Published online by Cambridge University Press: 05 August 2012
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.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.