Curvature of a connection

This curvature is also sometimes known as the Riemann curvature tensor. However, the latter term is usually reserved for situations where we have a linear connection, in particular, the Riemann curvature tensor arising from the Levi-Civita connection for a Riemannian or pseudo-Riemannian manifold

Definition

Given data

• A connected differential manifold $M$
• A vector bundle $E$ over $M$
• A connection $\mathrm{\nabla }$ for $E$

Definition part

The curvature of $\mathrm{\nabla }$ is defined as the map:

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

where $X,Y\in \mathrm{\Gamma }$

Note that $R(X,Y)$ itself outputs a linear map $\mathrm{\Gamma }(E)\to \mathrm{\Gamma }(E)$. We can thus write this as:

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

In local coordinates

Further information: curvature matrix of a connection

Consider a system of local coordinate charts for $M$ such that the vector bundle $E$ is trivial on each chart. For any connection $\mathrm{\nabla }$, we can write a matrix that, in local coordinates, describes the curvature of $\mathrm{\nabla }$. This matrix is sometimes denoted as $\mathrm{\Omega }$, and is defined by:

$\mathrm{\Omega }:=d\omega +\omega \wedge \omega$

In the linear case

In the special case where $E=TM$ (the case of a linear connection) we get that $X,Y,Z\in \mathrm{\Gamma }(TM)$. We can thus think of this map as a (1,3)-tensor because it takes as input three vector fields and outputs one vector field.

Properties

Tensoriality

Further information: Curvature is tensorial

The curvature is tensorial in all three arguments. This is best proved by proving $C^{\mathrm{\infty }}$-linearity in all arguments, via a computation.

Antisymmetry

Further information: Curvature is antisymmetric in first two variables

We have the following identity:

$R(X,Y)=-R(Y,X)$