Page 308 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 24: Two Fundamental Form

Definition 9. Let X : U   be a parametric surface, and let k be its second fundamental form. Notes
3
Denote the unit circle in the tangent space at u by S X  Y  T X : Y   1 . For u  U, define the
u
u
principal curvatures of X at u by:
k  min k(Y,Y), k  maxk(Y,Y).
2
1
Y  S u X Y SsX

The unit vectors Y  S X along which the principal curvatures are achieved are called the principal
u
directions. The mean curvature H and the Gauss curvature K of X at u are given by:
1
H  k  k 2  , K  k k .
1
2
1
2
If we consider the tangent space T X with the inner product g and the unique linear transformation
u
 : T X  T X satisfying:
u
u
Z
 g (Y),Z   k(Y,Z),   T X, ...(9)
u
then k  k are the eigenvalues of  and the principal directions are the eigenvectors of . If
2
1
k = k then k = g and every direction is a principal direction. A point where this holds is called
1
2
an umbilical point. Otherwise, the principal directions are perpendicular. We have that H is the
trace and K the determinant of . Let(g ) be the inverse of the 2 × 2 matrix (g ):
ij
ij
g g mj   i j .
im
Set    X i  j i Xj, then since k   g (X ),X  i  j  m i g , we find:
mj
ij
mj
 j i  k g .
im
mj
j
It is customary to say that g raises the index of k and to write the new object k  k g . Here
im
i
since k is symmetric, it is not necessary to keep track of the position of the indices, and hence we
ij
j
write:  j i  k . In particular, we have:
i
 
1 det k ij
i
H  k , K  . ...(10)
 
i
2 det g ij
il
im
ij
Now, k  g g k , and we have
lm
2
2
2
2
ij
k  k k  tr 2  k  k  4H  2K.
2
1
ij
Hence, we conclude
1 2
2
K  2H  k ...(11)
2
24.4 Examples
In this section, we use u = u, and u =  in order to simplify the notation.
1
2
24.4.1. Planes. Let U   be open, and let X : U   be a linear function:
3
2
X(u, ) = Au + B,
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