Components of Riemann Tensor on sphere Since the surface of a sphere is a 2D manifold, the Riemann Tensor should have only one component.  But if I calculate
according to
$$R^i_{~rkj}=
\frac{\partial\Gamma^i_{~jr}}{\partial u^k} 
-\frac{\partial\Gamma^i_{~kr}}{\partial u^j}
+\Gamma^i_{~ks}\Gamma^s_{~jr}
-\Gamma^i_{~js}\Gamma^s_{~kr}$$
I get $$R^\theta_{~\phi\theta\phi}=\sin^2\theta$$
and $$R^\phi_{~\theta\phi\theta}=1$$
which are two seemingly independent components (plus of course their negative counterparts because of symmetry reasons).
So, I assume, they are not independent and I should be able to transform one into the other, but I have no idea how.
Can somebody please give me a hint (for dummies, please!)?
Thanks in advance
Fuzzy
 A: Yep, I did the calculations:
Lowering the first index with the metric of the sphere
$(
g_{\vartheta\vartheta}=R^2, 
g_{\vartheta\varphi}= g_{\varphi\vartheta}= 0, 
g_{\varphi\varphi}= R^2\sin^2\vartheta
)$ (sorry, "array" does not work here)
gets the correct result:
$$
\begin{align*}
R_{~\varphi\vartheta\varphi\vartheta}&=g_{\alpha\varphi}R^\alpha_{~\vartheta\varphi\vartheta}
= g_{\vartheta\varphi}R^\vartheta_{~\vartheta\varphi\vartheta}+g_{\varphi\varphi}R^\varphi_{\vartheta\varphi\vartheta} 
=R^2\sin^2\vartheta \\
R_{~\vartheta\varphi\vartheta\varphi}&=g_{\vartheta\alpha}R^\alpha_{~\varphi\vartheta\varphi} 
=g_{\vartheta\vartheta}R^\vartheta_{~\varphi\vartheta\varphi}+g_{\vartheta\varphi}R^\varphi_{~\varphi\vartheta\varphi} 
=R^2\sin^2\vartheta
\end{align*}
$$
Symmetry takes care of the rest:
$$
\begin{align*}
R^\vartheta_{~\varphi\varphi\vartheta}&=-R^\vartheta_{~\varphi\vartheta\varphi}=- \sin^2\vartheta 
\rightarrow R_{~\vartheta\varphi\varphi\vartheta}=-R^2\sin^2\vartheta \\
R^\varphi_{~\vartheta\vartheta\varphi}&=-R^\varphi_{~\vartheta\varphi\vartheta}=-1
\rightarrow R_{~\varphi\vartheta\vartheta\varphi}=-R^2\sin^2\vartheta
\end{align*}
$$
Thanks to all who helped!
