Page 371 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes At this point, it is useful to introduce the Christoffel symbols (of the first kind) [ ; ], defined
such that
[ ; ] = X . X ,
 
where , ,   {u, v}. It is also more convenient to let u = u and v = u , and to denote [u v ; u ]
2
1



as [ ; ].
Doing so, and remembering that

.
kn X = EA + FB,
u

.
kn X = FA + GB,
v
we have the following equation:
'
'
"
 E F  A   E F   u  [   ; 1]u u 
1



       "    ' '  . 

 F G B   F G u 2   1,2 [ ; 2]u u  


 1,2
However, since the first fundamental form is positive definite,
EG – F > 0, and we have
2
1 G
 E F   1 2    F 

   (EG F )   .
 F G   F E 
Thus, we get
'
"
'
1 G
 A   u  2    F  [   ; 1]u u 
1



  =  u "  +  (EG F )    ' '  . 
 B   2   1,2   F E    [ ; 2]u u  

 1,2
,
k
It is natural to introduce the Christoffel symbols (of the second kind)  defined such that
ij
 1 ij  2    F  [ij ; 1]
1 G
   (EG F )    . 

  2    F E  [ij ; 2] 
 ij 
Finally, we get
1 
'
1
'
"
A = u  G u u , j
ij
i
i 1,2

j 1,2

2 
"
'
'
B = u   ij 2 u u , j
i
i 1,2

j 1,2

Lemma 1. Given a surface X and a curve C on X, for any point p on C, the tangential part of the
curvature at p is given by
   


' 
' 
'
1 
k g g n = u   1 ij u u X    u   2 ij u u X ,
'
2 
"
"
j
v

j


i
i
u
 i 1,2   i 1,2 


 j 1,2   j 1,2 


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