Frame bundle

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

On the irreducibility of the punctual Quotient Scheme of a Surface

Abstract: We prove that the Quot-scheme of finite quotients of a vector bundle which are of a given length and supported in one point, is irreducible and of the expected dimension

Aharonov-Bohm Effect, Flat Connections, and Green's Theorem

Abstract: The validity of Green's theorem, and hence of Stokes' theorem, when the involved vector field is differentiable but not continuously differentiable, is crucial for a theoretical explanation of the Aharonov–Bohm (A-B) effect; we review this theorem. We describe the principal bundle in which the A-B effect occurs, and give the geometrical description of the relevant connection. We study the set of gauge equivalence classes of flat connections on a product bundle with abelian structural group, and show that this set has a canonical group structure, which is isomorphic to a quotient of cohomology groups. We apply this result to the A-B bundle and calculate the holonomy groups of all flat connections.

Reduction in Path Integrals on a Riemannian Manifold with Group Action

Abstract: A new method for the factorization of the path-integral measure in path integrals for a particle motion on a compact Riemannian manifold with a free isometric unimodular group action is proposed. It is shown that path-integral measure is not invariant under the factorization. An integral relation between the path integral given on the total space of the principal fiber bundle and the path integral on the base space of this bundle (the orbit space of the group action) is obtained.

A survey on rankin-cohen deformations

Abstract: This is a survey about recent progress in Rankin-Cohen deformations. We explain a connection between Rankin-Cohen brackets and higher order Hankel forms. The famous Erlanger Programm of Klein says that geometry is about to study the transfor- mation groups of various spaces, or more precisely the properties invariant under the actions of such groups, i.e., the symmetries. Noncommutative geometry(NCG), which originated from Connes' study in operator alge- bras in 1970's, brought the landscape of geometry many new objects and some astonishing phenomena. Back to the early 1990s, Connes and Moscovici pointed out that in noncommutative geome- try(NCG) while noncommutative spaces are represented by the algebras (usually noncommuta- tive C ∗ -algebras) of "continuous functions" over noncommutative spaces, the local symmetries are reflected in some Hopf algebras. One of the first noncommutative spaces studied in NCG is the C ∗ -algebra of a foliated space. In the case of codimension n foliations, Connes and Moscovici discovered a Hopf algebra Hn, which governs the local symmetry of leaf spaces of foliations of codimension n. The Hopf algebra Hn is universal in the sense that it depends only on the codimension of a foliation. This family of Hopf algebras {Hn} is very useful in the study of transverse index theory, and later was found to have connections with various different areas of mathematics, c.f. (10), (13). In this paper, we review the application of the Hopf algebra H1 in Rankin-Cohen deformations, which was initiated by Connes and Moscovici (14). We start by recalling the general setting of transverse geometry. Let M be a smooth manifold and F be a foliation on M of codimension n. Let X be a complete flat transversal of F, and F + X be the oriented frame bundle of X. The holonomy pseudogroup acts on X and therefore F + X by transforming X parallelly along paths in leaves of F. The "transverse geometry" is to study the transversal X along with the action by the holonomy pseudogroup . In what follows we focus on the case when n = 1, and define Connes-Moscovici's Hopf algebra H1. Now the complete transversal X is a flat 1-dim manifold; and the oriented frame bundle F + X is diffeomorphic to X × R + , and is a discrete holonomy pseudogroup acting on X as local diffeomorphisms. We introduce coordinates x on X and y on R + . The lifted action of on F + X is

On the normal bundle to abelian surfaces embedded in ℙ4(ℂ)

Abstract: In this note one computes the cohomology of the normal bundle NX (and its twists) to any abelian surface X in ℙ4(ℂ) and one shows that NX is simple. As a by-product we reobtain the results of Decker about the smoothness of the irreducible component of the moduli scheme M(−1,4) of rank 2 stable vector bundles on ℙ4 with c1=−1,c2=4, along the orbit of the Horrocks-Mumford bundle by the action of SL5 (ℂ) (cf. [2]).