Local flatness

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.

Definition

Suppose a d dimensional manifold N is embedded into an n dimensional manifold M. If we say N is locally flat at x if there is a neighborhood of x such that the topological pair is homeomorphic to the pair, with the standard inclusion of That is, there exists a homeomorphism such that the image of coincides with. 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 is called locally flat, even if it is not an embedding, if every x in N has a neighborhood U whose image 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 of x such that the topological pair is homeomorphic to the pair, where is a standard half-space and is included as a standard subspace of its boundary.

Consequences

Local flatness of an embedding implies strong properties not shared by all embeddings. Brown proved that if d = n − 1, then N is collared; that is, it has a neighborhood which is homeomorphic to N × with N itself corresponding to N × 1/2 or N × 0.

Non-example

Let be a non-trivial knot in ; that is, a connected, locally flat one-dimensional submanifold of such that the pair is not homeomorphic to. Then the cone on from the center of is a submanifold of, but it is not locally flat at.