Differential of the Gauss map

Differential of Gauss map, can be expressed as $dN_p: T_p S \rightarrow T_p S$. Furthermore, it's a symmetric endomorphism. $dN_p$ on canonical basis, given a Local chart: vid. Useful formula $<N_u,X_v> = <N_v,X_u> = -<N,X_uv>$, where $N_u = dN(X_u)$ is the partial derivative of the normal vector $N$ with respect to coordinate $u$ (where we are being a little sloppy and considering $N$ as a function in the domain of the chart, rather than the surface itself).

We can now define the Second fundamental form, and the First fundamental form

Since the differential of the Gauss map is symmetric, it is diagnoalizable. The eigenvalues of the differential of the Gauss map give the Principal curvatures. The corresponding eigendirections give the Principal directions