DText 8.2b -

Calculations

The coefficients of the Weingarten Map were defined by the following two equations:

(i) N_u = -L₁¹X_u - L₁²X_v
(ii) N_v = -L₂¹X_u - L₂²X_v

Dot both sides of equation (i) with X_u to get N_u · X_u = -L₁¹g₁₁ - L₁²g₂₁.
Then, use the identity N_u · X_u = -N · X_uu = -L₁₁ to get:

L₁₁ = L₁¹g₁₁ + L₁²g₂₁

Dot both sides of equation (i) with X_v to get N_u · X_v = -L₁¹g₁₂ - L₁²g₂₂.
Then, use the identity N_u · X_v = -N · X_vu = -L₂₁ to get:

L₂₁ = L₁¹g₁₂ + L₁²g₂₂

Dot both sides of equation (ii) with X_u to get N_v · X_u = -L₂¹g₁₁ - L₂²g₂₁.
Then, use the identity N_v · X_u = -N · X_uv = -L₁₂ to get:

L₁₂ = L₂¹g₁₁ + L₂²g₂₁

Dot both sides of equation (ii) with X_v to get N_v · X_v = -L₂¹g₁₂ - L₂²g₂₂.
Then, use the identity N_v · X_v = -N · X_vv = -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]