scispace - formally typeset
Search or ask a question
Topic

Frame bundle

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


Papers
More filters
Journal ArticleDOI
TL;DR: Some new global invariants of a fiber bundle with a connection are cohomology classes in the principal fiber bundle that are defined when certain characteristic curvature forms vanish and give necessary conditions for conformal immersion of a riemannian manifold in euclidean space.
Abstract: We define some new global invariants of a fiber bundle with a connection. They are cohomology classes in the principal fiber bundle that are defined when certain characteristic curvature forms vanish. In the case of the principal tangent bundle of a riemannian manifold, they are invariant under a conformal transformation of the metric. They give necessary conditions for conformal immersion of a riemannian manifold in euclidean space.

139 citations

Journal ArticleDOI
TL;DR: In this article, the authors consider the space of flat connections on the trivial SU(2) bundle over a surface M, modulo the space for gauge transformations and prove that if the surfaceM is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle.
Abstract: Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe on this quotient space a natural hermitian line-bundle with connection and prove that if the surfaceM is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. We show heuristically how path-integral quantisation of the Chern-Simons action yields holomorphic sections of this bundle.

138 citations

Journal ArticleDOI
TL;DR: In this paper, the vortex equations on a line bundle over a compact Kahler manifold were studied and the existence of a Hermitian-Yang-Mills metric on a holomorphic bundle was proved.
Abstract: We study the vortex equations on a line bundle over a compact Kahler manifold. These are a generalization of the classical vortex equations over ℝ2. We first prove an invariant version of the theorem of Donaldson, Uhlenbeck and Yau relating the existence of a Hermitian-Yang-Mills metric on a holomorphic bundle to the stability of such a bundle. We then show that the vortex equations are a dimensional reduction of the Hermitian-Yang-Mills equation. Using this fact and the theorem above we give a new existence proof for the vortex equations and describe the moduli space of solutions.

135 citations

Journal ArticleDOI
TL;DR: In this paper, the index of a transversally elliptic operator on an arbitrary foliation is computed using Hopf algebras associated to the transverse frame bundle and its cyclic cohomology is defined and shown to be canonically isomorphic to the Gelfand-Fuks co-homology.
Abstract: We present the solution of a longstanding internal problem of noncommutative geometry, namely the computation of the index of a transversally elliptic operator on an arbitrary foliation. The new and crucial ingredient is a certain Hopf algebra associated to the transverse frame bundle. Its cyclic cohomology is defined and shown to be canonically isomorphic to the Gelfand-Fuks cohomology.

133 citations

Network Information
Related Topics (5)
Lie group
18.3K papers, 381K citations
89% related
Manifold
18.7K papers, 362.8K citations
88% related
Cohomology
21.5K papers, 389.8K citations
88% related
Moduli space
15.9K papers, 410.7K citations
87% related
Symplectic geometry
18.2K papers, 363K citations
86% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20236
202214
20214
202012
201911
201811