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Showing papers on "Frame bundle published in 1976"





Journal ArticleDOI
TL;DR: In this article, the authors studied the Riemannian metrics that arise when the manifold is R or S 1 with a constant connection and showed that the frame bundle admits a Riemanian metric which Schmidt introduced to construct the b-boundary.
Abstract: On a manifold with connection the frame bundle admits a Riemannian metric which Schmidt introduced to construct the b-boundary of the underlying manifold. Here we study the metrics that arise when the manifold is R or S1 with a constant connection.

6 citations


Book ChapterDOI
W. Drechsler1
01 Jan 1976
TL;DR: In this article, a formalism describing extended hadrons is presented using generalized wave functions defined on a fiber bundle constructed over space-time, where the structural group of the bundle is taken to be the (4+1) de Sitter group acting as a group of motion in a locally defined space of constant curvature [the fiber] possessing a radius of curvature of the order of one Fermi.
Abstract: A formalism describing extended hadrons is presented using generalized wave functions defined on a fiber bundle constructed over space-time The structural group of the bundle is taken to be the (4+1) de Sitter group acting as a group of motion in a locally defined space of constant curvature [the fiber] possessing a radius of curvature of the order of one Fermi A gauge theory of strong interaction is formulated in terms of the geometry in such a de Sitter fiber bundle This geometric description does not require the existence of any constituents for hadrons and leads to three basic nonlinear wave equations of integro-differential type for the hadronic matter wave function

3 citations


Book ChapterDOI
01 Jan 1976
TL;DR: In this article, the authors discuss the M-tensor and three types of connections on TM, highlight their properties, and explore the relationship between them, and present the proofs of two decomposition theorems, one of which is a sharpened version of a theorem of Grifone.
Abstract: Publisher Summary After the concept of tangent bundle was discovered, it occurred to few differential geometers, including Professor E. T. Davies, that the proper setting for the Finsler metrics and general paths on a smooth manifold M is the tangent bundle TM and not M itself. This chapter discusses the concept of M-tensor and three types of connections on TM, highlights their properties, and explores the relationship between them. Some of the results help to define the relationship between several related known concepts in the differential geometry of TM, such as the system of general paths of Douglas, the nonlinear connections of Barthel and Yano and Ishihara, and the nonhomogeneous connection of Grifone, while others are generalizations of known results. The chapter describes the structure of the tangent bundle TM and the slit tangent bundle STM. It explores the M-tensors and three types of connections on TM and STM. The chapter highlights a (1, l)-connection on TM as horizontal distribution on TM and discusses the relationship between a vector field on TM and the horizontal distribution associated with a (1, l)-connection. It presents the proofs of two decomposition theorems, one of which is a sharpened version of a theorem of Grifone.

2 citations


Journal ArticleDOI
TL;DR: In this article, a vector bundle map of a smooth Γq∞-stucture on a compact manifold Xn is presented, whose vanishing is shown to be a necessary condition for deforming F to a codimension-q foliation on Xn.
Abstract: LetF be a (smooth) Γq∞-stucture (often called a codimension-q Haefliger structure) on a compact manifoldXn. Cohomological invariants associated to the singularities ofF are defined whose vanishing is shown to be a necessary condition for deformingF to a codimension-q foliation onXn. An analagous approach to vector bundle maps is then utilized to prove a general theorem concerning the possibility of embedding a vector bundle in the tangent bundle ofXn, and applications to the planefield problem are given. In the final section geometric realizations of the singularity classes associated toF are constructed.

Journal ArticleDOI
01 Feb 1976
TL;DR: In this paper, it was shown that a minimal vector field flow admits a minimal extension beyond a toral extension on compact manifolds and compact Lie groups, where the minimal sets form a partition into imbedded submanifolds.
Abstract: Group extensions of vector field flows for which the orbit closure map forms a fiber bundle are constructed for the case of minimal flows on compact manifolds and compact Lie groups. Conditions for which minimal nontoral extensions exist are studied. The existence question for minimal sets and minimal extensions of minimal sets has been studied extensively in topological dynamics. In [5], Ellis considers the group extension question for minimal homeomorphisms on compact metric spaces. For real flows on compact manifolds it is well known that the torus admits a minimal flow but the Klein bottle does not. In fact, Markley [7] shows that recurrent orbits on the Klein bottle are circles or points. It is also known that a minimal vector field flow admits a minimal extension through any torus [1, p. 58]. The purpose of this paper is to prove that a minimal vector field flow admits a minimal extension beyond a toral extension. It will follow that any Lie group admits a vector field flow which is minimal on the product of two distinct maximal tori and that any minimal vector field flow admits a minimal compact group extension in which the group is a semidirect product with a torus. It is also shown that a minimal vector field flow can always be extended so that the minimal sets form a partition into imbedded submanifolds and the projection onto the orbit closure spaces forms a fiber bundle. 0. Preliminaries. Definitions and proofs omitted here can be found in [1], [3] and [4]. All manifolds, maps, and transformation groups will be assumed C X unless otherwise specified. If (M, G) is right transformation group, the set {n * gln E N C M, g E G} will be denoted by N G and the space {n. GIn C N C M} with the quotient topology will be denoted by N/ G. If G is the real line and X is the vector field generating (M, G), we denote (M, G) by (M, X), N G by N X and N/ G by N/X. The symbols G N and G \ N will be used in case G acts on the left. If (G, P, M, v) is a principal G-bundle with base M, bundle space P and projection v; (H, P, H \ P, rl) will denote the induced principal H-bundle in which 1: P -H \ P takesp to H p. (H \ G, H \ P, M, v2) will denote the induced fiber bundle with standard fiber H \ G and projection r2: H \ P -M which takes H p to r (p). Note that 7 = o2 ? 7 1 For any mapf, Received by the editors September 28, 1975. AMS (MOS) subject classifications (1970). Primary 54H20, 34H35; Secondary 58F99.

Journal ArticleDOI
01 Jan 1976
TL;DR: In this article, the authors considered the question of the existence of fibrewise fixed-point free maps f: P (E) -> P(E), i.e. continuous maps such that
Abstract: Let E be a vector bundle and P(E) its associated projective bundle. Some necessary conditions on the characteristic classes of E for existence of a fibrewise fixed-point free map P (E) -- P (E) are obtained. 1. Introduction. Let E -> B be an F-vector bundle, where F denotes either the reals R, the complex numbers C, or the quaternions H, and let p: P (E) -> B denote the associated projective bundle. We consider the question of existence of fibrewise maps f: P (E) -> P (E), i.e. continuous maps such that