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**We have ##R^{1}_{212}## as the single independent Riemann tensor component, and I'm after ##R##. From symmetry properties and contracting we can attain the other non-zero components.**

The solution then states that ##R_{11}=R^{1}_{111} + R^{2}_{121}=R^{2}_{121}## .

**##R_{11}=R^{1}_{111} + R^{2}_{121}+R^{2}_{112}+R^{2}_{211}##**

I thought it would have been

I thought it would have been

All I can think of then is that the upper indicie can only contract with the middle indice? I've never heard this before though, is that what's going on here?

Similarlly for ##R_{22}=R^{2}_{222} + R^{1}_{212}=R^{2}_{121}## is the solution.

The components ##R^{2}_{211}, R^{2}_{112}, R^{1}_{221}, R^{1}_{122}, ##are the unused components with two 2s, two 1s - if the upper indice only contracts with the middle, these would not come into the formula for any ricci vector...

**If someone could let me know if I'm on the right or wrong track here,**

cheers !

cheers !