Triple vector bundles in Differential Geometry

The triple tangent bundle T3M of a manifold M is a prime example of a triple vector bundle. The definition of a general triple vector bundle is a cube of vector bundles that commute in the strict categorical sense. We investigate the intrinsic features of such cubical structures, introducing systematic notation, and further studying linear double sections; a generalization of sections of vector bundles.

A set of three linear double sections on a triple vector bundle E yields a total of six different routes from the base manifold M of E to the total space E. The underlying commutativity of the vector bundle structures of E leads to the concepts of warp and ultrawarp, concepts that measure the noncommutativity of the six routes. The main theorem shows that despite this noncommutativity, there is a strong relation between the ultrawarps. The methods developed to prove the theorem rely heavily on the analysis of the core double vector bundles and of the ultracore vector bundle of E.

This theorem provides a conceptual proof of the Jacobi identity, and a new interpretation of the curvature of a connection on a vector bundle A.