Riemann curvature

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

Definition

Let $M$ be a smooth manifold with an affine connection $\nabla$. Given two vector fields, $X,Y\in \mathfrak{X}(M)$, define the map

$$\begin{array}{rl}R(X,Y):\mathfrak{X}(M)& ⟶\mathfrak{X}(M)\\ Z& ⟼{\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z.\end{array}$$

As a matter of notation, this is often written

$$R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z.$$

Note we could also think of this map as

$$\begin{array}{rl}R:\mathfrak{X}(M)\times \mathfrak{X}(M)\times \mathfrak{X}(M)& ⟶\mathfrak{X}(M)\\ (X,Y,Z)& ⟼R(X,Y)Z\end{array}$$

It’s a tensor

This map defines a $(3,1)$-tensor.

Proof

todo

In coordinates

First Bianchi identity

Curvature tensor using Levi-Civita

Symmetry properties

In coordinates (lowering the index)