Page 379 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes has the beautiful expression

p 4
K = ,
a b c
2
2 2
where p is the distance from the origin (0, 0, 0) to the tangent plane at the point (x, y, z).
There are also surfaces for which H = 0. Such surfaces are called minimal surfaces, and they show
up in physics quite a bit. It can be verified that both the helicoid and the catenoid are minimal
surfaces. The Enneper surface is also a minimal surface. We will see shortly how the classification
of points on a surface can be explained in terms of the Dupin indicatrix.
The idea is to dip the surface in water, and to watch the shorelines formed in the water by the
surface in a small region around a chosen point, as we move the surface up and down very
gently. But first, we introduce the Gauss map, i.e. we study the variations of the normal N as the
p
point p varies on the surface.
30.3 The Gauss Map and its Derivative dN

Given a surface X :   E , given any point p = X(u, v) on X, we have defined the normal Np at
3
p (or really N (u,v) at (u, v)) as the unit vector
X × X
u
N = X × X v v
p
u
Gauss realized that the assignment p  N of the unit normal Np to the point p on the surface
p
X could be viewed as a map from the trace of the surface X to the unit sphere S . If N is a unit
2
p
vector of coordinates (x, y, z), we have x + y + z = 1, and N corresponds to the point
2
2
2
p
N(p) = (x, y, z) on the unit sphere. This is the so-called Gauss map of X, denoted as N : X  S .
2
The derivative dN of the Gauss map at p measures the variation of the normal near p, i.e., how
p
the surface “curves” near p. The Jacobian matrix of dN in the basis (X , X ) can be expressed
v
p
u
simply in terms of the matrices associated with the first and the second fundamental forms
(which are quadratic forms).
Furthermore, the eigenvalues of dN are precisely –k and –k , where k and k are the principal
p
1
2
1
2
curvatures at p, and the eigenvectors define the principal directions (when they are well defined).
In view of the negative sign in –k and –k , it is desirable to consider the linear map S = –dN ,
p
p
1
2
often called the shape operator.

Then, it is easily shown that the second fundamental form II (t) can be expressed as
p
  
II (t) = S (t), t p ,
p
p
where –, – is the inner product associated with the first fundamental form.
Thus, the Gaussian curvature is equal to the determinant of S , and also to the determinant of
p
dN , since (–k )(–k ) = k k . We will see in a later section that the Gaussian curvature actually
p
1 2
2
1
only depends of the first fundamental form, which is far from obvious right now! Actually, if X
is not injective, there are problems, because the assignment p  N could be multivalued.
p
We can either assume that X is injective, or consider the map from  to S defined such that
2
(u, v)  N (u,v) .
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