In
geometry, the notion of a
connection makes precise the idea of transporting data along a curve or family of curves in a
parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an
affine connection, the most elementary type of connection, gives a means for transporting
tangent vectors to a
manifold from one point to another along a curve. An affine connection is typically given in the form of a
covariant derivative, which gives a means for taking
directional derivatives of vector fields: the infinitesimal transport of a
vector field in a given direction.