Page 381 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes This is recorded in the following lemma that also shows that the eigenvectors of J(dN (u,v) )
correspond to the principal directions:
Lemma 3. Given a surface X, for any point p = X(u, v) on X, the eigenvalues of the Jacobian matrix
J(dN (u,v) ) of the derivative dN (u,v) of the Gauss map are –k , –k , where k and k are the principal
1
1
2
2
curvatures at p, and the eigenvectors of dN (u,v) correspond to the principal directions (when they
are defined). The Gaussian curvature K is the determinant of the Jacobian matrix of dN (u,v) , and
1
the mean curvature H is equal to  trace J(dN ).
2 (u,v)
The fact that Nu = –kX when k is one of the principal curvatures and when Xu corresponds to the
u
corresponding principal direction (and similarly N = –kX for the other principal curvature) is
v
v
known as the formula of Olinde Rodrigues (1815).
The somewhat irritating negative signs arising in the eigenvalues –k and –k of dN (u,v) can be
2
1
eliminated if we consider the linear map S (u,v) = –dN (u,v) instead of dN (u,v) .
The map S (u,v) is called the shape operator at p, and the map dN (u,v) is sometimes called the
Weingarten operator.
The following lemma shows that the second fundamental form arises from the shape operator,
and that the shape operator is self-adjoint with respect to the inner product –,– associated with
the first fundamental form:
Lemma 4. Given a surface X, for any point p = X(u, v) on X, the second fundamental form of X at
p is given by the formula
  
II (u,v) (t) = S (u,v) (t), t ,

for every t   . The map S(u,v) = –dN (u,v) is self-adjoint, that is,
2
   
S (u,v) (x), y = x, S (u,v) (y) ,
 
for all x, y   .
2
Thus, in some sense, the shape operator contains all the information about curvature.
Remark: The fact that the first fundamental form I is positive definite and that S(u,v) is self-
adjoint with respect to I can be used to give a fancier proof of the fact that S(u,v) has two real
eigenvalues, that the eigenvectors are orthonormal, and that the eigenvalues correspond to the
maximum and the minimum of I on the unit circle.

30.4 The Dupin Indicatrix

The second fundamental form shows up again when we study the deviation of a surface from its
tangent plane in the neighborhood of the point of tangency.

A way to study this deviation is to imagine that we dip the surface in water, and 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.
The resulting curve is known as the Dupin indicatrix (1813).
Formally, consider the tangent plane T (u ,v ) (X) at some point p = X(u , v ), and consider the
0
0
0 0
perpendicular distance (u, v) from the tangent plane to a point on the surface defined by (u, v).

374 LOVELY PROFESSIONAL UNIVERSITY