Page 306 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Unit 24: Two Fundamental Form

24.3 The Second Fundamental Form Notes

We now turn to the second fundamental form. First, we need to prove a technical proposition.
Let Y and Z be vector fields along X, and suppose that Y = y X is tangential. We define the
i
i
directional derivative of Z along Y by:
i
 Y Z  y Z  y i Z .
i
 u i
Note that the value of  Z at u depends only on the value of Y at u, but depends on the values of
Y
Z in a neighborhood of u. In addition,  Z is reparametrization invariant, but even if Z is
Y
i 
tangent, it is not necessarily tangent. Indeed, if we write Y   y X , then we see that:
i
Z u
u 

i
i 
y  i Z   y i Z u  i  y j  i j u  u k i  y  j Z.

k


u 
The commutator of Y and Z can now be defined as the vector field:
[Y,Z] =  Z  Y.
Z
Y
3
Proposition 4. Let X : U   be a surface, and let N be its unit normal.
(1) If Y and Z are tangential vector fields then [Y,Z]  T X.
u
(2) If Y, Z  T X then  N Z =  N Y.
Z
Y
u
Proof. Note first that since X is smooth, we have X = X , where we have used the notation X =
ji
ij
ij
 X/u u. Now, write Y = y X and Z = z X, and compute:
i
j
j
i
2
j
i
 Z   Y = y zX + y  z X  y z X  zy X i
i j
i
j
i j
i
j
Y
j
ij
Z
i
j
ji
i
i
  y z i j  z  i y j  X . j
To prove (2), extend Y and Z to be vector fields in a neighborhood of u, and use (1):
 N Z   N Y = N  ( Z   Y) = 0.
Y
Z
Z
Y
Note that while proving the proposition, we have established the following formula for the
commutator:
i
i
[Y,Z]   y z i j  z  i y j  X j ...(5)
Definition 8. Let X : U   be a surface, and let N : U   be its unit normal. The second
2
3
fundamental form of X is the symmetric bilinear form k defined on each tangent space T X by:
u
k(Y, Z) =  N Z. ...(6)
Y
We remark that since N  N = 1, we have  N N = 0, hence  N is tangential. Thus, according to
Y
Y
(6), the second fundamental form is minus the tangential directional derivative of the unit
normal, and hence measures the turning of the tangent plane as one moves about on the surface.
Note that part (2) of the proposition guarantees that k is indeed a symmetric bilinear form. Note
that it is not necessarily positive definite. Furthermore, if we set k = k(X , X) to be the coordinate
i
j
ij
representation of the second fundamental form, then we have:
k = X  N. ...(7)
ij
ij
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