# 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.

Fuzzy

• Yes you should. $R{^\theta}_{\phi\theta\phi}=\sin^2\theta$ is correct. If the other one is correct it should only be a matter of using the symmetries of the Riemann tensor and raising/lowering indices using the (inverse) metric tensor. Commented Mar 11, 2022 at 21:43
• Hi, Kurt, thanks for the hint and that site. I have a hunch how it could go - I will try that one (after I got an eye of sleep...) Fuzzy Commented Mar 11, 2022 at 23:25
• @Fuzzy Did you manage to fulfill the gap? If so, you should post an answer in order to mark this question as solved for future users Commented Mar 15, 2022 at 10:45

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!