Page 387 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes 30.7 The Theorema Egregium of Gauss, the Equations of Codazzi-
Mainardi, and Bonnet’s Theorem

Here, we expressed the geodesic curvature in terms of the Christoffel symbols, and we also
showed that these symbols only depend on E, F, G, i.e., on the first fundamental form and
we expressed N and N in terms of the coefficients of the first and the second fundamental form.
v
u
At first glance, given any six functions E, F, G, L, M, N which are at least C -continuous on some
3
open subset U of  , and where E, F > 0 and EG – F > 0, it is plausible that there is a surface X
2
2
defined on some open subset  of U, and having Ex + 2Fxy + Gy as its first fundamental form,
2
2
and Lx + 2Mxy + Ny as its second fundamental form.
2
2
However, this is false! The problem is that for a surface X, the functions E, F, G, L, M, N are not
independent.
In this section, we investigate the relations that exist among these functions. We will see that
there are three compatibility equations. The first one gives the Gaussian curvature in terms of
the first fundamental form only. This is the famous Theorema Egregium of Gauss (1827).
The other two equations express M – L and N – M in terms of L, M, N and the Christoffel
u
u
v
v
symbols. These equations are due to Codazzi (1867) and Mainardi (1856).
Remarkably, these compatibility equations are just what it takes to insure the existence of a
surface (at least locally) with Ex + 2Fxy + Gy as its first fundamental form, and Lx + 2Mxy + Ny 2
2
2
2
as its second fundamental form, an important theorem shown by Ossian Bonnet (1867).
Recall that
"
'
"
'
2
'
X” = X u + X u + X (u ) + 2X u u + X (u )2,
'
uv
vv
1
2
v
2
2
uu
1
1
u

'
'
'
= (L(u ) + 2Mu u + N(u ) )N + k n ,
'
2
2
g
2
1
1
g
2
and since
   


' 
' 
k g g n = u   1 ij u u X    u   2 ij u u X ,
'
2 
'
"
"
1 
u
v


i
j

j
i
 i 1,2   i 1,2 


 j 1,2   j 1,2 


we get the equations (due to Gauss):
X =  1 11 X +  2 11 X + LN,
u
v
uu
X =  1 12 X +  2 12 X + MN,
uv
u
v
X =  1 12 X +  2 21 X + MN,
u
vu
v
X =  1 22 X +  2 22 X + NN,
v
u
vv
k
where the Christoffel symbols  are defined such that
ij
 1 ij   E F   1  [ij ; 1]
      , 
  2  F [ij ; 2
 ij   G  
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