Page 307 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry

Notes This equation leads to another representation. Consider the Taylor expansion of X at a point, say
0  U:
1 3
 
i
j
i
X(u)  X(0) X (0)u  2  ij X(0)u u  O u

i
Thus, the elevation of X above its tangent plane at u is given up to second-order terms by:
1 i j 3
  .
i
X(u) X(0) X (0)u  i  N  2 k (0)u u  O u
ij
The paraboloid on the right-hand side of the equation above is called the osculating paraboloid.
A point u of the surface is called elliptic, hyperbolic, parabolic, or planar, depending on whether
this paraboloid is elliptic, hyperbolic, cylindrical, or a plane.
In classical notation, the second fundamental form is:

k
      L M  .
ij
M N 
Clearly, the second fundamental form is invariant under orientation-preserving
reparametrizations. Furthermore, the k ’s, the coordinate representation of k, changes like the
ij
first fundamental form under orientation-preserving reparametrization:
m



k   k X ,X  k ml  u  u l j ,
u
 j
ij
i
 u  u
Yet another interpretation of the second fundamental form is obtained by considering curves on
the surface. The following theorem is essentially due to Euler.
3
,
3
Theorem 1. Let   X  :[a,b]  be a curve on a parametric surface X : U   where

 :[a,b]  U. Let k be the curvature of , and let  be the angle between the unit normal N of X,
and the principal normal e of . Then:
2
k cos   k    . ...(8)
, 
Proof. We may assume that  is parametrized by arclength. We have:
   i  X ,

i
and
i 
ke     i  X    j  X .

2
i
ij
The theorem now follows by taking inner product with N, and taking (7) into account.
The quantity k cos  is called the normal curvature of . It is particularly interesting to consider
normal sections, i.e., curves  on X which lie on the intersection of the surface with a normal
plane. We may always orient such a plane so that the normal e to  in the plane coincide with the
2
unit normal N of the surface. In that case, we obtain the simpler result:
k   k    .
, 
Thus, the second fundamental form measures the signed curvature of normal sections in the
normal plane equipped with the appropriate orientation.

300 LOVELY PROFESSIONAL UNIVERSITY