Hello,

We have a 2-dimensional Riemannian Manifold immersed in $\displaystyle \mathbb{R}^3$, which is complete and has constant negative curvature.

Then the statement is, that for every point $\displaystyle p\in M$, there is an asymptotic curve $\displaystyle c:\mathbb{R}\rightarrow M$, 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 $\displaystyle c: (-a,a)\rightarrow M$ into a geodesic on the whole real line $\displaystyle c:\mathbb{R}\rightarrow M$.

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 $\displaystyle c''(t)\in T_p M$. But for a geodesic we have that c''(t) is perpendicular to $\displaystyle T_p M$.

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.

Regards