
Abstract: We show that h0(X, L)≥2k for every k-spanned line bundle on a smooth complete complex surface X.


Journal ArticleDOI
TL;DR: In this paper, it was shown that symmetric wavefunctions defined on a fiber bundle describing a charge-Dirac monopole system cannot be transformed into antisymmetric ones by a gauge transformation, in contradiction to the well known statement first pointed out in connection with the dyon spin problem.
Abstract: The authors point out that the very notion of space reflection is ill-defined when physical states are defined on a fibre bundle describing a charge-Dirac monopole system. In other words, there is no lift of the space reflection operator to the total space of such a non-trivial bundle. They construct a well defined transposition operator of two dyons on the non-trivial two-dyon bundle; consequently, they can correctly define its action on local sections. It is shown that symmetric wavefunctions defined on this bundle cannot be transformed into antisymmetric ones by a gauge transformation, in contradiction to the well known statement first pointed out in connection with the dyon spin problem.


Journal ArticleDOI
TL;DR: In this paper, a natural map between group cohomology of the structure group of a principal fiber bundle with coefficients in the space of functions from the total space into an Abelian group and Cech-cohomology in the base space is defined.
Abstract: A natural map between group‐cohomology of the structure group of a principal fiber bundle with coefficients in the space of functions from the total space into an Abelian group and Cech‐cohomology of the base space is defined. A differential complex of local group‐cochains is constructed and an analog of the Poincare lemma for group‐cohomology is proven. By using the machinery of spectral sequences the cohomology of this complex is calculated and the connection between group‐cohomology and Cech‐cohomology of the given principal fiber bundle is elucidated. Finally, the non‐Abelian and Witten anomaly in this context is reviewed and the relevance of our results for lifting principal group actions is discussed.

Journal ArticleDOI
TL;DR: A generalization to the second order of the notion of the G1-structure, the so called generalized almost tangent structure, is introduced, which is constructed on Vm by an endorphism J acting on T2(Vm), satisfying the relation J2 = 0 and some hypothesis on its rank.
Abstract: We introduce a generalization to the second order of the notion of the G1-structure, the so called generalized almost tangent structure. For this purpose, the concepts of the second order frame bundle H2(Vm), its structural group Lm2 and its associated tangent bundle of second order T2(Vm) of a differentiable manifold Vm are described from the point of view that is used. Then, a G1-structure of second order -called G12-structure- is constructed on Vm by an endorphism J acting on T2(Vm), satisfying the relation J2 = 0 and some hypothesis on its rank. Its connection and characteristic cohomology class are defined.

Journal ArticleDOI
TL;DR: In this paper, a Yano-Ledger connection was constructed for vector bundles, and the connection was shown to be equivalent to the Mendes-Miron connection on vector bundles.
Abstract: K. Yano and A. J. Ledger [13] const ructed f rom a linear connection V on a manifold B a torsion-free linear connection on T B (called the Yano-Ledger connection). M. Ma t sumoto [7] proved tha t : a) V determines a Finsler connection (75, V) in the space V T B of Finsler vectors; b) the symmetr iza t ion of the extension V ~ of (75, V) to T T B is exactly the Yano-Ledger connect ion on TB; c) the Levi-Civi ta connection of the Riemannian metr ic on T B derived f rom a Riemannian metric g on B coincides with the Yano-Ledger connect ion derived f rom the Levi-Civi ta connection V of g iff the Riemannian curvature tensor of g vanishes (see also [2]). On the other hand R. Miron [8] developed a theory of Finsler connections on vector bundles. The purpose of this paper is to construct a Yano-Ledger connect ion on vector bundles, and then to prove the analogues of Matsumoto ' s theorems a), b), and c) for vector bundles and vector bundle Finsler connections. In our considerations and construct ions we apply pullback of pseudoconnections. They are developed and invest igated in §§1, 2, and 3. §4 yields the Yano-Ledger connect ion for vector bundles, and §§4, 5, and 6 present the ment ioned theorems analogous to those of Matsumoto . Concerning no ta t ion and terminology we refer to the monographs [1],

Journal ArticleDOI
TL;DR: In this paper, the axiomatic approach to geometric objects is used to classify (quasi) prolongation functors with compact fibres, and conditions which ensure that π : N → M is continuous.
Abstract: This paper is a contribution to the axiomatic approach to geometric objects. A collection of a manifold M , a topological space N , a group homomorphism E : Diff(M) → Homeo(N) and a function π : N → M is called a quasi-natural bundle if (1) π◦E(f) = f◦π for every f ∈ Diff(M) and (2) if f, g ∈ Diff(M) are two diffeomorphisms such that f |U = g|U for some open subset U of M , then E(f)|π−1(U) = E(g)|π−1(U). We give conditions which ensure that π : N →M is continuous. In particular, if (M,N,E, π) is a quasi-natural bundle with N Hausdorff, then π is continuous. Using this result, we classify (quasi) prolongation functors with compact fibres. 0. Introduction. Throughout this paper manifolds are assumed to be paracompact, finite-dimensional and without boundary. The concept of a natural bundle was introduced by A. Nijenhuis [10] as a modern approach to the classical theory of geometrical objects (see [1]). A natural bundle (over n-dimensional manifolds) is a covariant functor F from the category of n-dimensional C∞ manifolds and C∞ embeddings into the category of C∞ locally trivial fibre bundles and C∞ bundle mappings such that: (1) for every n-dimensional C∞ manifold M , FM is a locally trivial fibre bundle over M ; (2) for every C∞ embedding φ : M → N of n-dimensional manifolds, Fφ : FM → FN covers φ and for any x ∈ M , Fφ maps diffeomorphically the fibre FxM onto the fibre Fφ(x)N ; (3) F is regular in the following sense: If φ : U × M → N is a C∞ mapping (where U is an open subset of R) such that for every t ∈ U , φt : M → N , φt(x) = φ(t, x), is an embedding, then the mapping U × FM 3 (t, y)→ Fφt(y) ∈ FN is of class C∞. 1991 Mathematics Subject Classification: Primary 55R65.




Journal ArticleDOI
TL;DR: In this article, the concepts of connections, soldering form, curvature, bitorsion, generalized Bianchi identities, and associated bundles were developed for biframes on L 2 M.
Abstract: When one considers bivectors on space-time as the basic geometrical objects in place of vectors one is led to the principal bundle of biframes L 2 M in place of the linear frame bundle LM. L 2 M is thus the natural arena for studying the geometry of bivectors on space-time. In this paper we develop the geometry on L 2 M including the concepts of connections, soldering form, curvature, bitorsion, generalized Bianchi identities, and associated bundles

Journal ArticleDOI
TL;DR: In this article, a manifold M with semi-Riemannian almost product structure invariant relative to a transformation group G is considered and a connection with special G-invariance property is constructed in the corresponding bundle of frames.
Abstract: A manifold M with semi-Riemannian almost product structure invariant relative to a transformation group G is considered A connection with special G-invariance property is constructed in the corresponding bundle of frames