A Discrete Theory of Connections on Principal Bundles

TL;DR

This paper develops a discrete geometric framework for connections on principal bundles, enabling intrinsic coordinate systems and symmetry reduction in discrete mechanics through novel discrete analogues of classical geometric structures.

Contribution

It introduces a discrete theory of connections on principal bundles, including a discrete Atiyah sequence, and extends discrete exterior calculus to define discrete Levi-Civita connections and curvature.

Findings

Provides a discrete intrinsic method for symmetry reduction in mechanics

Introduces a discrete analogue of the Atiyah sequence and connections

Extends discrete exterior calculus to define discrete Levi-Civita connection and curvature

Abstract

Connections on principal bundles play a fundamental role in expressing the equations of motion for mechanical systems with symmetry in an intrinsic fashion. A discrete theory of connections on principal bundles is constructed by introducing the discrete analogue of the Atiyah sequence, with a connection corresponding to the choice of a splitting of the short exact sequence. Equivalent representations of a discrete connection are considered, and an extension of the pair groupoid composition, that takes into account the principal bundle structure, is introduced. Computational issues, such as the order of approximation, are also addressed. Discrete connections provide an intrinsic method for introducing coordinates on the reduced space for discrete mechanics, and provide the necessary discrete geometry to introduce more general discrete symmetry reduction. In addition, discrete analogues of the Levi-Civita connection, and its curvature, are introduced by using the machinery of discrete exterior calculus, and discrete connections.