Second fundamental form of asymptotic line

Question

Why the second fundamental form at a point of a surface should vanish in the asymptotic direction?

Answer 1

Second fundamental form is

$$ ds^2= L du^2+2 M du dv + N dv^2 $$

$ds^2=0\;$ because normal lengths vanish and tangents run along asymptotic directions themselves.

For a particular direction, of the asymptotic line, its normal curvature vanishes.

Euler's relation is derived from second fundamental form coefficients.

$$ \kappa_n=\dfrac{L}{E}\cos^2 \psi+\dfrac{N}{G}\sin^2 \psi$$

$$=\kappa_1\cos^2 \psi+\kappa_2\sin^2 \psi =0 $$

at particular

$$ \psi= \tan^{-1}\sqrt{\dfrac{k_1}{k_2}} $$

In one quadrant near a bisector of parametric lines $k_n$ is positive and in the other quadrant it is negative and there is smooth transition of tangent bundle.

The fundamental form or formula itself does not vaniish.