Local flatness

Local flatness, in the topological branch of mathematics, is a property of a submanifold in a manifold of larger dimension. If a d-manifold N is embedded in an n-manifold M (where d < n), and if $x\in N,$ we say N is locally flat at x if there is a neighborhood U of x in M such that $(U,U\cap N)$ is homeomorphic to the pair $(R^{n},R^{d}).$ However, if M has boundary that contains N, we make a special definition: $(U,U\cap N)$ should be homeomorphic to $(R^{n+},R^{d}),$ where $R^{n+}=\{y\in R^{n}:y_{n}\geq 0\}$ and $R^{d})=\{y\in R^{n}:y_{n-d+1}=\cdots =y_{n}=0\}.$ (The first definition assumes that, if M has any boundary, x is not a boundary point of M.) We call N locally flat in M if every point of N is locally flat.

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).

References

Brown, Morton (1962), Locally flat imbeddings of topological manifolds. Annals of Mathematics, Second series, Vol. 75 (1962), pp. 331-341.