Page 388 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

and where Notes

1 1
u [1 1; 2] = F –
[1 1; 1] = E , E ,
v
2 u 2
1 1
v [1 2; 2] = G ,
[1 2; 1] = E ,
2 2 u
1 1
v [2 1; 2] = G ,
[2 1; 1] = E ,
2 2 u
1 1
u [2 2; 2] = G .
[2 2; 1] = F  2 G , 2 v
v
Also, recall that we have the Weingarten equations

 N u   a c  X u 
  =   
 N v   b d X v 
From the Gauss equations and the Weingarten equations

X =  1 11 X +  2 11 X + LN,
u
v
uu
X =  1 12 X +  2 12 X + MN,
v
u
uv
X =  1 X +  2 X + MN,
vu 21 u 21 v
X =  1 21 X +  2 21 X + NN,
v
u
vv
N = aX + cX ,
u
v
u
N = bX + dX ,
u
v
v
We see that the partial derivatives of X , X and N can be expressed in terms of the coefficient E,
u
v
F, G, L, M, N and their partial derivatives.
Thus, a way to obtain relations among these coefficients is to write the equations expressing the
commutation of partials, i.e.,
(X ) – (X ) = 0,
uv u
uu v
(X ) – (X ) = 0,
vv u
vu v
N – N = 0.
vu
uv
Using the Gauss equations and the Weingarten equations, we obtain relations of the form
A X + B X + C N = 0,
u
v
1
1
1
A X + B X + C N = 0,
u
2
2
v
2
A X + B X + C N = 0,
3
u
v
3
3
where A , B , and C are functions of E, F, G, L,M,N and their partial derivatives, for i = 1, 2, 3.
i
i
i
However, since the vectors X , X , and N are linearly independent, we obtain the nine equations
u
v
A = 0, B = 0, C = 0, for i = 1, 2, 3.
i
i
i
Although this is very tedious, it can be shown that these equations are equivalent to just three
equations.
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