affine connection

Affine connections are ways of taking “derivatives” of vector fields with other ones. It is a different notion than the Lie derivative, though.

Definition

Let be a smooth manifold with vector fields . An affine connection is a map

that satisfies the following properties:

1. Tensorial in .

2. - linear in

3. Derivation in

Nonexample

The Lie bracket is NOT tensorial in X or Y.

Relationship with covariant derivative

Many authors consider covariant derivative to be just an alternate name for an affine connection.

In @docarmo1992Riemannian there is a distinction made.

Let be a manifold with an affine connection . Let be a curve on the manifold and consider vector fields along that curve and denote these by . There is a unique mapping called the covariant derivative

that has the following properties:

1. Linear in :

2. Liebniz rule: for

3. If is induced by a vector field (that is, ) then .

This means a couple of things. First, an affine connection at a point really only depends on values of the vector field on a curve running through that point. Next, there is an intrinsic idea of differentiation for a vector field that has familiar looking properties that is equivalent to an affine connection. Thus, we should think of an affine connection as the “derivative” of a vector field along curves.

The proof is in @docarmo1992Riemannian page 51.

Local Property

Affine connections are local. That means that if for vector fields and either or vanish identically on an open set then vanishes on .

Proof

todo

Corollary

If coincide with on some open set then .

In coordinates

An affine connection can be written in coordinates using the Christoffel symbols. Consider vector fields

on we have