Frame bundle

About: Frame bundle is a research topic. Over the lifetime, 1600 publications have been published within this topic receiving 23049 citations.

(Pseudo) generalized Kaluza–Klein G-spaces and Einstein equations

Abstract: Introducing the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle, we develop the theory of general distinguished linear connections for this space. In particular, using the Lie algebroid generalized tangent bundle of the Kaluza–Klein vector bundle, we present the (g, h)-lift of a curve on the base M and we characterize the horizontal and vertical parallelism of the (g, h)-lift of accelerations with respect to a distinguished linear (ρ, η)-connection. Moreover, we study the torsion, curvature and Ricci tensor field associated to a distinguished linear (ρ, η)-connection and we obtain the identities of Cartan and Bianchi type in the general framework of the Lie algebroid generalized tangent bundle of a Kaluza–Klein bundle. Finally, we introduce the theory of (pseudo) generalized Kaluza–Klein G-spaces and we develop the Einstein equations in this general framework.

Exotic characteristic classes of quaternionic bundles

Abstract: For aC ∞ quaternionic vector bundle, the odd-dimensional real Chern classes vanish, and this allows for a construction of secondary (exotic) characteristic classes associated with a pair of quaternionic structures of a given complex vector bundle. This construction is then applied to obtain exotic characteristic classes associated with an automorphismβ of the holomorphic tangent bundle of a Kahler manifold. These results are the complex analoga of those given for the higher order Maslov classes in [V2].

On projective space bundle with nef normalized tautological line bundle

Abstract: In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef. As an application, we get a characterization of abelian varieties up to finite étale covering. Introduction For a morphism between smooth projective varieties π : Y → X the relative anticanonical divisor −Kπ on Y is defined by the difference of anticanonical divisors −Kπ := −KY − π(−KX). J. Kollár, Y. Miyaoka and S. Mori proved that the relative anticanonical divisor of a non-constant generically smooth morphism cannot be ample in arbitrary characteristic [7], [11]. In the case where π : Y = PX(E) → X is a projectivization of vector bundle on X, we know that the relative anti-canonical divisor is positive proportion of the normalized tautological divisor. Miyaoka studied the case where Y is a curve and showed that the nefness of the normalized tautological divisor is equal to the semistability of vector bundle [10]. Nakayama generalized this to the arbitrary dimension in [13]. In this paper we study the more explicit structure of vector bundles with nef normalized tautological divisor. In Section 1, we review the definition and some known results. In Section 2, we treat semiample cases and show that a pullback of such a bundle by some finite unramified covering is trivial up to twist by some line bundle. In Section 3, we treat the case where X is a blow-up of a smooth variety Z along smooth subvariety or a projective bundle over a smooth variety Z. In these cases we show that the vector bundle with nef normalized tautological divisor on X is isomorphic to the pullback of vector bundle on K having the same property up to twist by the exceptional divisor. In Section 4 we study manifolds whose tangent bundle have a nef normalized tautological divisor. We prove such surfaces are isomorphic to a quotient of abelian surface by some finite étale morphism. Moreover under the assumption that such a divisor is semiample, we can show that finite étale covering of abelian varieties are all varieties satisfying this property. 2000 Mathematics Subject Classification. Primary 14J40; Secondary 14J10, 14J60.

Anatomy of malicious singularities

Abstract: As well known, the b boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation ρ, defined on the Cauchy completed total space E¯ of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class [p0] related to the singularity remains in close contact with all other equivalence classes, i.e., if p0∊cl[p] for every p∊E. We formulate conditions for which such a situation occurs. The differential structure of any space-time with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra on E¯, which turns out to be a von Neumann algebra of random operators,...