Approximating the parallel transport of an induced connection

Abstract

Efficient numerical methods to approximate the parallel transport operators of the induced connection on a sub-bundle of a vector bundle are presented. These methods are simpler than naive applications of a Runge–Kutta algorithm and have accuracy up to order 4. They have the desirable property of being insensitive to choices of trivialization of the sub-bundle. The methods were developed to solve a problem of computing skyrmions using the Atiyah–Manton–Sutcliffe and Atiyah–Drinfeld–Hitchin–Manin constructions, but are applicable to a broader range of problems in computational geometry.