Page 305 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes The classical (Gauss) notation for the first fundamental form is g = E, g = g = F, and G = g ,
21
12
11
22
i.e.,
g
      E F  
ij
F G
Clearly, F < EG, and another condition equivalent to the condition that X and X are linearly
2
2
1
independent is that det(g ) = EG  F > 0. The first fundamental form is also sometimes written:
2
ij
ds = g du du = E (du ) + 2F du du + G(du ) .
1
2
1 2
2
j
2 2
i
ij
Note that the g ’s are functions of u. The reason for the notation ds is that the square root of the
2
ij
3
first fundamental form can be used to compute length of curves on X. Indeed, if :[a,b]   is
,
a curve on X, then   X  where  is a curve in U. Let (t)   (t), 1  2 (t)  , and denote time

derivatives by a dot, then

i 
L [a,b]   a b   dt  a b g   j  dt.
ij
Accordingly, ds is also called the line element of the surface X.
Note that g contains all the intrinsic geometric information about the surface X. The distance
between any two points on the surface is given by:
d(p,q)  inf L : is a curve on X between p and q . 

Also the angle between two vectors Y, Z  T X is given by:
x
g(Y,Z)
cos  ,
g(Y,Y) g(Z,Z)


and the angle between two curves  and  on X is the angle between their tangents and .  
Intrinsic geometry is all the information which can be obtained from the three functions g and
ij
their derivatives.
Clearly, the first fundamental form is invariant under reparametrization. The next proposition
shows how the g ’s change under reparametrization.
ij
3

Proposition 3. Let X : U   be a parametric surface, and let X  X  be a reparametrization

of X. Let g be the coordinate representation of the first fundamental form of X, and let g  be the
ij
ij
coordinate representation of the first fundamental form of X. Then, we have:

u k u l
 g ij  g kl  u i  u j , ...(4)
where d   u / u . i  j 
Proof. In view of (2), we have:
k
k
u 



g   g X ,X  g    u u  k i X ,   u l j X l      u  u l j  g X ,X  g kl   u   u l j .

k
 j
u
l
k
ij
i
i
i

du du 
u 

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