The Gauss equation and intrinsic curvature

Try

A plane and a cylinder have the same local metric but different normal bending. Their Gaussian curvature is nevertheless the same: zero. A sphere patch has K=1/R² and cannot be unrolled isometrically onto a plane patch.

Theorem Egregium explains this distinction. Gaussian curvature, although introduced as the determinant of the extrinsic shape operator, is determined entirely by the intrinsic metric. Individual principal curvatures and mean curvature need not be.

Predict

Predict which information should enter intrinsic curvature: the metric connection and its derivatives, not a selected ambient normal. Reversing the normal changes S to −S but leaves its determinant unchanged in two dimensions.

Fix the convention R(X,Y)Z=∇_X∇_YZ−∇_Y∇_XZ−∇_[X,Y]Z. With this convention K=⟨R(e₁,e₂)e₂,e₁⟩ in an orthonormal tangent basis. Sign conventions must be stated before comparing formulas.

Experiment

Use a conformal metric g=e^(2φ)(dx²+dy²).

φ	Curvature formula outcome
Constant	K=0
x	K=0
−log y, y>0	K=−1

The formula is K=−e^(−2φ)(φ_xx+φ_yy). For the last row, the Laplacian of φ is 1/y² and e^(−2φ)=y². This verifies the upper-half-plane metric’s intrinsic curvature independently of its visual geodesics.

Understand

The ambient derivative splits as D_XY=∇_XY+b(X,Y)N, while D_XN=−SX. Euclidean ambient space is flat: D_XD_YZ−D_YD_XZ−D_[X,Y]Z=0. Expand and take the tangent component. The connection terms give R(X,Y)Z; differentiating the normal terms contributes −b(Y,Z)SX+b(X,Z)SY. Therefore R(X,Y)Z=b(Y,Z)SX−b(X,Z)SY. This is the Gauss equation.

Evaluate with X=e₁,Y=e₂,Z=e₂ and pair with e₁. The result is b₂₂b₁₁−b₁₂²=det S=K. Since the connection is computed from g, its curvature tensor and this scalar depend only on g. Local isometries preserve them. This proves the intrinsic nature of K.

For the conformal formula, substitute g_ij=e^(2φ)δ_ij into the Christoffel expression. The nonzero pattern is Γ¹_11=φ_x, Γ¹_12=φ_y, Γ¹_22=−φ_x and the corresponding swapped-index terms Γ²_11=−φ_y, Γ²_12=φ_x, Γ²_22=φ_y. In the curvature combination the first-derivative products cancel, leaving the displayed Laplacian formula.

Apply

Repair “Equal K implies a rigid motion.” K is one invariant, not the whole metric or second form. Plane and cylinder already refute global rigid equivalence. A local isometry obstruction works in the opposite direction: differing K values rule out the map, but matching values alone do not construct it.

For optional depth, curvature in higher dimensions is not captured by one scalar determinant. The full tensor carries sectional curvatures for different tangent planes. Here the tangent plane is two-dimensional, so one Gaussian curvature determines the algebraic curvature tensor.

The proof’s ambient calculation used an embedded patch, while the final metric formula also defines K on abstract Riemannian surfaces.

Continue

Track the curvature convention and evaluate it on a sphere or hyperbolic metric as a sign check. Continue to Codazzi and surface reconstruction to identify the remaining compatibility condition for prescribed metric and bending data.