The setup: splitting the ambient connection along a submanifold
We finally come full circle. Volume I studied surfaces sitting in R^3 with their first and second fundamental forms; the previous four guides built the intrinsic machine — metric, Levi-Civita connection, geodesics, the Riemann curvature tensor — entirely from the inside, with no ambient space at all. This guide reconnects the two viewpoints. The stage is a Riemannian manifold (M-bar, g-bar) — read 'bar' as the ambient or surrounding space — together with an embedded submanifold M sitting inside it, where M inherits its metric g by simply restricting g-bar to vectors tangent to M. The question is the same one Gauss asked for surfaces: how much of M's geometry is intrinsic, and how much records the way M curves inside M-bar?
Here is the one idea everything rests on. At each point p of M the ambient tangent space T_p(M-bar) splits orthogonally into the tangent space T_p M and the normal space (the orthogonal complement) N_p M — every ambient vector is a tangent part plus a normal part. Now take two vector fields X, Y tangent to M and form the ambient covariant derivative nabla-bar_X Y. There is no reason this should stay tangent to M, and in general it does not. Decompose it: its tangent component is, by a short check of the two defining axioms, exactly the intrinsic Levi-Civita connection nabla_X Y of M itself, while its normal component is a brand-new object. That normal leftover is the whole story of how M bends.
ambient covariant derivative, split into tangent + normal parts:
nabla-bar_X Y = nabla_X Y + II(X, Y)
(tangent) (normal)
nabla_X Y = ( nabla-bar_X Y )^T = intrinsic Levi-Civita connection of M
II(X, Y) = ( nabla-bar_X Y )^N = second fundamental form (vector-valued, normal)
This is the Gauss formula. ^T = tangent part, ^N = normal part.The second fundamental form and the shape operator
That normal leftover, II(X, Y) = (nabla-bar_X Y)^N, is the second fundamental form of M in M-bar. Three of its properties make it usable. It is tensorial — II(X, Y) at p depends only on the values of X and Y at p, not on how they extend, because the offending derivative terms live in the tangent part and get subtracted away. It is symmetric, II(X, Y) = II(Y, X), a direct consequence of torsion-freeness of nabla-bar (the bracket [X, Y] is tangent to M, so it has no normal part to spoil symmetry). And it is normal-vector-valued: for a hypersurface (codimension one) you can pick a unit normal nu and read II off as a scalar-valued form, II(X, Y) = h(X, Y) nu, recovering exactly the second fundamental form you computed for surfaces in Volume I.
There is a dual way to package the same data that is often more convenient: the shape operator (or Weingarten map). Fix a unit normal nu and differentiate it: -nabla-bar_X nu, projected back to the tangent space, defines a linear map S_nu(X) on T_p M. The shape operator and the second fundamental form are two faces of one object, tied by g(S_nu(X), Y) = h(X, Y) — pairing the operator with the metric gives the form. Because h is symmetric, the shape operator S_nu is self-adjoint with respect to g, so the spectral theorem applies: at each point it has real eigenvalues k_1, ..., k_n. These are the principal curvatures, their eigenvectors the principal directions, and their symmetric functions are the curvatures you already know — the average is the mean curvature H, the product is (for surfaces) the Gaussian curvature.
Gauss & Codazzi: how extrinsic bending controls intrinsic curvature
Now the payoff. We have two curvature tensors in play — the ambient R-bar of M-bar and the intrinsic R of M — and one bending tensor II linking them. The Gauss equation is the exact bookkeeping that relates them: it says R-bar and R differ by a precise quadratic expression in II. In its sectional form, for orthonormal tangent vectors X, Y spanning a plane, K(X,Y) = K-bar(X,Y) + [h(X,X) h(Y,Y) - h(X,Y)^2], where K is M's intrinsic sectional curvature, K-bar is the ambient sectional curvature along the same plane, and h is the scalar second fundamental form. Reading it the other way, M's own intrinsic curvature is the part of the ambient curvature it inherits, corrected by how it bends. This is the same accounting Gauss discovered for surfaces, now stated for any submanifold of any Riemannian curvature background.
Gauss does not act alone. Its partner is the Codazzi equation, which constrains how the second fundamental form varies from point to point: in a flat ambient it says (nabla_X II)(Y, Z) = (nabla_Y II)(X, Z), i.e. the covariant derivative of II is totally symmetric in its three slots, and in a curved ambient the mismatch is exactly the normal component of R-bar. The pair has a clean division of labor: Gauss converts bending into intrinsic curvature, while Codazzi is the integrability condition tying the rate of change of the bending to the ambient curvature. Together they are the full set of compatibility relations any honest second fundamental form must obey — a fact we cash in at the end of this guide.
The Gauss equation has a famous corollary. Take M a surface in flat M-bar = R^3. Then K-bar = 0, the ambient term vanishes, and the formula collapses to K = h(X,X)h(Y,Y) - h(X,Y)^2, which is exactly the product of principal curvatures k_1 k_2 — the determinant of the shape operator. But k_1 k_2 was defined extrinsically, through the embedding in space, whereas the left side K is purely intrinsic, computable from the metric alone. That an extrinsically-defined quantity turns out to be intrinsic is Theorema Egregium, Gauss's remarkable theorem from Volume I, here falling out as a one-line special case. Its consequence is unchanged and vivid: you cannot flatten a sphere onto a plane without distorting distances, because K = 1 on the sphere and K = 0 on the plane, and bending alone can never change K.
A worked example: the round sphere as a hypersurface
Nothing makes the formulas concrete like running them on the sphere S^n of radius r sitting in R^(n+1). Here every step is short, and the answer is exactly the famous one — that is the point of doing it. Follow the four moves below and you will have computed the second fundamental form, the shape operator, the principal curvatures, and the intrinsic sectional curvature of the sphere, the last via the Gauss equation rather than by grinding Christoffel symbols.
- Pick the outward unit normal at a point p on S^n of radius r: it is nu = p/r, the position vector rescaled to unit length. The ambient connection on R^(n+1) is just ordinary directional differentiation of components, with all Gamma-bar = 0.
- Compute the shape operator: S_nu(X) = -(nabla-bar_X nu)^T = -(nabla-bar_X (p/r)) = -(1/r) X, since differentiating the position field p in direction X just returns X. So S_nu = -(1/r) times the identity on every tangent vector.
- Read off the principal curvatures: S_nu is a scalar multiple of the identity, so every direction is principal and all principal curvatures equal -1/r (a point with all k_i equal is an umbilic point — the sphere is totally umbilic, the same at every point in every direction). The scalar second fundamental form is h(X,Y) = -(1/r) g(X,Y).
- Feed this into the Gauss equation with flat ambient K-bar = 0: for orthonormal X, Y the numerator is h(X,X)h(Y,Y) - h(X,Y)^2 = (1/r^2)(1)(1) - 0 = 1/r^2, so K = 1/r^2. The sphere of radius r has constant sectional curvature 1/r^2 — positive, shrinking as the sphere grows, exactly as intuition demands.
Space forms: the three constant-curvature models
The sphere computation hands us the first of the three most important Riemannian manifolds of all. A space form is a complete, simply connected Riemannian manifold of constant sectional curvature — meaning K is the same number at every point and in every two-plane. A foundational classification says there are exactly three, one for each sign of K, and every one of them is forced to be one of these models up to scaling and isometry. The space forms are the curved geometries Volume I introduced axiomatically, now realized as genuine Riemannian manifolds with a metric you can write down.
There is one model for each sign of K. For K > 0 it is the round sphere S^n(r) with K = 1/r^2 (spherical geometry); for K = 0 it is flat Euclidean space R^n (flat geometry); for K < 0 it is hyperbolic space H^n with K = -1/r^2 (hyperbolic geometry). Up to scaling we may normalize to K = +1, 0, -1. Strikingly, all three carry a single metric written in geodesic polar coordinates, ds^2 = dr^2 + f(r)^2 d(omega)^2 where d(omega)^2 is the round metric on the unit sphere of directions; only the radial profile f(r) changes — f(r) = sin r for K = +1, f(r) = r for K = 0, and f(r) = sinh r for K = -1. One template, three trigonometric flavors.
Three honest remarks keep this from becoming a slogan. First, the hypotheses are load-bearing: 'complete and simply connected' is what pins the model down. Drop simple connectivity and you get the same local geometry on many different global shapes — a flat torus and flat R^2 both have K = 0 but are not isometric, and a flat Klein bottle has K = 0 too; the constant-curvature CONDITION is local, the classification of the SPACE form is global. Second, the metrics above are written in geodesic polar coordinates, and the single profile function f(r) — sin r, r, sinh r — is precisely the magnitude of a Jacobi field solving f'' + K f = 0, which is why these three trigonometric flavors recur everywhere in comparison geometry. Third, conventions: many books normalize to K = +1, 0, -1 and others keep a radius r, so a formula quoted without its normalization is ambiguous.
Why this matters: the fundamental theorem and the road ahead
Gauss and Codazzi are not merely consequences of an embedding — together they are its complete blueprint. The fundamental theorem of submanifold theory says the first fundamental form (the metric) and the second fundamental form (the bending), provided they satisfy the Gauss and Codazzi equations as compatibility conditions, determine the submanifold uniquely up to a rigid motion of the ambient space. This is the higher-dimensional sibling of Volume I's fundamental theorem of surfaces and the theorema egregium circle: intrinsic data plus bending data, constrained by Gauss-Codazzi, is exactly enough to reconstruct the shape. Curvature is not an afterthought computed from a surface; the curvature equations ARE the integrability conditions that let a surface exist.
The space forms you just met are the silent reference points of the entire rung's sequel. Every comparison theorem of Riemannian geometry — Bonnet-Myers (Ricci bounded below by a positive constant forces a compact manifold with bounded diameter), Cartan-Hadamard (nonpositive curvature makes the exponential map a covering, so the universal cover is R^n), the sphere theorem (pinched positive curvature forces a topological sphere), Rauch and Toponogov comparison — measures an unknown manifold against the sphere, Euclidean space, or hyperbolic space. The constant-curvature models are the rulers; the whole comparison-geometry industry is the art of bounding a general manifold between two of these rulers and reading off topology. That is exactly where this rung's study of curvature was always heading.