A local corank-one singularity of a smooth map between manifolds in which the map, in suitable local coordinates, has the standard quadratic normal form in one transverse direction so that a hypersurface is mapped with a crease or fold; classically called the Whitney fold.

Principle

A fold singularity is the generic stable obstruction of corank one: the failure of a map to be an immersion is confined to a single transverse direction and is modeled by a nondegenerate quadratic vanishing, giving a canonical normal form under coordinate changes.

Demonstration

Consider the smooth map F: R^2 -> R^2 defined by F(x,y) = (x, y^2). The line y = 0 is the locus of nonimmersive points; near the origin the image folds along the x-axis and the singularity is a Fold Singularity in the Whitney sense.

Misapplication

Calling any crease, sharp corner, cusp, or higher-corank degeneracy a fold; for example, labeling a cusp singularity or a branch point as a Fold Singularity ignores the required corank-one quadratic normal form.

Consequence

Recognizing a fold gives a stable local classification, predicts two-sheeted behaviour on one side of the fold and single-sheetedness on the other, and enables local normal forms and perturbation analyses that persist under small smooth deformations.

Reversal

The reversal is a higher-order or higher-corank singularity such as a cusp or a swallowtail, where the transverse vanishing is of higher order or involves more than one direction; unlike a fold, those cannot be put into the quadratic corank-one normal form.

Boundary

Applies to smooth maps between differentiable manifolds (or manifolds with boundary) and to isolated corank-one loci; excludes isolated transverse double points, nonisolated accumulations of singularities, and singularities that require higher corank or more complicated jet data.

Semantic Tension

Differs from informal uses of 'fold' in geometric modeling or origami: the analytic Fold Singularity is a precise local normal form, while everyday 'fold' may denote any creasing or sharp feature independent of corank or quadratic behaviour.

Synthesis

A Fold Singularity is the canonical, generically stable corank-one crease of a smooth map characterized by a quadratic normal form in one transverse coordinate; it organizes local two-sheet vs one-sheet behaviour and is distinguished from cusps and higher-corank failures.