# Find all components of the Riemann curvature tensor, the Ricci tensor, and the Ricci scalar

The metric of the 2-sphere $S^2$ is

$ds^2 = d \theta^2 + sin^2 \theta d \phi^2$

Problem: Find all components of the Riemann curvature tensor, the Ricci tensor, and the Ricci scalar.

Solution: We order the coordinates as {θ, φ}. We have

$\Gamma ^\theta_{ \phi \phi} = \frac{1}{2} g^ {\theta \theta}[g_{ \theta \phi, \phi} +g_{ \theta \phi, \phi} - g_{ \phi \phi, \theta}] = -\frac{1}{2} g^ {\theta \theta}g_{ \phi \phi, \theta}= -\frac{1}{2} (2sin \theta cos \theta)=-sin \theta cos \theta = -\frac{1}{2} sin (2 \theta)$

$\Gamma ^\phi_{ \theta \phi} = \frac{1}{2} g^ {\theta \theta}[g_{ \phi \theta, \phi} +g_{ \phi \phi, \theta} - g_{ \theta \phi, \phi}] = \frac{1}{2} \frac{1}{sin^2 \theta} (2sin \theta cos \theta)= \frac{ cos \theta}{ sin \theta}= cot \theta$

$\Gamma ^\phi_{ \phi \theta} = cot \theta =\Gamma ^\phi_{ \theta \phi}$

Let us first look at the Riemann tensor:  the only component to compute is $R_{ \theta \phi \theta \phi}$.

The component will be

$R_{\theta \phi \theta \phi} = \Gamma^\theta_{ \phi \phi, \theta} -$ $\Gamma^\theta_{ \phi \theta, \phi} +$ $\Gamma^\theta_{ \theta f} \Gamma ^f_{ \phi \phi} -$ $\Gamma^\theta_{ \phi f} \Gamma^f_{ \phi \theta}$

Since $\Gamma ^\theta_{ \theta f}=0$ the first double $\Gamma$ term vanishes. So, we get

$R_{\theta \phi \theta \phi} = \Gamma^\theta_{ \phi \phi, \theta} -$  $\Gamma^\theta_{ \phi \phi} \Gamma^\phi_{ \phi \theta}= -$ $cos( 2\theta) + cos (\theta^2)= -$ $(2cos^2 \theta -1) + cos^2 \theta= sin^2 \theta$