Local flatness
In topology, a branch of mathematics, local flatness is smoothness condition that can be imposed on topological submanifolds. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Violations of local flatness describe ridge networks and crumpled structures, with applications to materials processing and mechanical engineering. Definition. Suppose a "d" dimensional manifold "N" is embedded into an "n" dimensional manifold "M" (where "d" < "n"). If formula_1 we say "N" is locally flat at "x" if there is a neighborhood formula_2 of "x" such  that the topological pair formula_3 is homeomorphic to the pair formula_4, with the standard inclusion of formula_5. That is, there exists a homeomorphism formula_6 such that the image of formula_7 coincides with formula_8. In diagrammatic terms, the following square must commute: We call "N" locally flat in "M" if "N" is locally flat at every point. Similarly, a map formula_9 is called locally flat, even if it is not an embedding, if every "x" in "N" has a neighborhood "U" whose image formula_10 is locally flat in "M". In manifolds with boundary. The above definition assumes that, if "M" has  a boundary, "x" is not a boundary point of "M". If "x" is a point on the boundary of "M" then the definition is modified as follows. We say that "N" is locally flat at a boundary point "x" of "M" if there is a neighborhood formula_11 of "x" such that the topological pair formula_3 is homeomorphic to the pair formula_13, where formula_14 is a standard half-space and formula_8 is included as a standard subspace of its boundary. Consequences. Local flatness of an embedding implies strong properties not shared by all embeddings. Brown (1962) proved that if "d" =  n − 1, then "N" is collared; that is, it has a neighborhood which is homeomorphic to "N" × [0,1] with "N" itself corresponding to "N" × 1/2 (if "N" is in the interior of "M") or "N" × 0 (if "N" is in the boundary of "M").