The Riemann curvature tensor is one of the most important tensors in general relativity. If it is zero then it means that the space is flat. If it is non-zero then we have a curved space.
Many authors instead write or .
We shall use either notation. In general it turns out that even though , however in general it is true that . Let us examine this in more detail. Firstly consider the second derivative of a scalar . A scalar does not change under parallel transport therefore .