[Differential Geometry] What does an Asymptotic Geodesic Line of Curvature look like?

How could we show that a curve CSC \subset S on a smooth, orientable surface is both an asymptotic curve and a geodesic if and only if C is a segment of a straight line?

\Rightarrow NTS: If CC is both asymptotic and a geodesic, then it is a straight line.

By definition, asymptotic curve of a regular surface is such that, the tangent line of C at each point of the curve is an asymptotic direction, and along asymptotic directions the normal curvature is zero. Hence kn=0k_n = 0. The relationship of normal, geodesic, and usual curvature of CC gives: (kn)2+(kg)2=k2.(k_n)^2 + (k_g)^2=k^2. And since we have kn=0k_n=0, kg=k.\mid k_g \mid = k.

Additionally, a curve is geodesic if and only if kg=0k_g = 0 at each point of the curve. So k=0k=0. Since the usual curvature of curve C is zero at all points, if we let α(s)\alpha(s) be a regular parametrisation of C, we have that α(s)0\vert \alpha ''(s) \vert \equiv 0. Then by integration, α(s)=bs+c\alpha(s) = bs + c, so the curve is a (segment of) a straight line.

\Leftarrow NTS: If CC is a straight line, then it is both an asymptotic curve and a geodesic.

If CC is a straight line, the usual curvature of CC is zero, hence k=0k=0. From the relation (kn)2+(kg)2=k2(k_n)^2+(k_g)^2=k^2, we see that (kn)20(k_n)^2 \geq 0, and (kg)20(k_g)^2 \geq 0, so knk_n and kgk_g must both be zero. Therefore, C is both asymptotic and a geodesic line of curvature.

Therefore, we conclude that CSC \subset S is asymptotic and geodesic curve if and only if (C) is a (segment of) a straight line.

Note by Tasha Kim
1 year, 6 months ago

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