On Double Vector Bundles

TL;DR

This paper establishes an equivalence between double vector bundles and short exact sequences of vector bundles, enabling the transfer of operations and duality theories, with applications to jet bundles and Atiyah algebroids.

Contribution

It introduces a categorical framework linking double vector bundles to short exact sequences, providing new tools for their analysis and duality theory.

Findings

Equivalence between double vector bundles and short exact sequences

Transfer of operations and duality theory via this equivalence

Applications to jet bundles and Atiyah algebroids

Abstract

In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theory of double vector bundles in term of the corresponding short exact sequences. Examples including the jet bundle and the Atiyah algebroid are discussed.