Flari, M.K. and Mackenzie, K. orcid.org/0000-0002-4697-2513 (2019) Warps, grids and curvature in triple vector bundles. Letters in Mathematical Physics, 109 (1). pp. 135-185. ISSN 0377-9017
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 (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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
Keywords: | Triple vector bundles; Double vector bundles; Connections; Curvature |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 07 Aug 2018 11:05 |
Last Modified: | 10 Aug 2020 11:02 |
Status: | Published |
Publisher: | Springer |
Refereed: | Yes |
Identification Number: | 10.1007/s11005-018-1103-y |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:133147 |
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