Showing papers on "Frame bundle published in 1977"

Moduli of vector bundles on curves with parabolic structures

Abstract: Let H be the upper half plane and F a discrete subgroup of AutH. When H mo d F is compact, one knows that the moduli space of unitary representations of F has an algebraic interpretation (cf. [7] and [10]); for example, if moreover F acts freely on H, the set of isomorphism classes of unitary representations of F can be identified with the set of equivalence classes of semi-stable vector bundles of degree zero on the smooth projective curve H modF, under a certain equivalence relation. The initial motivation for this work was to extend these considerations to the case when H m od F has finite measure. Suppose then that H modF has finite measure. Let X be the smooth projective curve containing H modF as an open subset and S the finite subset of X corresponding to parabolic and elliptic fixed points under F. Then to interpret algebraically the moduli of unitary representation of F, we find that the problem to be considered is the moduli of vector bundles V on X, endowed with additional structures, namely flags at the fibres of V at PeS. We call these quasi parabolic structures of V at S and, if in addition we attach some weights to these flags, we call the resulting structures parabolic structures on V at S (cf. Definition 1.5). The importance of attaching weights is that this allows us to define the notion of a parabolic degree (generalizing the usual notion of the degree of a vector bundle) and consequently the concept of parabolic semi-stable and stable vector bundles (generalizing Mumford's definition of semi-stable and stable vector bundles). With these definitions one gets a complete generalization of the results of [7, 10, 12] and in particular an algebraic interpretation of unitary representations of F via parabolic semi-stable vector bundles on X with parabolic structures at S (cf, Theorem 4.1). The basic outline of proof in this paper is exactly the same as in [12], however, we believe, that this work is not a routine generalization. There are some new aspects and the following are perhaps worth mentioning. One is of course the idea of parabolic structures; this was inspired by the work of Weil (cf. [16], p. 56). The second is a technical one but took some time to arrive at, namely when one

Structure of symmetric tensors of type (0,2) and tensors of type (1,1) on the tangent bundle

Abstract: The concepts of M-tensor and M-connection on the tangent bundle TM of a smooth manifold M are used in a study of symmetric tensors of type (0, 2) and tensors of type (1, 1) on TM. The constructions make use of certain local frames adapted to an M-connection. They involve extending known results on TM using tensors on M to cases in which these tensors are replaced by M-tensors. Particular attention is devoted to (pseudo-) Riemannian metrics on TM, notably those for which the vertical distribution on TM is null or nonnull, and to the construction of almost product and almost complex structures on TM.

Group contraction in a fiber bundle with Cartan connection

Abstract: A contraction of the structural group with respect to the stability subgroup is performed in a fiber bundle with Cartan connection. The relation of the connections in the original and in the contracted bundle is examined. As an example interesting for physics the contraction of the SO(4,1) de Sitter bundle over space–time to the affine tangent bundle over space–time is discussed with the latter bundle possessing the Poincare group as structural group.

Currents in a theory of strong interaction based on a fiber bundle geometry

Abstract: A fiber bundle constructed over spacetime is used as the basic underlying framework for a differential geometric description of extended hadrons. The bundle has a Cartan connection and possesses the de Sitter groupSO(4, 1) as structural group, operating as a group of motion in a locally defined space of constant curvature (the fiber) characterized by a radius of curvatureR≈10−13 cm related to the strong interactions. A hadronic matter field ω(x, ζ) is defined on the bundle space, withx the spacetime coordinate and ζ varying in the local fiber. The components of a generalized tensor current ℑ ab (M) (x) are introduced, involving a bilinear expression in the fields ω(x, ζ) and ωΔ(x, ζ) integrated over the local fiber at the pointx. This hadronic matter current is considered as a source current for the underlying fiber geometry by coupling it in a gauge-invariant manner to the curvature expressions derived from the bundle connection coefficients, which are associated here with strong interaction effects, i.e., play the role of meson fields induced in the geometry. Studying discrete symmetry transformations between the 16 differently soldered Cartan bundles, a generalized matter-antimatter conjugation Ĉ is established which leaves the basic current-curvature equations Ĉ-invariant. The discrete symmetry transformation Ĉ turns out to be the direct product of an ordinary charge conjugation for the Dirac point-spinor part of ω(x, ζ) and an internal $$\hat P\hat T$$ transformation applied globally on the bundle to the fiber (i.e., de Sitter) part of ω(x, ζ).

On the Yang-Mills structure of gravitation: A new issue of the final state

Abstract: The Yang-Mills approach to gravity is presented. This extension of general relativity is based on the structures of a gauge theory: the Lorentz frame bundle acts as gauge bundle, the connection on the Lorentz bundle is the basic dynamical field, and the Yang-Mills equations define the dynamics for the connection. As a consequence of these dynamics the equivalence principle will be broken on space-time regions of high curvature, while it remains strictly preserved for the solar system, for example, and for homogeneous and isotropic world models. They are explicitly used to illustrate the extension of the general relativistic dynamics.