Calculations
The coefficients of the Weingarten Map were defined by the following two equations:
(i) Nu = -L11Xu
- L12Xv
(ii) Nv = -L21Xu
- L22Xv
Dot both sides of equation (i) with Xu to get
Nu · Xu = -L11g11 - L12g21.
Then, use the identity Nu · Xu = -N · Xuu = -L11 to get:
L11 = L11g11
+ L12g21
Dot both sides of equation (i) with Xv to get
Nu · Xv = -L11g12 - L12g22.
Then, use the identity Nu · Xv = -N · Xvu = -L21 to get:
L21 = L11g12
+ L12g22
Dot both sides of equation (ii) with Xu to get
Nv · Xu = -L21g11 - L22g21.
Then, use the identity Nv · Xu = -N · Xuv = -L12 to get:
L12 = L21g11
+ L22g21
Dot both sides of equation (ii) with Xv to get
Nv · Xv = -L21g12 - L22g22.
Then, use the identity Nv · Xv = -N · Xvv = -L22 to get:
L22 = L21g12
+ L22g22
We now have four equations that relate the coefficients of the Weingarten map tp the coefficients of the second fundamental form and the metric coefficients. These equations can be written more concisely as a matrix product:
[L11 L21]
= [L11 L12]
[g11 g12]
[L12 L22]
[L21 L22]
[g21 g22]
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