Sectional curvature

Remarks

The value does not depend on the choice of basis $u, v$ of $\sigma$, because numerator and denominator both change by the square of the determinant of the change of basis.

Example: on the unit sphere $S^n$, $R(X, Y)Z = g(Y, Z)X - g(X, Z)Y$, so for orthonormal $u, v$ the numerator is $g(g(v, v)u - g(u, v)v, u) = 1$ and $K(\sigma) = 1$ for every plane.

Non-example: $K(u, v)$ is not defined when $u$ and $v$ are linearly dependent, since then they span no plane and the denominator is $0$.

For a surface, the only plane at $p$ is $T_pM$, and $K$ is the Gaussian curvature. A manifold has constant sectional curvature $k$ if and only if $R(X, Y)Z = k\bigl(g(Y, Z)X - g(X, Z)Y\bigr)$. With the opposite sign convention for $R$, the numerator is written $g(R(u, v)u, v)$.