# Local flatness

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{{Short description|Property of topological submanifolds}}
In [topology](/source/topology), a branch of [mathematics](/source/mathematics), '''local flatness''' is a smoothness condition that can be imposed on topological [submanifold](/source/submanifold)s. In the [category](/source/Category_(mathematics)) of topological manifolds, locally flat submanifolds play a role similar to that of [embedded submanifolds](/source/Submanifold) in the category of [smooth manifolds](/source/smooth_manifolds). Violations of local flatness describe ridge networks and [crumpled structures](/source/Crumpling), with applications to materials processing and [mechanical engineering](/source/mechanical_engineering).

== Definition ==
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](/source/topological_pair) <math>(U, U\cap N)</math> is [homeomorphic](/source/homeomorphic) to the pair <math>(\mathbb{R}^n,\mathbb{R}^d)</math>, with the standard inclusion of <math>\mathbb{R}^d\to\mathbb{R}^n.</math> That is, there exists a homeomorphism <math>U\to \mathbb{R}^n</math> such that the [image](/source/image_(mathematics)) of <math>U\cap N</math> coincides with <math>\mathbb{R}^d</math>.  In diagrammatic terms, the following [square must commute](/source/Commuting_square):
alt=Commutative diagram: {{math|''U''&amp;cap;''N''}} has a monomorphism to {{mvar|U}}, both of which have isomorphisms to <math>\mathbb{R}^d</math> and <math>\mathbb{R}^n</math> (respectively), and <math>\mathbb{R}^d</math> has a monomorphism to <math>\mathbb{R}^n.</math>|center|frameless|upright

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''.

=== In manifolds with boundary ===
The above definition assumes that, if ''M'' has a [boundary](/source/Boundary_(topology)), ''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](/source/Half-space_(geometry)) and <math>\mathbb{R}^d</math> is included as a standard subspace of its boundary.

== Consequences ==
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'' × [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'').

== Non-example ==
{{See also|Slice_knot#Cone_construction}}
Let <math>K</math> be a non-trivial knot in <math>S^3</math>; that is, a connected, locally flat one-dimensional submanifold of <math>S^3</math> such that the pair <math>(S^3, K)</math> is not homeomorphic to <math>(S^3, S^1)</math>. Then the cone on <math>K</math> from the center <math>\underline{0}</math> of <math>D^4</math> is a submanifold of <math>D^4</math>, but it is not locally flat at <math>\underline{0}</math>.<ref>{{citation|title=Differential and Low-Dimensional Topology|author=András Juhász|page=3}}</ref>

==See also==
*[Euclidean space](/source/Euclidean_space)
*[Neat submanifold](/source/Neat_submanifold)

==References==
{{Reflist}}
* [Brown, Morton](/source/Morton_Brown) (1962), Locally flat imbeddings{{sic}} of topological manifolds. ''Annals of Mathematics'', Second series, Vol. 75 (1962), pp.&nbsp;331–341.
* [Mazur, Barry](/source/Barry_Mazur). On embeddings of spheres. ''Bulletin of the American Mathematical Society'', Vol. 65 (1959), no. 2, pp.&nbsp;59–65. http://projecteuclid.org/euclid.bams/1183523034.

Category:Topology
Category:Geometric topology

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