Embedding and knotting of flat compact surfaces in 3-space

Abstract

In 3-space, any compact surface with nonempty boundary is isotopic to a flat (zero Gaussian curvature) surface and two such flat surfaces are isotopic through flat surfaces if and only if they are isotopic through ordinary surfaces. Hereby the isotopy classes of flat surfaces are in one-to-one correspondence with the isotopy classes of ordinary surfaces which have no constraint in their curvature. Applied to Seifert surfaces we get: Any simple closed space curve can be deformed until it bounds a flat orientable surface.