I'm trying to understand the relationship between geodesics and lines of principal curvature (to keep things simple, let's say Riemannian 2-manifolds embedded in $\mathbb{R}^3$). In my reading, I came across the following discussion in Cartan, Riemannian Geometry in an Orthogonal Frame (pp. 231-232):

Parallel transport preserves the principal directions, the principal curvature, the asymptotic tangents and the geodesic curvatures.

Let $M$ and $M^\prime$ be two neighboring points of a curvature line. Under parallel transport, the principal directions at the point $M$ are transferred into the principal directions at the point $M^\prime$. Therefore, the

curvature lines are geodesic lines.The latter have zero torsion and constant curvature.

However, I must not be understanding the context here. For instance, consider Monge's ellipsoid (an illustration can be found on page 3 of this paper). It seems to me (and in fact, I can confirm via numerical integration) that if I construct a geodesic starting with a tangent to a line of curvature (say, one near an umbilical point), the geodesic will look nothing like the line of curvature. What gives?

More generally, I am interested in the following question. For an embedded sphere $S$, consider a connection whose holonomy around any loop equals $2\pi$ times the sum of the indices of the umbilical points enclosed by that loop. (Here I mean indices in the sense of the principal foliation.) Are the geodesics defined by this connection lines of principal curvature? Again, numerical experiments say "yes," but Cartan says "no" --- according to the above statements, it would seem that curvature lines are simply geodesic with respect to the *Levi-Civita* connection.

We require thatthe normal curvature does not change [under parallel transport]" at the beginning of section 152. It is the 3rd sentence after the section title. The rest of the section is about what follows from this assumption. $\endgroup$ – Sergei Ivanov Apr 30 '10 at 4:17