Assume <u,u>=<v,v>=1. As given here: https://en.wikipedia.org/wiki/Sectional_curvature we have this relationship between the Riemann curvature tensor and sectional curvature:

K(u,v) = <R(u,v)v,u>/sin^{2}a

where a is the angle between u and v.

What is the intuition behind this identity?

Also, extend u to an orthonormal basis of the the tangent space {u,...,u_{n}}.

Then we have by above R(u,v)v = c_{1}u+...+c_{n}u_{n }where c_{1}=sin^{2}a K(u,v). Are there other identities for the other coefficients, that is, where i>1, c_{i}=<R(u,v)v,u_{i}>=?

Thanks!