The connection form arises when applying the exterior connection to a particular frame e. Upon applying the exterior connection to the eα, it is the unique k × k matrix (ωαβ) of one-forms on M such that
In terms of the connection form, the exterior connection of any section of E can now be expressed, for suppose that ξ = Σα eαξα. Then
Taking components on both sides,
where it is understood that d and ω refer to the exterior derivative and a matrix of 1-forms, respectively, acting on the components of ξ. Conversely, a matrix of 1-forms ω is a priori sufficient to completely determine the connection locally on the open set over which the basis of sections e is defined.
Read more about Connection Form: Structure Groups, Principal Bundles
Famous quotes containing the words connection and/or form:
“One must always maintain one’s connection to the past and yet ceaselessly pull away from it. To remain in touch with the past requires a love of memory. To remain in touch with the past requires a constant imaginative effort.”
—Gaston Bachelard (1884–1962)
“The man who is admired for the ingenuity of his larceny is almost always rediscovering some earlier form of fraud. The basic forms are all known, have all been practicised. The manners of capitalism improve. The morals may not.”
—John Kenneth Galbraith (b. 1908)