How could we show that a curve 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?
NTS: If 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 . The relationship of normal, geodesic, and usual curvature of gives: And since we have ,
Additionally, a curve is geodesic if and only if at each point of the curve. So . Since the usual curvature of curve C is zero at all points, if we let be a regular parametrisation of C, we have that . Then by integration, , so the curve is a (segment of) a straight line.
NTS: If is a straight line, then it is both an asymptotic curve and a geodesic.
If is a straight line, the usual curvature of is zero, hence . From the relation , we see that , and , so and must both be zero. Therefore, C is both asymptotic and a geodesic line of curvature.
Therefore, we conclude that is asymptotic and geodesic curve if and only if (C) is a (segment of) a straight line.