the Gauss-Codazzi equations
/ GOWSS koh-DAHT-see /
A submanifold has two layers of geometry: the curvature it inherits from sitting in the ambient space, and the curvature of the ambient space itself. The Gauss-Codazzi equations are the exact compatibility relations between them — the rules that say how a submanifold's intrinsic curvature, the ambient curvature, and the second fundamental form must fit together. They are the higher-dimensional, all-encompassing generalization of Gauss's surface theory.
Let M be a submanifold of (N, g-bar) with second fundamental form II and shape operator S. Decomposing the ambient Riemann tensor into tangential and normal parts gives three identities. The Gauss equation expresses M's intrinsic curvature: R(X,Y,Z,W) = R-bar(X,Y,Z,W) + g(II(X,W), II(Y,Z)) - g(II(X,Z), II(Y,W)) — intrinsic curvature equals ambient curvature plus a quadratic correction in II. The Codazzi(-Mainardi) equation governs the normal component, relating the covariant derivative of II to the normal part of the ambient curvature: (nabla_X II)(Y,Z) - (nabla_Y II)(X,Z) equals the normal projection of R-bar(X,Y)Z. The Ricci equation handles the curvature of the normal bundle. Together they package all the ways intrinsic and extrinsic geometry constrain each other.
Why it is central: for a surface in flat R^3 the Gauss equation reduces to K = det(S) = kappa_1 kappa_2 — the Gaussian curvature equals the product of principal curvatures — which is Gauss's theorema egregium, the astonishing fact that a quantity defined extrinsically (via II) is actually intrinsic. The fundamental theorem of surface theory says, conversely, that a first and second fundamental form satisfying Gauss-Codazzi locally determine a surface in R^3 up to rigid motion. Honest caveat: in a curved ambient space the ambient term R-bar does not vanish, so 'K = det S' is special to flat backgrounds; the full Gauss equation always carries the ambient curvature term.
For a surface in flat R^3, R-bar = 0, so the Gauss equation collapses to K = kappa_1 kappa_2. A sphere of radius a has kappa_1 = kappa_2 = 1/a, giving K = 1/a^2; a saddle has principal curvatures of opposite sign, giving K < 0. The point: K is computed here from extrinsic data (principal curvatures) yet is an intrinsic invariant — exactly theorema egregium.
In flat R^3 the Gauss equation is theorema egregium, K = kappa_1 kappa_2: extrinsic data yields an intrinsic invariant.
The clean K = det(S) is special to FLAT ambient spaces; in a curved background the Gauss equation always includes the ambient curvature term R-bar. Don't quote 'K equals product of principal curvatures' without checking the ambient space is flat.