Page 401 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes the angle  is defined as follows:
i
Let  be the angle between ’ (t ’) and ’ + (t ) such that 0 <| |  , its sign being determined as
i
i
–
i
i
follows:
If p is not a cusp, which means that | |  , we give i the sign of the determinant
i
i
(’ (t ), ’ (t ), N ).
–
p
i
+
i
i
If p is a cusp, which means that | | = , it is easy to see that there is some  > 0 such that the
i
i
determinant
(’(t – ), ’(t + ), N ) i
i
i
p
does not change sign for  ] – ,  [ , and we give  the sign of this determinant.
i
Let us call a region defined as above a simple region.
In order to state a simpler version of the theorem, let us also assume that the curve segments
between consecutive points p are geodesic lines.
i
We will call such a curve a geodesic polygon. Then, the local Gauss-Bonnet theorem can be
stated as follows:
Theorem 6: Given a surface X :   E , assuming that X is injective, F = 0, and that is an open
3
disk, for every simple region R of X() bounded by a geodesic polygon with n vertices p , . . . ,
1
p , letting  , . . . ,  be the exterior angles of the geodesic polygon, we have
n
1
n
n
 K dA     2 .

i
R i 1

Some clarification regarding the meaning of the integral  K dA is in order.
R
Firstly, it can be shown that the element of area dA on a surface X is given by
dA = X × X v dudv = EG – F dudv.
2
u
Secondly, if we recall from lemma that
1

 N u   L M  E F   X u 
 N        X  ,
 v   M N  F G  v 
it is easily verified that
LN – M 2
N × N = EG – F 2 X × X = K(X × X ).
u
v
v
u
v
u
Thus,
 KdA =  K X × X dudv
u
v
R R
=  N × N dudv,
v
u
R
the latter integral representing the area of the spherical image of R under the Gauss map.
This is the interpretation of the integral  KdA that Gauss himself gave.
R
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