I’ve always been confused with the combinatorial aspect of proving the properties of the Riemann curvature tensor. I want to record my proof of the fact that . This is different from the standard proof given in books. I have been unable to prove this theorem in the past, and hence am happy to write down my proof finally.
Define the function . We want to prove that this function is .
By simple usage of the facts that and that switching the first two or last two vector fields gives us a negative sign, we can see that
Now note that . This is obtained by switching the first two and last two indices. However,
As both positive and negative , we can conclude that it is .
It is not easy to prove this theorem because just manipulating the indices mindlessly (or even with some gameplan) can lead you down a rabbithole without ever reaching a conclusion. Meta-observations, like the above, are necessary to prove this assertion.