Why the principal curvature lines intersect in the sphere?

Question

I've been looking at the principal curvature lines of many surfaces and I realized that the principal curvature lines of the sphere intersect, whereas the principal curvature lines of other surfaces don't.

Also, this is not the case for all closed surfaces, for example, the one holed torus principal curvature lines don't intersect. However, these lines for the two-holed torus do.

I have also been wondering if this has any relation with the global or local parametrization of these surfaces, for example, there is a different "type" of discontinuity in the parametrization of the sphere at the poles.

Answer 1

Be careful: On the sphere, any curve whatsoever is a line of curvature. The phenomenon you're noticing on the two-holed torus is that these intersections can occur only at umbilic points. Otherwise, there are precisely two principal directions at each point and there can be no such intersections. Of course, every point on the sphere is an umbilic and you can get as many intersections as you want by taking crazy curves (which will be lines of curvature). (The particular picture you're drawing with Mathematica is based on the spherical coordinates parametrization, so it's just the singularity of the coordinate system that's being reflected in this picture.)

Answer 2

This is an example of the Hairy Ball theorem. Notice that the lines of longitude produce a continuous tangent vector field on the surface, which necessarily has at least one singularity. Likewise the lines of latitude. (The singularities are at the poles in both cases.)

In general, this is tied to the Euler characteristic of the surface. The two cylinders, the torus (one-holed), and the plane have Euler characteristic zero, so need not have a singularity. The two-holed torus you drew has non-zero Euler characteristic, so necessarily has at least one singularity in its tangent vector fields.

Regarding the "type" of singularity -- with your curves, the latitude of each pole is unambiguous, but the longitude is degenerate (similar to, in polar coordinates, the lack of a definite angular coordinate for points at the origin of the coordinate system). The type of singularity here is that at least one curve from one set of curves is crushed to a single point. (In the cited page, this is the curve (/these are the curves) corresponding to $f(p) =0 $, the tangent vectors on that curve project onto the sphere giving zero vectors, so the curve lies on one point.)

Answer 3

We have Euler's relation at regular points for curves of continuous differentiabilty $$ k_n=k_1 \cos^2\psi+ k_2\sin^2\psi $$

At umbilical points the (principal) curvature lines can be oriented in anyway or direction due to the singularity. We have all points of a shere as umbilicals. Euler $k_n$ is a constant because $k_1=k_2$.

Two principal directions cannot be defined at such points on axis of symmetry intersecting a surface of revolution.