I found the following equation on Wikipedia here: \begin{equation} K(u,v)={\langle R(u,v)v,u\rangle\over \langle u,u\rangle\langle v,v\rangle-\langle u,v\rangle^2} \end{equation} No explanation I could understand was given for where this formula comes from or why it represents sectional curvature.
Can someone just briefly summarize how this formula works in words? How does it let me measure sectional curvature?
On
The idea of sectional curvature is to assign curvatures to planes - the sectional curvature of a plane $\Pi$ in the tangent space is proportional to the Gaussian curvature of the surface swept out by geodesics with starting directions in $\Pi$. Intuitively you're taking a two-dimensional slice through a given plane, and measuring the classical two-dimensional curvature of this slice.
For the purpose of computation, the easiest way to represent a plane is by two vectors $u,v$ that span it. In the case where $u$ and $v$ are orthonormal, the sectional curvature is simply $\langle R(u,v)v,u\rangle$. You can determine this is the correct expression in the 2-dimensional case by showing it's equal to the Gaussian curvature, and this carries over to general dimension using the Gauss-Codazzi relations and the fact that the second fundamental form of the slice is zero at the base point of $\Pi$.
The formula you've given is in terms of an arbitrary basis for $\Pi$. It's clear that the formula must change in this case, since e.g. scaling one of the vectors linearly would change the curvature expression quadratically. To find the correct formula in an arbitrary basis, you can e.g. apply Gram-Schmidt to $u,v$ to get an orthonormal basis $e_1, e_2$, then write out $K = \langle R(e_1,e_2)e_2,e_1 \rangle$ in terms of $u,v$ and use the symmetries/bilinearity of $R$ to simplify.
Perhaps it's best just to look at what's going on here geometrically.
I said in one of your other questions that the Riemmann tensor and sectional curvature can be written directly in terms of the exterior algebra. Namely, given a simple 2-vector $B$,
$$K(B) = \frac{\langle R(B), B \rangle}{\langle B, B \rangle}$$
where the inner product $\langle,\rangle$ is extended as usual to the space of 2-vectors.
With this in mind, you can think of $R$'s matrix representation on the space of 2-vectors. We're taking one of the elements on the diagonal--$\langle R(B), B \rangle$--and normalizing based on the magnitude of $B$--that is, by $\langle B, B \rangle$--so that the function $K$ can be thought of as a function of planes, independent of the magnitude of the 2-vector $B$ plugged into it.
The key in comparing this to the regular formula is to recognize the denominator as the inner product of 2-vectors (of two planes with each other) and the numerator similarly.