Imagine in our manifold a very small closed loop whose four sides are the coordinate lines , , , .
Figure 6.3: Parallel transport around a closed loop ABCD.
A vector defined at A is parallel transported to B. From the parallel transport law
it follows that at B the vector has components
where the notation `` '' under the integral denotes the path AB. Similar transport from B to C to D gives
The integral in the last equation has a different sign because of the direction of transport from C to D is in the negative direction.
Similarly, the completion of the loop gives
The net change in is a vector , found by adding (93)-(96).
To lowest order we get
This involves derivatives of 's and of . The derivatives of can be eliminated using for example
To obtain this, one needs to relabel dummy indices in the terms quadratic in .
Notice that this just turns out to be a number times summed on . Now the indices 1 and 2 appear because the path was chosen to go along those coordinates. It is antisymmetric in 1 and 2 because the change would have the opposite sign if one went around the loop in the opposite direction.
If we use general coordinate lines and , we find
we can write
are the components of a 1/3 tensor. This tensor is called the Riemann curvature tensor.