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I was given this exercise but I don't even know where to start: to compute the Riemann tensor of the 2-dimensional sphere. The tensor acts on vector fields X,Y,Z like this: $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$ where $\nabla$ is the affine connection defined considering extensions $X_1$,$Y_1$ on $\mathbb{R}^3$ of the fields $X$,$Y$ and the Gauss map $N$ so that $\nabla_XY=\nabla_{X_1}Y_1-<\nabla_{X_1}Y_1,N>N$ (here $\nabla$ is the standard flat connection). Thanks for any help

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    Since you have not given to us your background, nor showed any your effort to solve your problem, it is extremely unclear what kind of help would be the most beneficial to you.2012-05-07
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    Also, because you have noted this is homework, I assumed that a hint would be sufficient to start the discussion. From your comment to my answer I realize that you need to understand quite a few things before you will be able to approach the calculation.2012-05-07
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    To explain why I am asking for your background (say, what are you reading or what book is your course based on), I'd like to add that in fact the result is almost obvious for those who know that the Riemannian curvature of 2-dimensional surfaces has only **one** independent component, and in coordinates is [expressed](http://en.wikipedia.org/wiki/Riemann_curvature_tensor) as $R_{abcd} = K (g_{ac}g_{db} - g_{ad}g_{cb})$, where $K$ is _the Gaussian curvature_.2012-05-07

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