
In topology, a branch of mathematics, local flatness is a property of a submanifold in a topological manifold of larger dimension. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' < ''n''). If $x\; \backslash in\; N,$ we say ''N'' is locally flat at ''x'' if there is a neighborhood $U\; \backslash subset\; M$ of ''x'' such that the topological pair $(U,\; U\backslash cap\; N)$ is homeomorphic to the pair $(\backslash mathbb^n,\backslash mathbb^d)$, with a standard inclusion of $\backslash mathbb^d$ as a subspace of $\backslash mathbb^n$. That is, there exists a homeomorphism $U\backslash to\; R^n$ such that the image of $U\backslash cap\; N$ coincides with $\backslash mathbb^d$. 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 $U\backslash subset\; M$ of ''x'' such that the topological pair $(U,\; U\backslash cap\; N)$ is homeomorphic to the pair $(\backslash mathbb^n\_+,\backslash mathbb^d)$, where $\backslash mathbb^n\_+$ is a standard halfspace and $\backslash mathbb^d$ is included as a standard subspace of its boundary. In more detail, we can set $\backslash mathbb^n\_+\; =\; \backslash $ and $\backslash mathbb^d\; =\; \backslash =\backslash cdots=y\_n=0\backslash \}$. We call ''N'' locally flat in ''M'' if ''N'' is locally flat at every point. Similarly, a map $\backslash chi\backslash colon\; N\backslash to\; M$ is called locally flat, even if it is not an embedding, if every ''x'' in ''N'' has a neighborhood ''U'' whose image $\backslash chi(U)$ is locally flat in ''M''. 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'' × () 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''). ==See also== *Neat submanifold 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「local flatness」の詳細全文を読む スポンサード リンク
