Page 380 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 30: Geodesic Curvature and Christoffel Symbols

Then, we have a map from  to S , where (u, v) is mapped to the point N(u, v) on S associated Notes
2
2
with N(u,v). This map is denoted as N :   S . It is interesting to study the derivative dN of the
2
Gauss map N :  S (or N : X  S ). As we shall see, the second fundamental form can be defined
2
2
in terms of dN. For every (u, v)  , the map dN (u,v) is a linear map dN (u,v) :    .
2
2
It can be viewed as a linear map from the tangent space T (u,v) (X) at X(u, v) (which is isomorphic to
 ) to the tangent space to the sphere at N(u, v) (also isomorphic to  ).
2
2
Recall that dN (u,v) is defined as follows: For every (x, y)   ,
2
dN (u,v) (x, y) = N x + N y.
u
v
.
Thus, we need to compute Nu and Nv. Since N is a unit vector, N N = 1, and by taking
.
derivatives, we have Nu N = 0 and N v . N = 0.
Consequently, N and N are in the tangent space at (u, v), and we can write
u
v
N = aX + cX ,
u
u
v
N = bX + dX .
v
u
v
Lemma 2. Given a surface X, for any point p = X(u, v) on X, the derivative dN (u,v) of the Gauss map
expressed in the basis (X , X ) is given by the equation
u
v
x

x  
dN (u,v)    a b   ,


 
y
y
   c d 
where the Jacobian matrix J(dN (u,v) ) of dN (u,v) is given by
1

 a b   E F   L M 
  =     
 c d  F G  M N 
1  MF LG NF MG 
= 2     .


EG F  LF ME MF NE 

The equations
J(dN ) =   a b 
(u,v) 
 c d
1  MF LG NF MG 


= 2   .



EG F  LF ME MF NE 
are know as the Weingarten equations (in matrix form).
If we recall the expressions for the Gaussian curvature and for the mean curvature
GL – 2FM + EN
H = ,
2(EG – F )
2
LN – M 2
K = ,
EG – F 2
we note that the trace a + d of the Jacobian matrix J(dN (u,v) ) of dN (u,v) is –2H, and that its determinant
is precisely K.
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