We have a 2-dimensional Riemannian Manifold immersed in , which is complete and has constant negative curvature.
Then the statement is, that for every point , there is an asymptotic curve , parametrized by arclength, s.t. c(0)=p.
I don't know how i can prove this. I mean the Surface M is complete, that is we can extend every geodesic into a geodesic on the whole real line .
And we know, locally there exists always a geodesic. But a geodesic is not an asymptotic curve. Since for an asymptotic curve we have that . But for a geodesic we have that c''(t) is perpendicular to .
So we have to search for another idea:
I could show, that for any point p, we have locally an asymptotic curve, parametrized by arclength.
But i have no idea, why we can extend such an asymptotic curve to the real line.....
I hope you can help me with this problem.