After learned some Riemann Geometry, I want to compute the curvature of $S^2$ by the way of Riemann curvature.
So, I assume the $S^2$ is $$ (\theta_1,\theta_2)\rightarrow (\cos\theta_1\sin\theta_2,\sin\theta_1\sin\theta_2,\cos\theta_2) $$ Then, $$ g_{11}=\sin^2\theta_2 \\ g_{22}=1 ~~~~~~~~~ $$ Then $$ \Gamma_{11}^2=-\sin\theta_2\cos\theta_2 \\ \Gamma_{12}^1=\Gamma_{21}^1=\frac{\cos\theta_2}{\sin\theta_2} $$ Then,......
I think if by the way ,it's very hard for computing the curvature of n-sphere of $r$-radius
So, I want to know what is the easy way to compute the curvature, at least, the easy way to compute the curvature of $S^n(r)$ (radius is $r$).