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  • Print publication year: 1987
  • Online publication date: August 2012

14 - Fibre bundles

Summary

As we have shown in Chapter 13, the manifold TM of tangent vectors to a given manifold M has a special structure which may be conveniently described in terms of the projection map which takes each tangent vector to the point of the original manifold at which it is tangent. The set of points of TM which are mapped by the projection to a particular point of the original manifold M is just the tangent space to M at that point: all tangent spaces are copies of the same standard space (Rm) but not canonically so, though a common identification may be made throughout a suitable open subset of the original manifold, for example a coordinate neighbourhood. These are the basic features of what is known as a fibre bundle: roughly speaking a fibre bundle consists of two manifolds, a “larger” and a “smaller”, the larger (the bundle space) being a union of “fibres”, one for each point of the smaller manifold (the base space); the fibres are all alike, but not necessarily all the same. A product of two manifolds (base and fibre) is a particular case of a fibre bundle, but in general a fibre bundle will be a product only locally, as is the case for the tangent bundle of a differentiable manifold. The projection map, from bundle space to base space, maps each fibre to the associated point of the base.