Showing papers on "Frame bundle published in 1981"

The geometrical structure of a complexified theory of gravitation

Abstract: Outlines the procedure for the complexification of the tangent bundle over a four-dimensional space-time manifold. By introducing a connection and metric compatible with the complex structure, the authors form the geometrical basis for a new (complexified) theory of gravitation whose fundamental gauge group is U(3,1). They further prove that the Lagrangian for the theory is necessarily real when the connection is compatible with the metric.

Holomorphic Structures Modeled After Compact Hermitian Symmetric Spaces

Abstract: Let M be a compact m-dimensional complex manifold and F(M) the holomorphic frame bundle over M. Then π:F(M)→M is a holomorphic principal GL(m;c)-bundle over M. Let G be a complex Lie subgroup of GL(m;c). A holomorphic principal G-subbundle tt:P + M of Fin) is called a holomorphic G-structure on M.

A continuous version of the Borsuk-Ulam theorem

Abstract: ABSTRAcr. Letp: E -B be an n-sphere bundle, q: V-B be an RI-bundle and f: E -V be a fibre preserving map over a paracompact space B. Letp: E-B be the projectivized bundle obtained from p by the antipodal identification and let Af be the subset of E consisting of pairs (e, -e) such thatfe = f(-e). If the cohomology dimension d of B is finite then the map (PIA)*: Hd(B; Z2) -Hd(Af; Z2) is injective for a continuous cohomology theory H*. Moreover, if thejth Stiefel-Whitney class of q is zero for 1 d r. If all the Stiefel-Whitney classes of q are zero then (PlAf)* is injective in every degree.

A nonmanifold theory of space–time

Abstract: A Zeeman topology is defined in the general framework of any set W of events which has been equipped with an acyclic signal relation ∼ →. The Wssumption that the ∼ → structure of W is locally that of Minkowski space and that the ’’piecing together’’ maps are smooth in an appropriate sense, allows a tangent bundle p:E→W to be defined. This bundle has, as structure group, the group G of linear causal automorphisms of Minkowski space.

The spectrum of vector bundle flows with invariant subbundles

Abstract: A vector bundle flow (', ') on the vector bundle E over a compact metric space M induces a one-parameter group {$%} of bounded operators acting on the continuous sections of E, with infinitesimal generator L. An example is given by the tangent flow (T', '), if ' is a flow on a smooth manifold. In this article, the spectrum of the generator L is used to study the exponential growth rates of bundle trajectories in the neighborhood of a fixed invariant subbundle, e.g. the tangent bundle of a submanifold of M. Auxiliary normal and tangential spectra are introduced, and their relationship and fine structure are explored. Introduction. The aim of this paper is to link linear dynamics in vector bundles to the spectra of certain Banach space operators associated with a linear vector bundle flow. Of particular interest is the behavior of such systems at a fixed invariant subbundle, e.g. the tangent bundle of an invariant submanifold, discussed below. What sets this article apart from related work of, say, Sacker and Sell [8], [9] is a faithfulness to the operator-theoretic point of view, which allows one to sieve out the "easy" functional analysis from the harder dynamics. The payoff comes if difficult results become natural and easy-to-state facts about operators, or, conversely, statements about the spectrum lead to new dynamical considerations. Specific examples of such dividends are given in [1], [5] as well as in this paper. In §1 we introduce the pertinent notation and terminology (see also [1]); while in §2, we isolate results from abstract operator theory which are then applied, in §3, to vector bundle flows with a fixed invariant subbundle. As in [9], we identify the tangential spectrum along the subbundle, a(LF), and the normal spectrum a(L); but as operator spectra, yielding the inclusion a(L) c o(LF) u o(L) (Theorem 3.2), which implies Theorem 1 of [9]. Next, sufficient conditions are given for the crucial equality o(L) = o(LF) u o(L), which means, roughly, that the growth rates of the bundle trajectories can be determined from the normal and tangential dynamics. We describe a flow example showing that the equality can fail. Theorem 3.5 generalizes the result in [9] asserting that equality always holds if the base flow is chain recurrent. Finally, we discuss the existence of complementary invariant subbundles in terms of operator theory. 1. The spectrum of a vector bundle flow. Let M denote a compact metric space which admits a one-parameter group of homeomorphisms {'},eR. Suppose A is a Received by the editors June 30, 1980. 1980 Mathematics Subject Classification. Primary 58F15, Secondary 58F19.

Tetrad fields and metric tensor in the gauge theory of gravitation

Abstract: The most relevant geometrical aspects of the gauge theory of gravitation are considered. A global definition of the tetrad fields is given and emphasis is placed on their role in defining an isomorphism between the tangent bundle of space-time and an appropriate vector bundle B associated to the gauge bundle. It is finally shown how to construct the fundamental geometrical objects on space-time, starting from B.

The domain of definition of bundle representations

Abstract: It is proved that the equivalence class of bundle representations of a group G on the product bundle B0 with total space B0=X×Y includes all representations of G on bundles B which are homeomorphic (but not necessarily naturally homeomorphic) to the product X×Y, provided the G has the same action on the fibres of B0 and B. The group A of the bundle B is immaterial.

Algebraically special Yang–Mills solutions without sources

Abstract: We consider source‐free Yang–Mills solutions for which the curvature is decomposable in the sense that the curvature 2‐form is the product of a single Lie‐algebra‐valued function and a real 2‐form. If the curvature is everywhere nonnull (or null with twisting rays), then the solution is a connection in a principal fiber bundle, which is reducible to a source‐free Maxwell principal bundle. All such solutions are therefore readily obtained, locally or globally, from Maxwell solutions. Our analysis uses the Ambrose–Singer theorem to show that the holonomy group is one‐dimensional. A principal bundle‐with‐connection is reducible to the holonomy subbundle of any point, and, in this case, since the holonomy group is one‐dimensional, the reduced bundle has the structure of a Maxwell bundle. On the other hand, if the curvature is null and twist‐free on a full neighborhood of some point, then the bundle need not be reducible. The holonomy group is generally the entire gauge Lie group. The solutions can still be co...

Normal bundle to subvarieties in quadratics II

Abstract: In this paper we prove that the restriction of the tangent bundle of a nonsingular quadrix Q to a subvariety X is ample if and only if X does not contain a straight line. This implies that the normal bundle of a locally complete intersection, reduced and irreducible curve C is ample if and only if C is not a straight line. The result gives information also for higher dimensional subvarieties of Q.