Bundles and Connections

Ivan Kolář, Jan Slovák and Peter W. Michor
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Ivan Kolář: Masaryk University, Department of Algebra and Geometry, Faculty of Science
Jan Slovák: Masaryk University, Department of Algebra and Geometry, Faculty of Science
Peter W. Michor: Universität Wien, Institut für Mathematik

Chapter Chapter III in Natural Operations in Differential Geometry, 1993, pp 76-115 from Springer

Abstract: Abstract We begin our treatment of connections in the general setting of fiber bundles (without structure group). A connection on a fiber bundle is just a projection onto the vertical bundle. Curvature and the Bianchi identity is expressed with the help of the Frölicher-Nijenhuis bracket. The parallel transport for such a general connection is not defined along the whole of the curve in the base in general — if this is the case for all curves, the connection is called complete. We show that every fiber bundle admits complete connections. For complete connections we treat holonomy groups and the holonomy Lie algebra, a subalgebra of the Lie algebra of all vector fields on the standard fiber.

Keywords: Vector Bundle; Fiber Bundle; Parallel Transport; Principal Bundle; Holonomy Group (search for similar items in EconPapers)
Date: 1993
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-662-02950-3_3

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DOI: 10.1007/978-3-662-02950-3_3

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