Riemann curvature

The Riemann curvature gives a way to measure how non-commutative a affine connection is.

Definition

Let be a smooth manifold with an affine connection . Given two vector fields, , define the map

As a matter of notation, this is often written

Note we could also think of this map as

It’s a tensor

This map defines a -tensor.

Proof

todo

In coordinates

First Bianchi identity

Curvature tensor using Levi-Civita

Symmetry properties

In coordinates (lowering the index)