Let $p$ be a hyperbolic point of a surface $S$. Let $\alpha_1$ and $\alpha_2$ be two asymptotic curves passing through $p$ (in two different asymptotic directions) and assume that they have nonzero curvatures at $p$. Prove that if $\tau_1$ is the torsion of $\alpha_1$ at $p$ and $\tau_2$ is the torsion of $\alpha_2$ at $p$, then $\tau_1$ = $-\tau_2$.
I'm thinking this has to do with the fact that a hyperbolic point has negative Gaussian curvature (per http://mathworld.wolfram.com/HyperbolicPoint.html) and possibly relating the principal curvatures to the asymptotic curves, but I'm not quite sure where to go from there.
Best Answer
HINT: See Exercises 17 and 18 (along with the hints) on p. 55 of my differential geometry text. It might help in the latter problem to write the unit tangent of the asymptotic curve as $\cos\theta\mathbf e_1 +\sin\theta\mathbf e_2$, where $\mathbf e_1,\mathbf e_2$ are the principal directions. How are the $\theta$'s related for the two asymptotic curves?