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Showing papers on "Frame bundle published in 1985"


Journal ArticleDOI
TL;DR: In this article, the right-hand side of the integral of t/(M) for hyperbolic 3-manifolds is calculated explicitly, in a special case, for Riemannian manifolds of dimension 3.
Abstract: In this paper, we use the formula in the above theorem to study t/(M) for hyperbolic 3-manifolds and calculate the right-hand side of the formula explicitly, in a special case. Our main task is to represent the integral of/]1 by some more tractible ones. In Sect. 1, we deal with general compact oriented Riemannian manifolds M of dimension 3. Let F(M) be the SO(3) oriented frame bundle of M. Let L be a

109 citations


Journal ArticleDOI
TL;DR: In this paper, a U1-valued latice gauge field u defined on a periodic, 2-dimensional lattice satisfies the generic continuity condition uuuu ≠ − 1, it can be used to construct a principal U 1-bundle over the torus and in that bundle a connection such that parallel transport along bonds is given by u. The characteristic classes and numbers of this bundle can then be calculated from u in a straightforward way.

40 citations



Journal ArticleDOI
TL;DR: In this paper, the Ferrand construction for smooth surfaces in ℙ4 was proposed, where if the normal bundle of a surface X has a suitable 1-subbundle, then a 2-vector bundle can be constructed, which has a section vanishing doubly on X.
Abstract: There is an analogue of the Ferrand construction for smooth surfaces in ℙ4: if the normal bundle of such a surface X has a suitable 1-subbundle, then a 2-vector bundle can be constructed, which has a section vanishing doubly on X. In this way the Horrocks-Mumford bundle is recovered from the quintic scroll.

29 citations



Journal ArticleDOI
TL;DR: In this article, a mathematical analysis of the space of globally hyperbolic solutions of the Einstein-Dirac field equations is presented, which is useful for controlling their gauge freedom and for studying the symmetries of solutions.

15 citations


Journal ArticleDOI
TL;DR: In this article, it is shown that whenever a (1, 1)-type tensor field, defined through two alternative Lagrangians, implies complete integrability for a second-order Lagrangian vector field Γ on the tangent bundle of a given configuration manifold, then a bundle atlas can be found in which both Γ and a class of equivalent Lagrangia are completely separated into a sum of second-Order vector fields and Lagrangias, each one corresponding to a single degree of freedom.
Abstract: It is shown that whenever a (1, 1)-type tensor field, defined through two alternative Lagrangians, implies complete integrability for a second-order Lagrangian vector field Γ on the tangent bundle of a given configuration manifold, then a bundle atlas can be found in which both Γ and a class of equivalent Lagrangians are completely separated into a sum of second-order vector fields and Lagrangians, each one corresponding to a single degree of freedom

13 citations


Journal ArticleDOI
01 Jul 1985
TL;DR: In this paper, the universal connection on the bundle of principal connections is used to stabilize the base manifold under perturbations of the chosen section, which reinforces the belief that general relativistic singularities cannot be removed by quantization.
Abstract: We make use of the universal connection on the bundle of principal connections; the bundle structure is governed by the action of the group on the first jet bundle. Each section determines a connection in the principal bundle, which in the case of the frame bundle allows a metric completion projecting onto the corresponding b- completion. It is shown that b -incompleteness of the base manifold is stable under perturbations of the chosen section. Hence, for instance, for (pseudo) Riemannian manifolds including spacetimes, b -incompleteness is a stable condition under conformal deformations of the metric. This reinforces the belief that general relativistic singularities cannot be removed by quantization.

11 citations


Journal ArticleDOI
Naoto Abe1
TL;DR: In this article, general connections on vector bundles over a manifold were defined and some algebraic properties of the space of covariant derivatives of general connections were studied, and two types of induced general connections which are induced by a pair of vector bundle homomorphisms and by a bundle map were defined.
Abstract: In this paper, general connections in the sense of T. Otsuki are dealt with. The general connections were defined by T. Otsuki in [01] as a generalized notion of usual connections. Recently they are called Otsuki connections in Europe. He defined the general connections on the tangent tensor bundles of a manifold and defined associating geometrical objects analogous to those of usual ones, for example, their torsion forms and curvature forms. In his papers [Ol]-[O11], several results about general connections were obtained. The purpose of this paper is to define general connections on vector bundles over a manifold and to study the fundamental properties. In § 1, we will prepare notations used in this paper and fundamental facts on the 1-jet bundle of a vector bundle. In §2, the general connections will be defined and some algebraic properties of the space of covariant derivatives of general connections will be studied. In § 3, we will define two types of induced general connections which are induced by a pair of vector bundle homomorphisms and by a bundle map. In § 4, using given general connections, we will construct general connections on the dual bundle and on the tensor product bundles. In § 5, we will define the curvature form of a general connection. The author would like to express his hearty thanks to Professor T. Otsuki for his helpful advice. He also would like to acknowledge the constant encouragement of Professor S. Yamaguchi.

10 citations





Journal ArticleDOI
TL;DR: In this paper, a gauge theory of supergravity which allows one to obtain first-order Lagrangians that are locally gauge invariant by construction is presented, which makes use of supertwistors as a representation space for the construction of a typical fiber of a vector bundle associated with a principal bundle.
Abstract: A gauge theory of supergravity which allows one to obtain first‐order Lagrangians that are locally gauge invariant by construction is presented. The formalism makes use of supertwistors as a representation space for the construction of a typical fiber of a vector bundle associated with a principal bundle, where the structural group is the super‐Poincare group. The approach proposed provides a means of resolving one of the central problems of gauge field theories of external symmetry groups, that is the satisfactory treatment of translations.



Journal ArticleDOI
TL;DR: In this paper, the inherited fibration structure of the fiber torus has been studied in terms of the curvature form of a connection when C-spaces are considered as the toral bundle space over an algebraic variety.
Abstract: Analytic subvarieties inC-spaces are discussed. First, a certain kind of closed 2-formdω is constructed. Then, the subvarieties are studied by means of this 2-form.dω may be considered as the curvature form of a connection whenC-spaces are considered as the toral bundle space over an algebraic variety. This 2-form is horizontal in its nature with respect to the bundle structure and indicates, in general, how different the bundle is from the trivial bundle. Because of this twist in the bundle space, subvarieties in theC-spaces tend to inherit the same structure. In this paper, the inherited fibration structure is studied. The most concrete results are obtained when the fiber torus has complex dimension 2.

Journal ArticleDOI
Lining Zhang1
TL;DR: In this paper, the sum bundle of tangent bundle and vector bundle, T(M)⊕E(M, F), is taken as the mathematical framework of grand unified theory.

Journal ArticleDOI
TL;DR: The problem of solving the combined gravitational and Yang-Mills field systems is regarded as a purely geometrical problem of determining a linear connection on the principal frame bundle L(M) from a connection on a SU(2) principal bundle over a space-time M.
Abstract: The problem of solving the combined gravitational and Yang–Mills field systems is regarded as a purely geometrical problem of determining a linear connection on the principal frame bundle L(M) from a connection on a SU(2) principal bundle over a space‐time M. It is suggested that mapping theorems of connections on bundles may provide a means of actually solving ‘‘field equations.’’