Local flatness

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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 $(U, U\cap N)$ 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 $R^{n+} = \{y \in R^n: y_n \ge 0\}$ 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).