Page 392 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

Such a curve is also characterized by the fact that the geodesic curvature k is null. Notes
g
As we will see shortly, such curves are called geodesics, which explains the name geodesic
torsion for T .
g
Lemma 10: can be used to give a quick proof of a beautiful theorem of Dupin (1813).
Dupin’s theorem has to do with families of surfaces forming a triply orthogonal system.

Given some open subset U of E , three families F , F , F of surfaces form a triply orthogonal
3
3
2
1
system for U, if for every point p  U, there is a unique surface from each family F passing
i
through p, where i = 1, 2, 3, and any two of these surfaces intersect orthogonally along their
curve of intersection.
Theorem 11: The surfaces of a triply orthogonal system intersect each other along lines of
curvature.
A nice application of theorem 11 is that it is possible to find the lines of curvature on an ellipsoid.
We now turn briefly to asymptotic lines. Recall that asymptotic directions are only defined at
points where K < 0, and at such points, they correspond to the directions for which the normal
curvature k is null.
N
Definition 4: Given a surface X, an asymptotic line is a curve C : t  X(u(t), v(t)) on X defined on
some open interval I where K < 0, and having the property that for every t  I, the tangent vector
C0(t) is collinear with one of the asymptotic directions at X(u(t), v(t)).
The differential equation defining asymptotic lines is easily found since it expresses the fact that
the normal curvature is null:

L(u’) + 2M(u’v’) + N(v’) = 0.
2
2
Such an equation generally does not have closed-form solutions.

Notes The u-lines and the v-lines are asymptotic lines iff L = N = 0 (and F  0).
Perseverant readers are welcome to compute E, F, G, L, M, N for the Enneper surface:

u 3
x = u   uv 2
3
u 3
2
y = v   u v
3
z = u – v .
2
2
Then, they will be able to find closed-form solutions for the lines of curvatures and the asymptotic
lines.

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