Warps, grids and curvature in triple vector 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 3 M of a manifold and deduce (as earlier) the Jacobi identity.We further apply the result to the triple vector bundle T 2 A for a vector bundle A using a connection in A to define a grid in T 2 A. In this case the curvature emerges from the warp theorem. * Mathematics Subject Classification (MSC2000): 53C05 (primary), 18D05, 18D35, 55R65 (secondary). Now given a triple vector bundle E, it is possible to build a triple vector bundle E from E 1 , E 2 , E 3 , E 12 , E 23 , E 13 and E 123 by taking pullbacks of Whitney sums in a similar way. The decomposition condition referred to in Definition 5 is that there exists an isomorphism of triple vector bundles from E to E which is the identity on the seven bundles listed above. See [5] for full details.This condition ensures that in a triple vector bundle E (nontrival) grids, as defined below always exist. Notation for zero sectionsThe zero section of E 1 is denoted by 0 E 1 : M → E 1 , m → 0 E 1 m , with similar notations for E 2 and E 3 .(T (µ)(X(m))) = T 2 (µ) (J M (T (Z)(X(m)))) -J A (T (Z H )(T (µ)(X(m)))). (61) Since J A interchanges the structures p T A and T (p A ), we can rewrite the last expression in (61) as J A T 2 (µ)(T (Z)(X(m))) -

Quotients of double vector bundles and multigraded bundles

“…, which is the warp pairing of Flari-Mackenzie [5]. Note that this pairing determines the bracket, for degree reasons.…”

“…Sections of C define vector fields on D of homogeneity −1, −1, these are the core vector fields. The action C × M D → D of the corresponding subgroup bundle C ∼ = exp(g −1,−1 ) ⊆ G on D is known as the core action [5]. It is a vector bundle action, given in terms of a linearization by…”

“…Mackenzie also studied Lie algebroid and Poisson structures on double vector bundles, and unearthed a wealth of interesting relationships among these structures. In later work, he and his co-authors explored the more difficult theory of triple vector bundles [5,9,16], and even n-fold vector bundles [8].…”

“…Here C → M is the core of the double vector bundle, given as the set of elements mapping to M under both side projections, while A, B → M are the fat bundles of Gracia-Saz and Mehta [10], given as certain extensions of the side bundles A, B → M by B * ⊗ C and A * ⊗ C, respectively. The fiberwise bracket on (3) is defined by the bilinear pairing, due to Flari and Mackenzie [5] (see also [20])…”

Preface to special issue in honor of Kirill C. H. Mackenzie

“…These objects play a central role in recent developments in the theory of representations of Lie algebroids and Lie groupoids [17,19,51] with connections to higher Lie theory and stacks. Mackenzie went further and began the investigation of triple structures [16,18,46]. The deep mathematical perspective underlying this body of work is increasingly appreciated and used as time passes.…”