Double vector bundles and duality

TL;DR

This paper explores the structure of double vector bundles, focusing on duality concepts, canonical isomorphisms, and key examples like iterated tangent and cotangent bundles, advancing the theoretical framework in differential geometry.

Contribution

It introduces the notions of dual double vector bundles and their morphisms, providing new theorems and examples that deepen understanding of duality in this context.

Findings

Canonical isomorphisms between dual double vector bundles established

New notions of dual double vector bundle and morphism introduced

Several illustrative examples provided

Abstract

The most important examples of a double vector bundle are provided by iterated tangent and cotangent functors: TTM, TT^*M, T^*TM, and T^*T^*M. We introduce the notions of the dual double vector bundle and the dual double vector bundle morphism. Theorems on canonical isomorphisms are formulated and proved. Several examples are given.