Page 393 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Figure 30.4: The Enneper Surface

Parabolic lines are defined by the equation
LN – M = 0,
2
where L + M + N > 0.
2
2
2
In general, the locus of parabolic points consists of several curves and points.
We now turn briefly to geodesics.
30.9 Summary

We now show that k can be computed only in terms of the first fundamental form of X, a
 n
result first proved by Ossian Bonnet circa 1848.
The computation is a bit involved, and it will lead us to the Christoffel symbols, introduced
in 1869.


Since n is in the tangent space T (X), and since (X , X ) is a basis of T (X), we can write
g
v
p
u
p

k g g n = AX + BX ,
v
u
form some A, B  .
However,
 
kn = k N + k g g n ,
N
and since N is normal to the tangent space,
.
.
N X = N X = 0, and by dotting
v
u
In general, we will see that the normal curvature has a maximum value k and a minimum
 1
value k , and that the corresponding directions are orthogonal. This was shown by Euler
2
in 1760.
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