Let's say I want to calculate the Ricci tensor, [tex]R_{bd}[/tex], in terms of the contractions of the Riemann tensor, [tex]{R^a}_{bcd}[/tex]. There are two definitions of the Riemann tensor I have, one where the [tex]a[/tex] is lowered and one where it is not, as above.(adsbygoogle = window.adsbygoogle || []).push({});

To change between the two all that I have ever seen written is 'we lower the indices' but I don't think I fully understand this. Does this mean...

[tex]R_{abcd} = g_{aa} {R^a}_{bcd}[/tex]

So the answer to my original question of finding the Ricci tensor is...

[tex]R_{bd} = g^{ac} g_{aa} {R^a}_{bcd}[/tex]

Following this, I also have written before me that...

[tex]{R^b}_{bcd} = 0[/tex] since [tex]R_{abcd}[/tex] is symmetric on a and b. Shouldn't this be antisymmetric on a and b?

Sorry if these are basic questions but I'm finding the vagueness of 'lowering the indices' a bit confusing...

Cheers.

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# Quick question to clear up some confusion on Riemann tensor and contraction

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