The most fundamental example of a double vector bundle, the tangent manifold TE of a vector bundle, has already been met in §3.4. This object, it will be remembered, provides a systematic way of dealing with connections, derivative endomorphisms, and linear vector fields, and does so because of the two vector bundle structures on TE and the interaction between them. In this chapter we study in detail the general concept of double vector bundle, which will be used repeatedly in the rest of the book.
A certain amount of the general theory of double vector bundles consists of a straightforward (but necessary) upgrading of constructions well–known for ordinary vector bundles. What is decisively different however, is the theory of duality. The duality of double vector bundles behaves in entirely new and unexpected ways and is the principal reason for the importance of double vector bundles whenever the relations between Lie algebroids and Poisson structures are considered.
In §9.1 we give the basic definitions, including the fundamental concept of core. The general theory of duality for double vector bundles is in §9.2. In §9.3, §9.4 and §9.5, we treat the two duals of TA, and the canonical isomorphisms which relate them.
Considered as a double vector bundle, the double tangent bundle is unusual in two respects: because the side bundles and the core bundle are equal to each other, it is easy to confuse the different roles which they play; and whereas the general concept of double vector bundle does not include any notion of bracket structure, it is precisely the bracket structures on the double tangent bundle which are the main focus of its interest.