In this paper we introduce a generalisation of the notion of holonomy for connections over a bundle map on a principal fibre bundle. We prove that, as in the standard theory on principal connections, the holonomy groups are Lie subgroups of the structure group of the principle fibre bundle and we also derive a straightforward generalisation of the Reduction Theorem.

We consider a vector bundle E → M and the principal bundle PE of frames of E. Let K be a principal connection on PE and let Λ be a linear connection on M . We classify all principal connections on W 2PE = P 2M ×M J2PE naturally given by K and Λ.

Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding Riemannian (or unitary) vector bundles with parallel curvature then reduces ...

In this paper, by using of Frobenius theorem a relation between Lie subalgebras of the Lie algebra of a top space T and Lie subgroups of T(as a Lie group) is determined. As a result we can consider these spaces by their Lie algebras. We show that a top space with the finite number of identity elements is a C^{∞} principal fiber bundle, by this method we can characterize top spaces.

Let G be a simple linear algebraic group defined over the field of complex numbers. Fix a proper parabolic subgroup P of G, and also fix a nontrivial antidominant character χ of P . We prove that a holomorphic principal G–bundle EG over a connected complex projective manifold M is semistable satisfying the condition that the second Chern class c2(ad(EG)) ∈ H (M, Q) vanishes if and only if the l...

In [Bi2] we defined connections on a parabolic principal bundle. While connections on usual principal bundles are defined as splittings of the Atiyah exact sequence, it was noted in [Bi2] that the Atiyah exact sequence does not generalize to the parabolic principal bundles. Here we show that a twisted version of the Atiyah exact sequence generalize to the context of parabolic principal bundles....