While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence:
For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$
which is the oposite of the usual convention, but more symmetric. The Ricci curvature is the contraction $$R_{ik}=g^{jl}R_{ijkl},$$ and on the sphere we have $$R(u,u)=R_{ij}u^{i}u^{j}.$$
What I'm strugling with is that part where
"for the sphere we have $R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0$."
Could some one explain me why is that so? Or any link that explains this? Is it just saying how curvature tensor acts on the tangent vectors of the sphere?