Warps, grids and curvature in triple vector bundles

A triple vector bundle is a cube of vector bundle structures which commute in the (strict) categorical sense. A grid in a triple vector bundle is a collection of sections of each bundle structure with certain linearity properties. A grid provides two routes around each face of the triple vector bundle, and six routes from the base manifold to the total manifold; the warps measure the lack of commutativity of these routes. In this paper we first prove that the sum of the warps in a triple vector bundle is zero. The proof we give is intrinsic and, we believe, clearer than the proof using decompositions given earlier by one of us. We apply this result to the triple tangent bundle (Formula presented.) of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle (Formula presented.) for a vector bundle A using a connection in A to define a grid in (Formula presented.). In this case the curvature emerges from the warp theorem.