Properties of the Riemann curvature tensor
since i.e the first derivative of the metric vanishes in a local inertial frame. Hence
Using the fact that partial derivatives always commute so that , we get
in a local inertial frame. Lowering the index with the metric we get
So in a local inertial frame the result is
Thus is antisymmetric on the final pair and second pair of indices, and symmetric on exchange of the two pairs.
Since these last two equations are valid tensor equations, although they were derived in a local inertial frame, they are valid in all coordinate systems.
A flat manifold is one which has a global definition of parallelism: i.e. a vector can be moved around parallel to itself on an arbitrary curve and will return to its starting point unchanged. This clearly means that
i.e. the manifold is flat [ Assignment 6.5: try a cylinder! ].
As usual we can simplify things by working in a local inertial frame. So in this frame we get
The third term of this is zero in a local inertial frame, so we obtain
Consider the same formula with the and interchanged:
If we subtract these we get the commutator of the covariant derivative operators and :
The terms involving the second derivatives of drop out because [ partial derivatives commute ].
Since in a local inertial frame the Riemann tensor takes the form
This is closely related to our original derivation of the Riemann tensor from parallel transport around loops, because the parallel transport problem can be thought of as computing, first the change of in one direction, and then in another, followed by subtracting changes in the reverse order.