Page 336 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 26: Lines of Curvature

Taking note of the fact that X = X , we now interchange j and l and subtract to obtain (18) and Notes
ilj
ijl
(19).
Equation (18) is called the Gauss Equation, and Equation (19) is called the Codazzi Equation. The
Gauss Equation has the following corollary which has been coined Theorema Egregium. It’s
discovery marked the beginning of intrinsic geometry, the geometry of the first fundamental
form.

Corollary 1. Let X : U   be a parametric surface. Then the Gauss curvature K of X can be
3
g
computed in terms of only its first fundamental form   and its derivatives up to second
ij
order:
1
n
n
K  g ij   m   m    m nm    m  ,
nj
im
2
ij,m
ij
im,j
where  are the Christoffel symbols of the first kind.
m
ij
Proof.
We now show, in a manner quite analogous to Theorem 6, that provided they satisfy the Gauss-
Codazzi Equations, the first and second fundamental form uniquely determine the parametric
surface up to rigid motion.
g
2

Theorem 7 (Fundamental Theorem). Let U   be open and simply connected, let   : U  S 2 2
ij

k

and   : U  S 2 2 be smooth, and suppose that they satisfy the Gauss-Codazzi Equations
ij
k
g
(18)–(19). Then there is a parametric surface X : U   such that   and   are its first and
3
ij
ij
second fundamental forms. Furthermore, X is unique up to rigid motion: if X is another

parametric surface with the same first and second fundamental forms, then there is a rigid
motion R of  such that X R X.

3


Proof. We consider the following over-determined system of partial differential equations for
X ,X ,N: 6
2
1
X   m ij X  k N, ...(20)
m
i,j
ij
im
N   k g X , ...(21)
m
i
ij
m
where  is defined in terms of   by (16). The integrability conditions for this system are:
g
ij
ij
  m X  k N  m il X  k N  j ...(22)
 
ij
il
m
m
ij
l
 k g X m   k g X lj jm m  . ...(23)
jm
ij
l
i
The proof of Theorem 18 also shows that the Gauss-Codazzi Equations (18)–(19) imply (22) if Xi
and N satisfy (20) and (21). We now check that (19) also implies (23). First note that since  is
m
ij
defined by (16), we have
1
 m g mn  2  g ni,j  g nj,i  g ij,n  .
ij
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