How to show that a Riemannian manifold $M$ has constant sectional curvature $k$ iff $R_{ijkl}=k(g_{ik}g_{jl}-g_{il}g_{jk})$ ?
I think it's just compute directly, but when compute $R^l_{kil}=\partial_i\Gamma_{jk}^l....$, it's too complex.
May I have a standard answer? Thanks.
Here is a sketch of proof. Let $K$ denote the tensor on the right hand side. First, you show that (i) $K$ has all the symmetries of a Riemannian curvature tensor, and (ii) if you define sectional curvatures for $K$, you get $k$ for every two-plane.
Then you prove the general result that each tensor $R$ with the symmetries of a Riemannian curvature tensor is determined by $R(v,w,w,v)$ for all pairs of vectors. You can start by proving that $R(u,w,w,v)=\frac12\,(R(u+v,w,w,u+v)-R(u,w,w,u)-R(v,w,w,v))$. Next, using all symmetries of $R$, you get a (more complicated) formula for $R(u,w,x,v)$ in terms of things like $R(u,w,w,v)$.
Now, if $R$ is a Riemman curvature tensor with all scalar curvatures equal to $k$, then $R(v,w,w,v)=K(v,w,w,v)$ for all pairs of vectors, and you are done.