\begin{definition}[Definition:Riemann-Christoffel Tensor]
A '''Riemann-Christoffel tensor''' is a tensor field which expresses the curvature of a Riemannian manifold.
The '''Riemann-Christoffel tensor''' is given in terms of the Levi-Civita connection $\nabla$ by:
:$R \left({u, v}\right) w = \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{\left[{u, v}\right]} w$
where $\left[{u, v}\right]$ is the Lie bracket of vector fields.
It measures the extent to which the metric tensor is not locally isometric to that of Euclidean space.
\end{definition}