Related papers: On Connections and their Curvatures

Any leafwise connection on a fibre bundle over a foliated manifold is proved to come from a connection on this fibre bundle.

A theory of principal bundles possessing quantum structure groups and classical base manifolds is presented. Structural analysis of such quantum principal bundles is performed. A differential calculus is constructed, combining differential…

The general problem for consistency between arbitrary transports along paths in fibre bundles and bundle morphisms between them is formulated and investigated. The special case of one fibre bundle, its morphism and transport along paths…

This paper is based on my talk at ICM on recent progress in a number of classical problems of linear algebra and representation theory, based on new approach, originated from geometry of stable bundles and geometric invariant theory.

This is a review of the basic concepts of the theory of real and complex smooth vector bundles with finite rank. Besides, the concept of a tensor field is studied within the general framework of a smooth vector bundle rather than a smooth…

Extended spinor connections associated with composite spin-tensorial bundles are considered. Commutation relationships for covariant and multivariate differentiations and corresponding curvature spin-tensors are derived.

We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article…

The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a…

We study the existence of a natural `linearisation' process for generalised connections on an affine bundle. It is shown that this leads to an affine generalised connection over a prolonged bundle, which is the analogue of what is called a…

We define locally trivial quantum vector bundles (QVB) and QVB associated to locally trivial quantum principal fibre bundles. There exists a differential structure on the associated vector bundle coming from the differential structure on…

A direct, bundle-theoretic method for defining and extending local isometries out of curvature data is developed. As a by-product, conceptual direct proofs of a classical result of Singer and a recent result of the authors are derived.

A generalised notion of connection on a fibre bundle E over a manifold M is presented. These connections are characterised by a smooth distribution on E which projects onto a (not necessarily integrable) distribution on M and which, in…

We review the notions of symplectic and orthogonal vector bundles over curves, and the connection between principal parts and extensions of vector bundles. We give a criterion for a certain extension of rank 2n to be symplectic or…

We show how the amplitude of holonomies on a vector bundle can be controlled by the integral of the curvature of the connection on a surface enclosed by the curve.

We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties…

In these lecture notes we will try to give an introduction to the use of the mathematics of fibre bundles in the understanding of some global aspects of gauge theories, such as monopoles and instantons. They are primarily aimed at beginning…

We compare the behaviour of entire curves and integral sets, in particular in relation to locally trivial fiber bundles, algebraic groups and finite ramified covers over semi-abelian varieties.

A review of the parallel transport (translation) in fibre bundles is presented. The connections between transports along paths and parallel transports in fibre bundles are examined. It is proved that the latter ones are special cases of the…

Let $L$ be a (semi)-positive line bundle over a Kahler manifold, $X$, fibered over a complex manifold $Y$. Assuming the fibers are compact and non-singular we prove that the hermitian vector bundle $E$ over $Y$ whose fibers over points $y$…

This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. In the…