I wish to solve the following problem using the matrix of the shape operator $S_{P}$. Suppose $K= \text{det} S_{P} <0$, and $C$ is an asymptotic curve with curvature $\kappa (P)$ nonzero. I want to show that the torsion $\tau$ at $P$ is $\pm \sqrt{-K}$.
I understand that $S_{P}$ is a symmetric linear map, and so its matrix with respect to an orthonormal basis will be symmetric. Now, let $\lbrace \textbf{U},\text{V} \rbrace$ be an orthonormal basis for $T_{P}M$, where $\textbf{U}$ is tanget to $C$. If I can find the $S_{P}$ matrix, then I think I can do the rest (I hope).
I also know if that if $C$ is asymptotic, and $\textbf{U} \in T_{P}M$, then $S_{P}(\textbf{U}) \cdot \textbf{U}=0$. I'd like to get an idea of where to start.
Well known Beltrami- Enneper theorem. May be treated in:
A Treatise on the Differential Geometry of Curves and Surfaces
by Luther Pfahler Eisenhart
Normal curvature vanishes, total curvature is entirely in tangent plane, so that geodesic torsion is
$$ \tau_g = \sqrt {- K} $$