Hypersurfaces

A hypersurface is a submanifold of codimension one in an ambient manifold. In Euclidean space R^n, a hypersurface is an (n−1)-dimensional submanifold, equivalently the regular level set of a smooth function f: R^n → R with ∇f ≠ 0 on the set.

Locally every point on a hypersurface has a neighborhood diffeomorphic to an open set in R^{n−1}. The tangent space at a point is an (n−1)-dimensional subspace of R^n, and a normal vector provides a transversal direction. Hypersurfaces can be oriented if a global normal field exists.

Common examples include planes and spheres in R^3, and more generally graphs of functions or level sets defined by a single equation. They carry curvature data: the second fundamental form, principal curvatures, mean curvature, and Gaussian curvature in appropriate dimensions. Minimal hypersurfaces are those with zero mean curvature.

In algebraic geometry, a hypersurface is defined by a single polynomial equation in affine or projective space; a smooth hypersurface has a nonvanishing gradient on its points. In complex geometry, a hypersurface in a complex manifold has real codimension two and interacts with divisors and line bundles.

Applications appear in geometry processing, general relativity (as spacelike or null slices), and differential topology. Construction methods include level sets of smooth functions, graphs of functions, and implicit equations, as well as more global techniques for embedding and deformation.