Affine connection

Summary

An affine connection on a smooth manifold is a connection on its tangent bundle. It lets one differentiate vector fields along vector fields, and so defines parallel transport of tangent vectors, geodesics, torsion and curvature. On a Riemannian manifold there is a preferred affine connection, the Levi-Civita connection. The name comes from the idea that a connection identifies nearby tangent spaces as pieces of one affine space.