bifolds

Bifolds are a proposed class of geometric objects that generalize manifolds and orbifolds by incorporating a distinguished two-sheeted folding along a codimension-1 fold set. Informally, a bifold space locally resembles two smooth sheets of Euclidean space glued together along a common boundary.

Definition: A bifold structure on a space X consists of a closed subset F called the fold set, together with an atlas of charts modeled on the double of R^n across R^{n-1}. In neighborhoods outside F, points have charts homeomorphic to R^n. In neighborhoods of points on F, charts look like two copies of R^n glued along R^{n-1} via a reflection or a similar identification.

Relation to known objects: If the two sheets are identified trivially, the structure reduces to a smooth manifold. If a finite group action governs the local identifications, one obtains an orbifold-like object. A simple construction is to double a manifold with boundary along its boundary, producing a bifold whose fold set is precisely the boundary.

Maps and morphisms: A bifold map between bifolds preserves the fold set and is locally compatible with the two-sheet description, acting as a pair of smooth maps on the sheets that agree along F under the local identifications.

Invariants and structure: One may study topological invariants by treating the bifold as a stratified space with a codimension-1 singular locus F. Differential and metric structures can be defined piecewise on each sheet with compatibility along the fold.

Applications: The bifold framework has potential uses in modeling creased or folded surfaces, in computer graphics for folded textures, and in certain areas of singularity theory. It remains a developing concept with varying definitions in the literature.

Note: The term is not universally standardized in mainstream geometry, and definitions may differ among authors. See also manifold, orbifold, folded manifold, creased surface.