Warps and duality for double vector bundles

A linear section of a double vector bundle is a parallel pair of sections which form a vector bundle morphism; examples include the complete lifts of vector fields to tangent bundles and the horizontal lifts arising from a connection in a vector bundle. A grid in a double vector bundle consists of two linear sections, one in each structure, and thus provides two paths from the base manifold to the top space; the warp of the grid measures the lack of commutativity of the two paths. For a double vector bundle $D$, a linear section induces a linear map on the relevant dual of $D$ and thence a linear section of the iterated dual; we call this the corresponding squarecap section. This notion makes precise the relationships between various standard $1$-forms and vector fields that are not dual in the usual sense. A grid in $D$ induces squarecap sections in the two iterated duals. These iterated duals are themselves in duality and we obtain an expression for the warp of a grid in terms of a pairing. This result extends the relationships between various standard formulas, and clarifies the iterated dualization involved in the concept of double Lie algebroid. In the first part of the paper we review the nonstandard pairing of the two duals of a double vector bundle, showing in particular that it arises as a direct generalization of the Legendre-type antisymplectomorphism $T^*(A^*)\to T^*(A)$ for any vector bundle $A$ of Mackenzie and Xu.