Page 390 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

Theorem 7: Let E, F, G, L, M, N be any C3-continuous functions on some open set U   , and Notes
2
such that E > 0, G > 0, and EG – F > 0. If these functions satisfy the Gauss formula (of the
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Theorema Egregium) and the Codazzi-Mainardi equations, then for every (u, v)  U, there is an
open set   U such that (u, v)  , and a surface X :   E such that X is a diffeomorphism, and
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E, F, G are the coefficients of the first fundamental form of X, and L, M, N are the coefficients of
the second fundamental form of X. Furthermore, if  is connected, then X() is unique up to a
rigid motion.

30.8 Lines of Curvature, Geodesic Torsion, Asymptotic Lines

Given a surface X, certain curves on the surface play a special role, for example, the curves
corresponding to the directions in which the curvature is maximum or minimum.
Definition 3: Given a surface X, a line of curvature is a curve C : t  X(u(t), v(t)) on X defined on
some open interval I, and having the property that for every t  I, the tangent vector C’(t) is
collinear with one of the principal directions at X(u(t), v(t)).

Notes we are assuming that no point on a line of curvature is either a planar point or
an umbilical point, since principal directions are undefined as such points.

The differential equation defining lines of curvature can be found as follows:
Remember from lemma that the principal directions are the eigenvectors of dN (u,v) .
Therefore, we can find the differential equation defining the lines of curvature by eliminating k
from the two equations from the proof of lemma:

MF – LG u'  NF – MG v'   ku',
EG – F 2 EG – F 2

LF – ME MF – NE kv.
EG – F 2 u'  EG – F 2 v'  

It is not hard to show that the resulting equation can be written as

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 (v') 2  u'u' (u') 
 
det E F G   0.
 
 L M N 
 
From the above equation, we see that the u-lines and the v-lines are the lines of curvatures iff F
= M = 0.
Generally, this differential equation does not have closed-form solutions.
There is another notion which is useful in understanding lines of curvature, the geodesic torsion.
Let C : S  X(u(s), v(s)) be a curve on X assumed to be parameterized by arc length, and let X(u(0),
v(0)) be a point on the surface X, and assume that this point is neither a planar point nor an
umbilic, so that the principal directions are defined.

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