##  [Fold Singularity](/fold-singularity-0) 

 Definition

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

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

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

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

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

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

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

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

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.