Page 334 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 26: Lines of Curvature

Proof. The proof uses the following transformation introduced by H. Lewy: Notes

  (u,v)   ,   (u p,v q)  
where p = h , and q = h . Clearly,  is continuously differentiable, and its Jacobian is:
u
v

 1 r s 
d    s 1 t  ,

 



where r = h , s = h , and t = h . Since det d  2 r t 0, it follows from the inverse function
uu
vv
uv
theorem that  is a local diffeomorphism, i.e., each point has a neighborhood on which  is a
diffeomorphism. In particular,  is open.
In view of the convexity of the function h, we have :
u  u 1    1  v 2  v 1    1 
2
2
2
2
2
 u  u 1  v 2  v 1  u 2  u 1 p  p 1  v 2  v 1 q  q 1 
2
2
2
2
2
 u  u 1  v 2  v 1  ,
2
and therefore:
2
2
2
2
u  u 1  v 2  v 1     1     1  ,


2
2
2
i.e.,  is an expanding map. This implies immediately that  is one-to-one. Thus,  has an inverse
(u, v) =  (, ) which is also a diffeomorphism. Consider now the function
–1




 f   i   u p i(v q) 2u     i  2v   ,
1.
where i   In view of
1

s  
1


d     u   u    2 r t    1 t 1 r  ,
s
v




v 

it is straightforward to check that f satisfies the Cauchy-Riemann equations, and consequently
f is analytic. In fact, f is an entire functions and so is f. Furthermore,
(t r) 2is 2 4


1
f ( )  , f ( )    1,




2 r t 2 r t




and Liouville’s Theorem gives that f is constant. Finally, the relations:
2 2

1 f   i f   f 1 f


f  2 , s  2 , t  2 ,
1  f 1  f 1  f
show that r, s, t are constants.
26.5 Theorema Egregium
In this section, we prove that the Gauss curvature can be computed in terms of the first fundamental
form and its derivatives. We then prove the Fundamental Theorem for surfaces in  , analogous
3
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