Let $\alpha(s)$ be an asymptotic curve on a surface $S$ with Gaussian curvature $\kappa$ so that the curvature of $\alpha$ is never zero. Prove that the torsion $\tau(s)$ of α satisfies $|\tau(s)| = \sqrt{- \kappa}$.
Curve $\alpha$ is called asymptotic if its normal curvature is everywhere zero. Note that $\kappa \leq 0$ at all points $\alpha$ passes through because otherwise there wouldn’t be any asymptotic directions at those points.)
