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Let's have the metric for a sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's components, but I got problems with it.

The Ricci curvature must be $$ R_{ij}=\frac{2}{R^{2}}g_{ij}. $$ But when I use definition of Ricci tensor, I can't turn the expression into the expression for the metric tensor

Maybe, there are siome hints, which can help?

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What problems are you having exactly? It will make a much better question if you show what you've tried and ask about the specific concept that is giving you trouble (and I'll be happy to reopen this as soon as you edit accordingly). –  David Z Apr 29 '13 at 21:32
    
Ok. The Ricci curvature must be $R_{ij} = \frac{2}{R^{2}}g_{ij}$. But when I use definition of Ricci tensor, I can't turn the expression into the expression for the metric tensor. –  PhysiXxx Apr 29 '13 at 22:07
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Maybe this would be better on the Math SE. –  John Rennie Apr 30 '13 at 6:31
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David Zaslavsky, why didn't you reopen my question? –  PhysiXxx Apr 30 '13 at 16:21
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@PhysiXxx I don't think I noticed that this was edited, but anyway: while the edit is, strictly speaking, an improvement, the question still isn't really asking anything specific. I'm not clear on what you mean by "turn the expression into the expression for the metric tensor," or more importantly, what conceptual problem you have when you try to do it. If you'd still like the question reopened, could you elaborate on that i.e. show a bit more of your work including the specific step that you're stuck on? –  David Z Sep 20 '13 at 16:06
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closed as too localized by David Z Apr 29 '13 at 21:32

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