Thread: Riemann curvature tensor

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>/sin²​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 = c₁u+...+c_nu_n where c₁=sin²​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!

What is the intuition behind this identity?

For the denominator, you mean?
$\displaystyle <u,u><v,v>-<u,v>^2=|u{\rm x} v|^2=|u||v|sin^2(a)$.