In topology, a branch of mathematics, local flatness is a 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.
Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n). If
x\inN,
U\subsetM
(U,U\capN)
(Rn,Rd)
Rd\toRn.
U\toRn
U\capN
Rd
We call N locally flat in M if N is locally flat at every point. Similarly, a map
\chi\colonN\toM
\chi(U)
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
U\subsetM
(U,U\capN)
d) | |
(R | |
+,R |
n | |
R | |
+ |
Rd
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).