(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have the metric of a three sphere:

[itex]g_{\mu \nu} =

\begin{pmatrix}

1 & 0 & 0 \\

0 & r^2 & 0 \\

0 & 0 & r^2\sin^2\theta

\end{pmatrix}[/itex]

Find Riemann tensor, Ricci tensor and Ricci scalar for the given metric.

2. Relevant equations

I have all the formulas I need, and I calculated the necessary Christoffel symbols, by hand and by mathematica and they match. There are 9 non vanishing Christoffel symbols. Some I calculated and for others I used the symmetry properties and the fact that the metric is diagonal (which simplifies things).

But when I go and try to calculate Riemman tensor via:

[itex]R^{a}_{bcd}=\partial_d \Gamma^a_{bc}-\partial_c\Gamma^a_{bd}+\Gamma^m_{bc}\Gamma^a_{dm}-\Gamma^m_{bd}\Gamma^a_{cm}[/itex]

I get all zeroes for components :\

And I kinda doubt that every single component is zero.

The Christoffel symbols are:

[itex]

\begin{array}{ccc}

\Gamma _{\theta r}^{\theta } & = & \frac{1}{r} \\

\Gamma _{\phi r}^{\phi } & = & \frac{1}{r} \\

\Gamma _{r\theta }^{\theta } & = & \frac{1}{r} \\

\Gamma _{\theta \theta }^r & = & -r \\

\Gamma _{\phi \theta }^{\phi } & = & \cot (\theta ) \\

\Gamma _{r\phi }^{\phi } & = & \frac{1}{r} \\

\Gamma _{\theta \phi }^{\phi } & = & \cot (\theta ) \\

\Gamma _{\phi \phi }^r & = & -r \sin ^2(\theta ) \\

\Gamma _{\phi \phi }^{\theta } & = & -\cos (\theta ) \sin (\theta )

\end{array}

[/itex]

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# Riemann tensor, Ricci tensor of a 3 sphere

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