Covariant derivatives and connections

This chapter examines the related notions of covariant derivative and connection. It covers the space of covariant derivatives. It also gives a relatively straightforward construction of a covariant derivative on a given vector bundle E → M with fiber 𝕍n = ℝnRn or ℂn. It looks at principal bundles and connections; connections and covariant derivatives; and horizontal lifts. The chapter gives an application to the classification of principal G-bundles up to isomorphism and explains connections, covariant derivatives, and pull-back bundles.