Local flatness

In topology, a branch of mathematics, local flatness is a property of a submanifold in a topological manifold of larger dimension.

Suppose a "d" dimensional manifold "N" is embedded in an "n" dimensional manifold "M" (where "d" < "n"). If $x\; in\; N,$ we say "N" is locally flat at "x" if there is a neighborhood $U\; subset\; M$ of "x" such that $(U,\; Ucap\; N)$ is homeomorphic to the pair $(R^n,R^d)$. However, if "M" has boundary that contains "N", we make a special definition: $(U,\; Ucap\; N)$ should be homeomorphic to $(R^\{n+\},R^d),$ where $R^\{n+\}\; =\; \{y\; in\; R^n:\; y\_n\; ge\; 0\}$ 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.