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Examples ofprincipal bundle
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It also provides a basic example of aprincipalbundle, by identifying the fiber with the circle group.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In particular, the product of transition functions on a three-way intersection is not always equal to one, as is required for aprincipalbundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In particular, one studies the wavefunctions in a tangent space known as aprincipalbundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.But for the purposes of an overview, it is convenient to stick with theprincipalbundleviewpoint.
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This example is from Wikipedia and may be reused under a CC BY-SA license.From any vector bundle, one can construct the frame bundle of bases which is aprincipalbundle(see below).
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This example is from Wikipedia and may be reused under a CC BY-SA license.Mathematically, a "gauge" is just a choice of a (local) section of someprincipalbundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Aprincipalbundleis trivial if and only if it admits a global cross section.
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This example is from Wikipedia and may be reused under a CC BY-SA license.It should be emphasized that, in the physical literature on gauge theory, a structure group of aprincipalbundleoften is called the gauge group.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In particular, on aprincipalbundle, a principal connection is a natural reinterpretation of the connection form as a tensorial object.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Thisprincipalbundleis called the holonomy bundle (through "p") of the connection.
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This example is from Wikipedia and may be reused under a CC BY-SA license.In many cases, however, the prolongation is aprincipalbundlein its own right, and its structure group can be identified with a subgroup of a higher-order jet group.
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This example is from Wikipedia and may be reused under a CC BY-SA license.This then specializes to the case when the specified fibre is a principal homogeneous space for the left action of the group on itself, yielding the associatedprincipalbundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.On the other hand, the connection form has the advantage that it is a differential form defined on the differentiable manifold, rather than on an abstractprincipalbundleover it.
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This example is from Wikipedia and may be reused under a CC BY-SA license.It turns out that these properties completely characterize smooth principal bundles.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The same fact applies to local trivializations of principal bundles.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Connections (gauge connection) define thisprincipalbundle, yielding a covariant derivative in each associated vector bundle.
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This example is from Wikipedia and may be reused under a CC BY-SA license.However, it is not, and the notion can be described precisely using the language of principal bundles.
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This example is from Wikipedia and may be reused under a CC BY-SA license.The classical examples include vector bundles, principal bundles and sheaves over topological spaces.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Essentially they are a smooth version of abelian gerbes belonging more to the hierarchy starting with principal bundles than sheaves.
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This example is from Wikipedia and may be reused under a CC BY-SA license.Note that definitions of principal superbundles and principal superconnections straightforwardly follow that of smooth principal bundles and principal connections.
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This example is from Wikipedia and may be reused under a CC BY-SA license.These examples are from corpora and from sources on the web. Any opinions in the examples do not represent the opinion of the Cambridge Dictionary editors or of Cambridge University Press or its licensors.
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