# Geodesic math help

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.

Homework Helper
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"?

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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?