Related papers: Higher Order Connections
The aim of this paper and its sequel is to introduce and classify the holonomy algebras of the projective Tractor connection. After a brief historical background, this paper presents and analyses the projective Cartan and Tractor…
We give a geometric characterisation of plus-one generated projective line arrangements that are next-to-free. We present new succinct proofs, via associated line bundles, for some properties of plus-one generated projective line…
The main purpose of this paper is to study the vector groupoids. This is an algebraic structure which combines the concepts of Brandt groupoid and vector space such that these are compatible.
The term higher gauge theory refers to the generalization of gauge theory to a theory of connections at two levels, essentially given by 1- and 2-forms. So far, there have been two approaches to this subject. The differential picture uses…
The past decade has witnessed two important new developments in the study of linear series on algebraic varieties. First, vector bundles have emerged as powerful tools for analyzing linear series on curves and surfaces. More recently, the…
We describe a notion of (abstract) projective line over a field as a set equipped with a certain first order structure, and a projectivity between projective lines as a bijection preserving this structure. The structure in question is that…
In this paper, we will study on some topologies induced by order convergences in a vector lattice. We will investigate the relationships of them.
In this Note, we propose a line bundle approach to odd-dimensional analogues of generalized complex structures. This new approach has three main advantages: (1) it encompasses all existing ones; (2) it elucidates the geometric meaning of…
In our [Higher-order preconnections in synthetic differential geometry of jet bundles, Beitr\"{a}ge zur Algebra und Geometrie, 45 (2004), 677-696] we have established the affine bundle theorem in the synthetic approach to jet bundles in…
For a principal bundle $P\to M$ equipped with a connection ${\bar A}$, we study an infinite dimensional bundle ${\mathcal P}^{\rm dec}_{\bar A}P$ over the space of paths on $M$, with the points of ${\mathcal P}^{\rm dec}_{\bar A}P$ being…
Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing…
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…
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 develop a combinatorial theory of vector bundles with connection on locally ordered simplicial complexes. This is a first step towards a discrete exterior calculus for bundle-valued forms. The basic building block is the discrete…
The notion of virtual global generation (VGG) for a vector bundle has multiple possible generalization from the case of curves to higher dimensional normal projective varieties. We study relationship between these notions. All these notions…
The geometry of graded principal bundles is discussed in the framework of graded manifold theory of Kostant-Berezin-Leites. In particular, we prove that a graded principal bundle is globally trivial if and only if it admits a global graded…
For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genus g, for line bundles of degree g equivalence is shown to the height on…
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie…
In this paper we discuss various philosophical aspects of the hyperstructure concept extending networks and higher categories. By this discussion we hope to pave the way for applications and further developments of the mathematical theory…
The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H.…