the shape operator
Stand on a curved surface in space and watch the unit normal vector — the arrow pointing straight out — as you walk in some tangent direction. On a flat plane the normal never tilts; on a sphere it swings the more sharply curved the sphere is. The shape operator is the precise machine that records how fast and in which direction the normal turns, repackaging the same data as the second fundamental form into a tidy linear map you can diagonalize.
For a hypersurface M in (N, g-bar) with a chosen unit normal field nu, the shape operator S (also called the Weingarten map) is the tangent-space endomorphism S(X) = -nabla-bar_X nu, the tangential rate of change of the normal in direction X. It is self-adjoint with respect to the induced metric, and it encodes the second fundamental form by h(X, Y) = g(S(X), Y). Being self-adjoint, S has real eigenvalues kappa_1, ..., kappa_(n-1) (the principal curvatures) and an orthonormal eigenbasis (the principal directions). Its trace is the mean curvature H = kappa_1 + ... + kappa_(n-1) (some authors average), and its determinant is the Gauss-Kronecker curvature (in 2D, the intrinsic Gaussian curvature K = kappa_1 kappa_2).
Where it lives: principal curvatures, mean curvature, and the classification of surface points (elliptic, hyperbolic, parabolic) all read off the shape operator's spectrum. Honesty: the shape operator is extrinsic and depends on a choice of normal orientation — flip nu and S flips sign (so mean curvature changes sign, but the Gaussian curvature det S in 2D does not, since the dimension-2 determinant is unchanged). It is the same content as the second fundamental form, just viewed as an operator rather than a bilinear form; do not treat them as independent data.
On a sphere of radius a with the outward normal, the normal moves exactly parallel to your motion, so S = (1/a) times the identity: both principal curvatures equal 1/a, mean curvature 2/a, and Gaussian curvature det S = 1/a^2. On a cylinder of radius r the principal curvatures are 1/r (around) and 0 (along), so det S = 0 — extrinsically curved, intrinsically flat.
The shape operator's eigenvalues are the principal curvatures; its determinant in 2D is the Gaussian curvature.
The shape operator depends on the choice of unit normal: reversing nu reverses S (and the sign of mean curvature). It carries exactly the same information as the second fundamental form — they are two views of one object, not two pieces of data.