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>/sin2a
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,...,un}.
Then we have by above R(u,v)v = c1u+...+cnun where c1=sin2a K(u,v). Are there other identities for the other coefficients, that is, where i>1, ci=<R(u,v)v,ui>=?
Thanks!