1. ## 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,...,un}.

Then we have by above R(u,v)v = c1u+...+cnun where c1=sin2​a K(u,v). Are there other identities for the other coefficients, that is, where i>1, ci=<R(u,v)v,ui>=?

Thanks!

2. ## Re: Riemann curvature tensor

What is the intuition behind this identity?

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