View source for Bullough–Dodd model

In [[topology]], a branch of [[mathematics]], '''local flatness''' is a property of a [[submanifold]] in a [[topological manifold]] of larger [[dimension]]. In the [[Category (mathematics)|category]] of topological manifolds, locally flat submanifolds play a role similar to that of [[Submanifold#Embedded submanifolds|embedded submanifolds]] in the category of [[smooth manifolds]].

Suppose a ''d'' dimensional manifold ''N'' is embedded into an ''n'' dimensional manifold ''M'' (where ''d'' &lt; ''n'').  If <math>x \in N,</math> we say ''N'' is '''locally flat''' at ''x'' if there is a neighborhood <math> U \subset M</math> of ''x'' such that the [[topological pair]] <math>(U, U\cap N)</math> is [[homeomorphic]] to the pair <math>(\mathbb{R}^n,\mathbb{R}^d)</math>, with a standard inclusion of <math>\mathbb{R}^d</math> as a subspace of <math>\mathbb{R}^n</math>. That is, there exists a homeomorphism <math>U\to R^n</math> such that the [[image (mathematics)|image]] of <math>U\cap N</math> coincides with <math>\mathbb{R}^d</math>.

The above definition assumes that, if ''M'' has a [[Boundary (topology)|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 <math>U\subset M</math> of ''x'' such that the topological pair <math>(U, U\cap N)</math> is homeomorphic to the pair <math>(\mathbb{R}^n_+,\mathbb{R}^d)</math>, where <math>\mathbb{R}^n_+</math> is a standard [[Half-space (geometry)|half-space]] and <math>\mathbb{R}^d</math> is included as a standard subspace of its boundary. In more detail, we can set 
<math>\mathbb{R}^n_+ = \{y \in \mathbb{R}^n\colon y_n \ge 0\}</math> and <math>\mathbb{R}^d = \{y \in \mathbb{R}^n\colon y_{d+1}=\cdots=y_n=0\}</math>.

We call ''N'' '''locally flat''' in ''M'' if ''N'' is locally flat at every point. Similarly, a map <math>\chi\colon N\to M</math> is called '''locally flat''', even if it is not an embedding, if every ''x'' in ''N'' has a neighborhood ''U'' whose image <math>\chi(U)</math> 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'' &minus; 1, then ''N'' is [[collared]]; that is, it has a neighborhood which is homeomorphic to ''N'' &times; [0,1] with ''N'' itself corresponding to ''N'' &times; 1/2 (if ''N'' is in the interior of ''M'') or ''N'' &times; 0 (if ''N'' is in the boundary of ''M'').

==See also==
*[[Neat submanifold]]

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

[[Category:Topology]]
[[Category:Geometric topology]]

{{topology-stub}}