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I am trying to calculate the Riemann tensor for a 4D sphere. In D'inverno's book, I have this equation [itex]R^{a}_{bcd}=\partial_{c}\Gamma^{a}_{bd}-\partial_{d}\Gamma^{a}_{bc}+\Gamma^{e}_{bd}\Gamma^{a}_{ec}-\Gamma^{e}_{bc}\Gamma^{a}_{ed}[/itex]

But the exercise asks me to calculate [itex]R_{abcd}[/itex]. Do I lower it with a metric? if so, a metric with what indices?

An lastly, in the above equation, there is an extra index e. Is it another dummy variable that I have to sum over with?

too much work

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# I Riemann Tensor

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