DText 8.8a -

Derivation of the Intrinsic Formula for Κ(u,v)

In lab 7 we saw thath the determinant of the Weingarten matrix is equal to the Gaussian curavture (i.e. det(L) = Κ(u,v)). We then related the Weingarten matrix to the coefficients of the first and second fundamental forms and arrived at yet another formula for Gaussian curvature: Κ(u,v) = det(L_ij)/det(g_ij. We begin with this equation in our proof that Κ(u,v) is intrinsic, despite the fact that the coefficients of the second fundamental form are not intrinsic.

X_uu = Γ₁₁¹X_u + Γ₁₁²X_v + L₁₁N (1)
X_vv = Γ₂₂¹X_u + Γ₂₂²X_v + L₂₂N (2)
X_uv = Γ₁₂¹X_u + Γ₁₂²X_v + L₁₂N (3)
X_vu = Γ₂₁¹X_u + Γ₂₁²X_v + L₂₁N (4)

Take the dot product of equations (1) and (2), and assume that g₁₂ = g₂₁ = 0. This means that X_u · X_v = 0. Also use the fact that X_u · N = X_v · N = 0. So,

X_uu · X_vv = Γ₁₁¹Γ₂₂¹(X_u · X_u) + Γ₁₁²Γ₂₂²(X_v · X_v) + L₁₁L₂₂(N · N)
X_uu · X_vv = Γ₁₁¹Γ₂₂¹g₁₁ + Γ₁₁²Γ₂₂²g₂₂ + L₁₁L₂₂, or

L₁₁L₂₂ = X_uu · X_vv -Γ₁₁¹Γ₂₂¹g₁₁ - Γ₁₁²Γ₂₂²g₂₂ (5)

Take the dot product of equations (3) and (4):

X_uv · X_vu = Γ₁₂¹Γ₂₁¹(X_u · X_u) + Γ₁₂²Γ₂₁²(X_v · X_v) + L₁₂L₂₁(N · N)
X_uv · X_vu = Γ₁₂¹Γ₂₁¹g₁₁ + Γ₁₂²Γ₂₁²g₂₂ + L₁₂L₂₁, or

L₁₂L₂₁ = X_uv · X_vu - Γ₁₂¹Γ₂₁¹g₁₁ - Γ₁₂²Γ₂₁²g₂₂ (6)

Now, subtract equation (6) from equation (5) to get:

L₁₁L₂₂ - L₁₂L₂₁ = X_uu · X_vv - X_uv · X_vu -Γ₁₁¹Γ₂₂¹g₁₁ + Γ₁₂¹Γ₂₁¹g₁₁ - Γ₁₁²Γ₂₂²g₂₂ + Γ₁₂²Γ²g₂₂
= X_uu · X_vv - X_uv · X_vu - (Γ₁₁¹Γ₂₂¹ - Γ₁₂¹Γ₂₁¹)g₁₁ - (Γ₁₁²Γ₂₂² - Γ₁₂²Γ₂₁²)g₂₂

The term (X_uu · X_vv - X_uv · X_vu) can be expressed intrinsically using g₁₁ and g₂₂. Since X(u,v) is smooth, we make use of the equality of mixed partials (X_uv = X_vu) in the following calculations.

g₁₁ = X_u · X_u
g₁₁/v = X_uv · X_u + X_u · X_uv
= 2X_u · X_uv
²g₁₁/v² = 2X_uv · X_uv + 2X_u · X_uvv (7)

g₁₂ = X_u · X_v
g₁₂/u = X_uu · X_v + X_u · X_uv
²g₁₂/uv = X_uv · X_uv + X_u · X_uvv + X_uuv · X_v + X_uu · X_vv (8)

g₂₂ = X_v · X_v
g₂₂/u = X_uv · X_v + X_v · X_uv
= 2X_v · X_uv
²g₁₁/u² = 2X_uv · X_uv + 2X_v · X_uuv (9)

Combining equations (7), (8), and (9), gives us

²g₁₂/uv - (1/2)(²g₁₁/v² + ²g₂₂/u²) = X_uuX_vv - X_uv²

This expression can be simplified further by noting that g₁₂ = 0 and so

²g₁₂/

v = 0 as well. So,

X_uu · X_vv - X_uv² = -(1/2)(²g₁₁/v² + ²g₂₂/u²)

Substituting this result into the expression for L₁₁L₂₂ - L₁₂L₂₁ yields:

L₁₁L₂₂ - L₁₂L₂₁ = -(1/2)(²g₁₁/v² + ²g₂₂/u²) - (Γ₁₁¹Γ₂₂¹ - Γ₁₂¹Γ₂₁¹)g₁₁ - (Γ₁₁²Γ₂₂² - Γ₁₂²Γ₂₁²)g₂₂

With this last result, we can now calculate the Gaussian curvature.

Κ(u,v) = (L₁₁L₂₂ - L₁₂L₂₁)/g
Κ(u,v) = -(1/2g)[(g₁₁)₂₂ + (g₂₂)₁₁] - (Γ₁₁¹Γ₂₂¹ - Γ₁₂¹Γ₂₁¹)g₁₁/g - (Γ₁₁²Γ₂₂² - Γ₁₂²Γ₂₁²)g₂₂/g