Relative connections on principal bundles and relative equivariant structures

Equivariant bundles and adapted connections

Given a complex manifold M equipped with a holomorphic action of a connected complex Lie group G, and a holomorphic principal H–bundle EH over X equipped with a G–connection h, we investigate the connections on the principal H–bundle EH that are (strongly) adapted to h. Examples are provided by holomorphic principal H– bundles equipped with a flat partial connection over a foliated manifold.

Connections on Principal Fibre Bundles

Complete definitions and motivations are given for principal fibre bundles, connections on these, and associated vector bundles. The relationships between various concepts of differentiation in these settings are proved. Finally, we briefly give an outline of the theory of curvature in terms of connections.

Principal Bundles, Groupoids, and Connections

We clarify in which precise sense the theory of principal bundles and the theory of groupoids are equivalent; and how this equivalence of theories, in the differentiable case, reflects itself in the theory of connections. The method used is that of synthetic differential geometry. Introduction. In this note, we make explicit a sense in which the theory of principal fibre bundles is equivalent t...

Complex Structures on Principal Bundles

Holomorphic principal Gbundles over a complex manifold M can be studied using non-abelian cohomology groups H(M,G). On the other hand, if M = Σ is a closed Riemann surface, there is a correspondence between holomorphic principal G-bundles over Σ and coadjoint orbits in the dual of a central extension of the Lie algebra C∞(Σ, g). We review these results and provide the details of an integrabilit...

The Equivariant Brauer Groups of Principal Bundles