Warps and grids for double and triple vector bundles

TL;DR

This paper studies the structure of triple vector bundles, proving a key warp sum property, and applies it to tangent bundles and curvature, providing new insights into their geometric and algebraic properties.

Contribution

It proves that the sum of warps in a triple vector bundle is zero using an intrinsic approach, and applies this to tangent bundles and curvature analysis.

Findings

Sum of warps in a triple vector bundle is zero

Application to the triple tangent bundle yields the Jacobi identity

Curvature emerges from the warp theorem in specific bundles

Abstract

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 $T^3M$ of a manifold and deduce (as earlier) the Jacobi identity. We further apply the result to the triple vector bundle $T^2A$ for a vector bundle $A$ using a connection in $A$ to define a grid in $T^2A$. In this case the curvature emerges from the warp theorem.