Connection theory on differentiable fibre bundles: A concise introduction

This is a partial review of the connection theory on differentiable fibre bundles. From different view points, this theory can be found in many works, like [2–6, 9, 13–15, 18, 19, 24, 26, 27, 30, 31, 35–40, 42]. The presentation of the material in Sections 2–5, containing the grounds of the connection theory, follows some of the main ideas of [30, Chapters 1 and 2], but their realization here is quite different and follows the modern trends in differential geometry. Since in the physical literature one can find misunderstanding or not quite rigorous applications of known mathematical definitions and results, the text is written in a way suitable for direct application in some regions of theoretical physics. The work is organized as follows. In Section 2 some introductory material is collected, like the notion of Lie derivatives and distributions on manifolds needed for our exposition. Here some of our notations are fixed too. Section 3 is devoted to the general connection theory on bundles whose base and bundles spaces are differentiablemanifolds. In Section 3.1 some coordinates and frames/bases on the bundle space which are compatible with the fibre structure of a bundle are reviewed. Section 3.2 deals with the general connection theory. A connection on a bundle is defined as a distribution on its bundle space which is complimentary to the vertical distribution on it. The notion of parallel transport generated by connection and of specialized frame is introduced. The fibre coefficients and fibre components of the curvature of a connection are defined via part of the components of the anholonomicity object of a specialized frame. Frames adapted to local bundle coordinates are introduced and the local (2-index) coefficients in them of a connection are defined; their transformation law is