Geodesic & Asymptotic Curves: Proving Segment of Straight Line

[Dragonfall]

Homework Statement

Show that a regular curve on a smooth surface is a geodesic and an asymptotic curve if and only if it is a segment of a straight line.

The Attempt at a Solution

I did the <= implication, which is quite easy. I can't get the other one.

 

[AKG]

A unit speed curve $\gamma$ is a segment of a straight line iff $\ddot{\gamma} = 0$, and all regular curves can be reparametrised to be unit speed. What are the definitions of "geodesic" and "asymptotic curve"?

 

[Dragonfall]

Suppose $$\alpha$$ a unit speed curve, then it is a geodesic if the covariant derivative $$D\alpha/dt=0$$ and it is an asymptotic curve if $$II(\alpha '(s))=0$$ for all s, where II is the second fundamental form and the normal curvature k_n at a point on the curve. Since $$k_n=k<n,N>=0$$ where k is the curvature of $$\alpha$$ at a point, n the curve's normal and N the surface's normal, we have to conclude that k=0 or <n,N>=0. This is where I am stuck. What if <n,N>=0?

 

[AKG]

My book says that a curve is a geodesic if it's second derivative is zero or if it's second derivative is perpendicular to the surface. Since the second derivative is kn, this is equivalent to saying that kn = aN for some real a.

k<n,N> = <kn,N> = <aN,N> = a<N,N>

if the curve is asymptotic, then k<n,N> = 0, which would imply a = 0, implying that the second derivative is 0, which I claimed to be equivalent to saying that the curve is part of a line segment. So somehow, you need to show that your book's definition of "geodesic" is equivalent to my book's.