I am looking for the most straightforward way of showing that the Gaussian curvature of the unit sphere is 1. How could I go about this?
I found a way for this which involves showing that $${R^{\theta}}_{\phi} = \sin(\theta) \; e^{\theta} \wedge e^{\phi}$$ however how do I relate the Ricci tensor to the Gaussian curvature?
The most straightforward way is using that $K(p) = \det(-{\rm d}N_p)$, where $N$ is a unit normal field to $\Bbb S^2$. For the unit sphere, the position vector $N(p) = p$ is normal to the sphere, so the result follows from noting that $\det({\rm Id}_2)=1$. Which allows you to compute the curvature like this is Gauss' Theorema Egregium.