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Tensors and Relativity: Chapter 6

# The curvature tensor

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 and 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 This gives 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 Defining we can write  are the components of a 1/3 tensor. This tensor is called the Riemann curvature tensor .