There is an exercise in Manfredo do Carmo's book,page172,that aks about the asymptotic curves of $x^2+y^2-z^2=1$. I used the formula $e(u')^2+2fu'v'+g(v´)^2=0$ where $e,f,g$ are the coefficients of the second fundamental form.I got this
$$ (u^2+v^2-1)((u´)^2+(v')^2)=u(u')^2-2uu'vv'+v(v')^2$$
but I have no idea how to proceed to find $u$ and $v$. Any ideas?
First of all, what parametrization are you using? Second of all, there is a fundamental geometric fact you should know about the hyperboloid of one sheet (or the hyperbolic paraboloid, saddle surface) — it is doubly ruled. Through each point of the surface there are two lines contained in the surface.