8.8: Theorema Egregium

Properties of surfaces in space that only depend on the metric coefficients g_ij are called intrinsic . We have seen that such properties include the angle between two different smooth curves through a point, the area of a region, and the lengths of parametrized curves. We have also introduced several extrinsic properties, which depend on how a surface lies in space and cannot be calculated from the metric coefficients alone. The coefficients of the Weingarten Map, the coefficients of the second fundamental form, and the principal curvatures are all extrinsic. It turns out, however, that some of the properties initially defined using extrinsic concepts are in fact intrinsic. This leads us to one of the major theorems in differential geometry, Gauss' Theorema Egregium.

In lab 6, we defined two properties of a surface: the Gaussian curvature Κ(u,v), and the mean curvature H(u,v). The natural question to ask is whether these are extrinsic or intrinsic properties of a surface. Note that surfaces that have the same metric coefficients also share the same intrinsic proprties. Take, for example, the case of the catenoid X(u,v) = (cosh(v)cos(u), cosh(v)sin(u), v) and the helicoid Y(u,v) = (-sinh(v)sin(u), sinh(v)cos(u), -u) , which are two minimal surfaces that have the same metric coefficients. If we calculate the Gaussian and mean curvatures, we find that Κ(u,v) = -1/cosh⁴(v) and H(u,v) = 0 for both surfaces. Looking at this example by itself, we might be led to believe that Κ(u,v) and H(u,v) are both intrinsic quantities. A second example, however, proves that this is not a valid conclusion. Consider this time the case of a plane X(u,v) = (u, v, 0) and a circular cylinder Y(u,v) = (cos(u), sin(u), v). These two surfaces have the same metric coefficients and the same Gaussian curvature Κ(u,v) = 0. However, the mean curvature for the plane is H(u,v) = 0, while the mean curvature for the circular cylinder is H(u,v) = 1/2. This means that mean curvature does not depend solely on the metric coefficients and must, therefore, be an extrinsic property. The Gaussian curvature, on the other hand, is the same for both surfaces, which gives further evidence to support the claim that:

The Gaussian curvature Κ(u,v) is intrinsic and can be computed using the metric coefficients

The Christoffel Symbols

Consider a regular surface X(u,v). The partial derivative vectors and the normal vector form a basis for 3-space. Because of this, the second partial derivative vectors can be expressed as linear combinations of X_u(u,v), X_v(u,v), and N(u,v).

X_uu = Γ₁₁¹X_u + Γ₁₁²X_v + L₁₁N
X_uv = Γ₁₂¹X_u + Γ₁₂²X_v + L₁₂N
X_vv = Γ₂₂¹X_u + Γ₂₂²X_v + L₂₂N

X_ij = Γ_ij¹X₁ + Γ_ij²X₂ + L_ijN

Lemma: The Christoffel symbols are intrinsic, dependent on g_ij

Γ_{ij, k} · X_k = (1/2)(∂g_jk/∂uⁱ + ∂g_ik/∂u^j - ∂g_ij/∂u^k)

Γ_ij^mg_mk = Γ_{ij, k}

Γ₁₁¹ = (1/2g₁₁)(∂g₁₁/∂u)                               Γ₁₁² = (-1/2g₂₂)(∂g₁₁/∂v)
Γ₁₂¹ = (1/2g₁₁)(∂g₁₁/∂v)                               Γ₁₂² = (1/2g₂₂)(∂g₂₂/∂u)
Γ₂₂¹ = (-1/2g₁₁)(∂g₂₂/∂u)                              Γ₂₂² = (1/2g₂₂)(∂g₂₂/∂v)  

The Intrinsic Formula for Gaussian Curvature

Having defined the Christoffel symbols in terms of the metric coefficients, we derive the following intrinsic formula for Gaussian curvature:

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

Κ(u,v) = - (1/2√g)[∂/∂v(1/√g · (∂g₁₁/∂v)) + ∂/∂u(1/√g · (∂g₂₂/∂u))]

Exercises

1. Show that the two intrinsic formulas given for Gaussian curvature are equivalent.