Show that if a curve $C ⊂ S$ is both a line of curvature and a geodesic, then $C$ is a plane curve.
Give an example of a line of curvature which is a plane curve and not a geodesic. (My thoughts: Take a sphere for $S$ and let the curve $C$ be any of the latitudes of $C$ that are not the equator or the poles. Then $C$ will be a plane curve. Additionally, since the curvature is constant on the whole of $S$, every direction is a principle direction (of sorts; this might be the flaw in my argument), so any curve is a line of curvature. But, $C$ is not a segment of a great circle on $S$, so it is not a geodesic.)
Prove that a curve $C ⊂ S$ is both an asymptotic curve and a geodesic if and only if $C$ is a (segment of a) straight line.
Can I have some hints? I am not very good at covariant derivatives yet, so hand-holding with detailed examples there would be nice.
Best Answer
Let $\alpha$ be an arc length parametrization of $C$. If $C$ is both a line of curvature and a geodesic then we must have $n$ parallel to $N$ where $n$ is the unit normal of $C$ and $N$ is the unit normal of $S$. Why? This tells us that $b$ the binormal vector is constant. Why? Thus $\tau=0$ where $\tau$ is the torsion of $C$ so $C$ must be a plane curve.
Suppose that $C$ is an asymptotic curve and a geodesic. Then $k_{n}=0$ and $k_{g}=0$. What does this tell us about the curvature of $C$?