Parallel transport

Summary

Parallel transport moves vectors along a curve in a manifold while keeping them "constant" in the sense of a given connection. It gives, for each curve, a linear isomorphism between the fibres over its endpoints. The result usually depends on the curve, not only on its endpoints, and this dependence is measured by curvature and holonomy. A connection can be recovered from its parallel transports, so the two describe the same data.