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
The curvature of is defined as the map:
Note that itself outputs a linear map . We can thus write this as:
In local coordinates
Further information: curvature matrix of a connection
Consider a system of local coordinate charts for such that the vector bundle is trivial on each chart. For any connection , we can write a matrix that, in local coordinates, describes the curvature of . This matrix is sometimes denoted as , and is defined by:
Here, is a matrix of connection forms.
In the linear case
In the special case where (the case of a linear connection) we get that . 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.
Further information: Curvature is tensorial
The curvature is tensorial in all three arguments. This is best proved by proving -linearity in all arguments, via a computation.
Further information: Curvature is antisymmetric in first two variables
We have the following identity: