Related papers: The Berwald-type linearisation of generalised conn…
We develop an alternative view on the concept of connections over a vector bundle map, which consists of a horizontal lift procedure to a prolonged bundle. We further focus on prolongations to an affine bundle and introduce the concept of…
A linear connection is associated to a nonlinear connection on a vector bundle by a linearization procedure. Our definition is intrinsic in terms of vector fields on the bundle. For a connection on an affine bundle our procedure can be…
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…
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 address the recently introduced notions of generalized principal bundle and generalized principal connection by keeping track of global geometric properties through local coordinate transformation laws. This approach leads us to…
The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The…
We present the notion of a filtered bundle as a generalisation of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more…
A generalized notion of a Lie algebroid is presented. Using this, the Lie algebroid generalized tangent bundle is obtained. A new point of view over (linear) connections theory on a fiber bundle is presented. These connections are…
In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties. We study what are called the Verlinde bundles in order to obtain information…
We study the linearization of line bundles and the local structure of actions of connected linear algebraic groups, in the setting of seminormal varieties. We show that several classical results about normal varieties extend to that…
This paper studies linear generalised complex structures over vector bundles, as a generalised geometry version of holomorphic vector bundles. In an adapted linear splitting, a linear generalised complex structure on a vector bundle $E\to…
A theory of double affine and special double affine bundles, i.e. differential manifolds with two compatible (special) affine bundle structures, is developed as an affine counterpart of the theory of double vector bundles. The motivation…
We review the concept of a graded bundle as a natural generalisation of a vector bundle. Such geometries are particularly nice examples of more general graded manifolds. With hindsight there are many examples of graded bundles that appear…
The construction of a linear connection on a pullback bundle from a connection on a vector bundle is explained in terms of fiberwise linear approximation. This procedure clarifies the geometric meaning of the linearized connection as well…
Projective spaces for finite-dimensional vector spaces over general fields are considered. The geometry of these spaces and the theory of line bundles over these spaces is presented. Particularly, the space of global regular sections of…
Graded bundles are a particularly nice class of graded manifolds and represent a natural generalisation of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids we define the notion of a weighted…
The paper contains a review on the general connection theory on differentiable fibre bundles. Particular attention is paid to (linear) connections on vector bundles. The (local) representations of connections in frames adapted to holonomic…
Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed,…
Generalized are the investigated in other works of the author transports along paths in fibre bundles to transports along arbitrary maps in them. Their structure and some properties are studied. Special attention is paid to the linear case…
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…