Page 325 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes Now, fix 1  k  2, and consider the over-determined system:

 u  k  a , i  1,2.
k
 u i i
The integrability condition for this system is exactly (2), hence there is a solution in a
a
1
k
neighborhood of u . Furthermore, since the Jacobian of the map   u ,u 2   u ,u  1  2  is d    ,
0
i
and det a k i 0, it follows from the inverse function theorem, that perhaps on yet a smaller
  
neighborhood,  is a diffeomorphism. Let     1 , then  is a diffeomorphism on a
,


neighborhood V of (u ), and if we set X  X  then:

0
0
u
j

X  X j   u i j  X b  Y . i
i
i
j
Proposition 2. Let X : U   be a parametric surface, and let Y and Y be linearly independent
3
1
2
vector fields. Then for any point u  U there is a neighborhood of u and a reparametrization
0
0



X  X  such that X  f Y  for some functions f . i

i
i
i
Proof. By Lemma 1 is suffices to show that there are function f such that f Y and f Y commute.
i
1
2
1
2

Write Y ,Y  a Y  a Y , and compute:
1
1
2
1
2
2
f Y ,f Y   f f a Y  a Y  f  Y f Y   f  Y f Y . 
 1 1 2 2 1 2 1 1 2 2 1 1 2 2 2 2 1 1
f Y ,f Y  vanishes if and only if the following two equations are satisfied:
Thus, the commutator  1 1 2 2
Y f – a f = 0
2 1
1 1
Y f – a f = 0.
1 2
2 2
We can rewrite those as:
Y log f = a 1
2
1
Y log f = a .
2
2
1
Each of those equation is a linear first-order partial differential equation, and can be solved for
a positive solution in a neighborhood of u .
0
In a neighborhood of a non-umbilical point, the principal directions define two orthogonal unit
vector fields. Thus, we obtain the following Theorem as a corollary to the above proposition.
Theorem 1. Let X : U   be a parametric surface, and let u be a non-umbilical point. Then
3
0


there is neighborhood U of u and a diffeomorphism  : U  U such that X  X  is

0
0
0
0
parametrized by lines of curvature.
If X is parametrized by lines of curvature, then the second fundamental form has the coordinate
representation:
0 
1
k
      k g 11 k g 22  
0
ij
2
Definition 2. A curve  on a parametric surface X is called an asymptotic line if it has zero normal
 
, 
curvature, i.e.,  k    0.
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