Calculating Riemann Tensor for S^2 with Pull-Back Metric from Euclidean Space

[Mathman_]

Find the Riemann tensor of the 2-sphere of radius r

S$$^{2}_{r}$$={(x,y,z) $$\in$$$$\Re^{3}$$|x$$^{2}$$ + y$$^{2}$$ + z$$^{2}$$ = r$$^{2}$$}

with metric g obtained as the pull-back of the Euclidean metric g_R3 by the inclusion
map S$$^{2}$$ $$\hookrightarrow$$$$\Re^{3}$$.

Any help would be appreciated. Thanks

 

[AEM]

This seems to be a pretty straight forward problem. Rewrite your metric in spherical coordinates. Identify your $g_{\mu \nu}$. Go to a book on General Relativity and find the definition of the Riemann tensor in terms of the Christoffel symbols and calculate it out. It will take a little time, but it's not hard.