In chapter 7 we introduced connections in Riemannian manifolds which enable us to compare vectors in different tangent spaces. In the present chapter connections on fibre bundles are defined in an abstract though geometrical way.
We first define a connection on a principal bundle. Our abstract definition is realized concretely by introducing the connection one-form whose local form is well known to physicists as a gauge potential. The Yang-Mills field strength is defined as the curvature associated with the connection. A connection on a principal bundle naturally defines a covariant derivative in the associated vector bundle. We reproduce the results obtained in chapter 7, applying our approach to tangent bundles. We conclude this chapter with a few applications of connections to physics: to gauge field theories and Berry’s phase. We follow the line of Choquet-Bruhat et al (1982), Kobayashi (1984) and Nomizu (1981). Details will be found in the classic books by Kobayashi and Nomizu (1963, 1969). See also Daniel and Viallet (1980) for a quick review.
Problems
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正文：英文 · OCR 机器稿 · 待校对
核对状态：OCR 机器稿 · 待校对
来源版本：second-edition-local
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