Page 372 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

where the Christoffel symbols  are defined such that Notes
k
ij
 1 ij   E F   1  [ij; 1]
  =    , 
  2  F [ij; 2]
 ij   G  
and the Christoffel symbols [i j; k] are defined such that
[i j; k] = X ij . X .
k
Note that
[i j; k] = [j i; k] = X ij . X .
k
Looking at the formulae
[ ; ] = X . X
 
for the Christoffel symbols [ ; ], it does not seem that these symbols only depend on the first
fundamental form, but in fact they do!
After some calculations, we have the following formulae showing that the Christoffel symbols
only depend on the first fundamental form:

1 1
[1 1; 1] = Eu, [1 1; 2] = F  E ,
v
u
2 2
1 1
[1 2; 1] = E , [1 2; 2] = G ,
u
v
2 2
1 1
[2 1; 1] = E , [2 1; 2] = G ,
2 v 2 u
1 1
[2 2; 1] = Fv – G , [2 2; 2] = G .
2 u 2 u
Another way to compute the Christoffel symbols [ ; ], is to proceed as follows. For this
computation, it is more convenient to assume that u = u and v = u , and that the first fundamental
2
1
form is expressed by the matrix
 g 11 g 12   E F 
 g g     ,
 21 22   F G
where g = X . X . Let
  

g  |   g  .
 u 

Then, we have
 g
g    X  X  X  X  [ ; ] [ ; ].




|
 u     
From this, we also have
g = [ ; ] + [ ; ],
|

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