Connection on a vector bundle

Summary

A connection on a vector bundle is a rule for differentiating sections of the bundle in the direction of tangent vectors. It is needed because the fibres over different points are different vector spaces. So there is no built-in way to compare a section at nearby points. A connection gives covariant derivatives, parallel transport, curvature and holonomy. Connections on the tangent bundle are the affine connections, such as the Levi-Civita connection. Connections on other bundles are the gauge fields of physics.