Page 335 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes to curves, which states that a parametric surface is uniquely determined by its first and second
fundamental form. Partial derivatives with respect to u will be denoted by a subscript i following
i
a comma, unless there is no ambiguity in which case the comma may be omitted.
Proposition 10. Let X : U   be a parametric surface. Then the following equations hold:
3
X   m ij X  k N, ...(15)
ij
ij
m
where,
1
m
  2 g mn  g ni,j  g nj,i  g ij,n  , ...(16)
ij
  are the coordinate representations of its first and second fundamental form.
g
and   and k ij
ij
Proof. Clearly, X can be expanded in the basis X , X , N of  . We already saw, that the component
3
ij
1
2
m
of X along N is k , hence Equation (15) holds with the coefficients  given by
ij
ij
ij
X  X   n ij g mn .
m
ij
In order to derive (16), we differentiate g = X  X, and substitute the above equation to obtain:
i
ij
j
g ij,m   n im g   n jm g . ...(17)
ni
nj
Now, permute cyclically the indices i, j,m, add the first two equations and subtract the last one:
g ij,m  g mi,j  g im,i  2 n jm g .
ni
Multiplying by g and dividing by 2 yields (16).
il
The coefficients  are called the Christoffel symbols of the second kind. It is important to note
m
5
ij
that the Christofell symbols can be computed from the first fundamental form and its first
derivatives. Furthermore, they are not invariant under reparametrization.
Theorem 6. Let X : U   be a parametric surface. Then the following equations hold:
3
n
m
m
n
 m ij,l   m il,j        g mn  k k  k k jn  , ...(18)
il
ln
il
nj
ij
nl
ij
k ij,l  k il,j   m ij k lm   m il k jm  0. ...(19)
Proof. If we differentiate (15), we get:
X    m X m   k N ij    m X   m ij X ml  k N k N .

ij
m
ij,l
ij
ij,l
l
ijl
l
l
Substituting X from (15) and N from (5), and decomposing into tangential and normal
l
ml
components, we obtain:
m
X  A X  B N,
m
ijl
ijl
ijl
where:
m
m
n
A   m ij,l     g mn k k ,
ln
ij
nl
ijl
ij
lm
B  k ij,l   m ij k .
ijl
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