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

How could we show that a curve $C \subset S$ 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 $C$ 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 $k_n = 0$ . The relationship of normal, geodesic, and usual curvature of $C$ gives: $(k_n)^2 + (k_g)^2=k^2.$ And since we have $k_n=0$ , $\mid k_g \mid = k.$

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

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

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

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

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
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`\frac{2}{3}`	$\frac{2}{3}$
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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$