video

$II_p(v,w) = -<dN_p(v), w>$, where $dN_p$ is the Differential of the Gauss map (see Curvature).

Geometric meaning of the Second fundamental form. One can define the Normal curvature of a curve, defined as $k_n = k <n,N>$, where $k$ is the curvature of the curve, $n$ is the normal vector of the curve, and $N$ is the normal vector of the surface.

It turns out that $II_p (v,v) \equiv II_p (v) = k_n(p)$ , for any curve with tangent $v$ at point $p$. This is known as Euler's theorem