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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):

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

  2. 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.

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This is indeed strange, of course curvature lines are almost never geodesics. As for the second question, defining holonomy does not seem to define a connection uniquely. Do you mean any connection with this holonomy or some specific one? –  Sergei Ivanov Apr 29 '10 at 19:05
    
Ah, yes, sorry --- uniqueness is given by the fact that the connection I'm considering is also closest to the Levi-Civita connection in the sense of the norm induced by the Hodge inner product. –  TerronaBell Apr 29 '10 at 19:32
    
In which chapter and section of Cartan's is this discussion found? I only have a Russian translation here - page numbers are different. –  Sergei Ivanov Apr 29 '10 at 20:39
    
In my copy it's in Chapter 24 (Forms of Laguerre and Darboux), Section 152 (Invariance of normal curvature under parallel transport of a vector). –  TerronaBell Apr 29 '10 at 21:28
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Russian translation has a sentence "We require that the 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. –  Sergei Ivanov Apr 30 '10 at 4:17

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