Related papers: Superconnections and Parallel Transport
A vector bundle with connection over a supermanifold leads naturally to a notion of parallel transport along superpaths. In this note we show that {\it every} such parallel transport along superpaths comes form a vector bundle with…
In this note we make use of some properties of vector fields on a manifold to give an alternate proof to [3] for the equivalence between connections and parallel transport on vector bundles over manifolds. Out of the proof will emerge a new…
The present work provides a mathematically rigorous account on super fiber bundle theory, connection forms and their parallel transport, that ties together various approaches. We begin with a detailed introduction to super fiber bundles. We…
The axiomatic approach to parallel transport theory is partially discussed. Bijective correspondences between the sets of connections, (axiomatically defined) parallel transports, and transports along paths satisfying some additional…
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…
Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we…
The linear transports along paths in vector bundles introduced in Ref. [1] are applied to the special case of tensor bundles over a given differentiable manifold. Links with the transports along paths generated by derivations of tensor…
In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby…
The (parallel) linear transports along paths in vector bundles are axiomatically described. Their general form and certain properties are found. It is shown that these transports are locally (i.e. along every fixed path) always Euclidean…
The theory of linear transports along paths in vector bundles, generalizing the parallel transports generated by linear connections, is developed. The normal frames for them are defined as ones in which their matrices are the identity…
In this addendum to our article "Superconnections and Parallel Transport" we give an alternate construction to the parallel transport of a superconnection contained in Corollary 4.4 of \cite{D1}, which has the advantage that is independent…
The (parallel linear) transports in tensor spaces generated by derivations of the tensor algebra along paths are axiomatically described. Certain their properties are investigated. Transports along paths defined by derivations of the tensor…
We give in this paper which is the fifth in a series of eight a theory of covariant derivatives of multivector and extensor fields based on the geometric calculus of an arbitrary smooth manifold M, and the notion of a connection extensor…
In this paper, we introduce the notion of a super tangent bundle of a manifold, and extend the basic notions of differential geometry such as differential forms, exterior derivation, connection, metric and divergence on manifolds that…
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,…
Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties 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…
Tangent categories provide an axiomatic approach to key structural aspects of differential geometry that exist not only in the classical category of smooth manifolds but also in algebraic geometry, homological algebra, computer science, and…
The motivation for this paper stems \cite{CR} from the need to construct explicit isomorphisms of (possibly nontrivial) principal $G$-bundles on the space of loops or, more generally, of paths in some manifold $M$, over which I consider a…
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…