Riemann tensor on a sphere

Question

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

Answer 1

Hint. Try to understand why the Weingarten map $L: T_{p}{\mathbb{S}^2} \rightarrow T_{p}{\mathbb{S}^2}$ on the sphere is given by $$L=-\frac{1}{r}\operatorname{id}$$ (I assume that $r$ is the radius of your sphere), and then use the Gauss equation $$ R(X,Y,Z,W) = <II(Y,Z),II(X,W)> - <II(X,Z),II(Y,W)> $$ where $II(X,Y) = <L(X),Y>$ is the second fundamental form.

Answer 2

If you do not want to use Vytakin's nice suggestion, you can directly compute the Christoffel symbols for the sphere $\Gamma^\mu_{\alpha \beta}=\frac{g^{\mu\gamma}}{2}(g_{\gamma \alpha, \beta} + g_{\gamma \beta, \alpha} - g_{\alpha \beta, \gamma})$ . Few of them survive, and then you can plug them into d$x^\rho(R(\partial_{\mu},\partial_{\nu})\partial_{\sigma})=:R^\rho{}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho{}_{\nu\sigma} - \partial_\nu\Gamma^\rho_{\mu\sigma} + \Gamma^\rho{}_{\mu\lambda}\Gamma^\lambda{}_{\nu\sigma} - \Gamma^\rho{}_{\nu\lambda}\Gamma^\lambda{}_{\mu\sigma}$.