In a surface you can compute Gaussian curvature by means of holonomy per unit area, being holonomy the rotation of a parallel transported vector which, necessarily, lies within the tangent plane of the loop (the loop is infinitesimally small, so we can look at it like contained in a plane, I guess).
In an $n$-manifold, however, even if we insist that the initial vector $w_0$ lie within the plane $\Pi(u,v)$ generated by $u,v$ in order to compute the Riemann curvature tensor $R(u,v)(w)$, the parallel transported vector $w_{\|}(o)$ along a loop contained in $\Pi(u,v)$ could stick out of $\Pi(u,v)$.
But if we focus in the projection $P(w_{\|})$ of $w_{\|}$ over $\Pi(u,v)$ then the rotation per unit area of $P(w_{\|})$ is independent of the choice of $w_0$. This quantity is called the sectional curvature of $\Pi(u,v)$.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: