Local flatness

<h2 id="definition">Definition</h2>
Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n). If 
 
 
 
 x
 ∈
 N
 ,
 
 
 {\displaystyle x\in N,}
 
 we say N is locally flat at x if there is a neighborhood 
 
 
 
 U
 ⊂
 M
 
 
 {\displaystyle U\subset M}
 
 of x such that the <a href="/facts/Topological_pair/npyJshKf">topological pair</a> 
 
 
 
 (
 U
 ,
 U
 ∩
 N
 )
 
 
 {\displaystyle (U,U\cap N)}
 
 is <a href="/facts/Homeomorphic/jMfWg3Mn">homeomorphic</a> to the pair 
 
 
 
 (
 
 
 R
 
 
 n
 
 
 ,
 
 
 R
 
 
 d
 
 
 )
 
 
 {\displaystyle (\mathbb {R} ^{n},\mathbb {R} ^{d})}
 
, with the standard inclusion of 
 
 
 
 
 
 R
 
 
 d
 
 
 →
 
 
 R
 
 
 n
 
 
 .
 
 
 {\displaystyle \mathbb {R} ^{d}\to \mathbb {R} ^{n}.}
 
 That is, there exists a homeomorphism 
 
 
 
 U
 →
 
 
 R
 
 
 n
 
 
 
 
 {\displaystyle U\to \mathbb {R} ^{n}}
 
 such that the <a href="/facts/Image_(mathematics)/KANefMAl">image</a> of 
 
 
 
 U
 ∩
 N
 
 
 {\displaystyle U\cap N}
 
 coincides with 
 
 
 
 
 
 R
 
 
 d
 
 
 
 
 {\displaystyle \mathbb {R} ^{d}}
 
. In diagrammatic terms, the following <a href="/facts/Commuting_square/bdG8xuer">square must commute</a>:

We call N locally flat in M if N is locally flat at every point. Similarly, a map 
 
 
 
 χ
 :
 N
 →
 M
 
 
 {\displaystyle \chi \colon N\to M}
 
 is called locally flat, even if it is not an embedding, if every x in N has a neighborhood U whose image 
 
 
 
 χ
 (
 U
 )
 
 
 {\displaystyle \chi (U)}
 
 is locally flat in M.

<h3>In manifolds with boundary</h3>
The above definition assumes that, if M has a <a href="/facts/Boundary_(topology)/WgdH18kp">boundary</a>, 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
 ⊂
 M
 
 
 {\displaystyle U\subset M}
 
 of x such that the topological pair 
 
 
 
 (
 U
 ,
 U
 ∩
 N
 )
 
 
 {\displaystyle (U,U\cap N)}
 
 is homeomorphic to the pair 
 
 
 
 (
 
 
 R
 
 
 +
 
 
 n
 
 
 ,
 
 
 R
 
 
 d
 
 
 )
 
 
 {\displaystyle (\mathbb {R} _{+}^{n},\mathbb {R} ^{d})}
 
, where 
 
 
 
 
 
 R
 
 
 +
 
 
 n
 
 
 
 
 {\displaystyle \mathbb {R} _{+}^{n}}
 
 is a standard <a href="/facts/Half-space_(geometry)/AGknBgnw">half-space</a> and 
 
 
 
 
 
 R
 
 
 d
 
 
 
 
 {\displaystyle \mathbb {R} ^{d}}
 
 is included as a standard subspace of its boundary.

<h2 id="consequences">Consequences</h2>
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).

<h2 id="non-example">Non-example</h2>
See also: <a href="/facts/Slice_knot/2IUXAXhc">Slice_knot § Cone_construction</a>
Let 
 
 
 
 K
 
 
 {\displaystyle K}
 
 be a non-trivial knot in 
 
 
 
 
 S
 
 3
 
 
 
 
 {\displaystyle S^{3}}
 
; that is, a connected, locally flat one-dimensional submanifold of 
 
 
 
 
 S
 
 3
 
 
 
 
 {\displaystyle S^{3}}
 
 such that the pair 
 
 
 
 (
 
 S
 
 3
 
 
 ,
 K
 )
 
 
 {\displaystyle (S^{3},K)}
 
 is not homeomorphic to 
 
 
 
 (
 
 S
 
 3
 
 
 ,
 
 S
 
 1
 
 
 )
 
 
 {\displaystyle (S^{3},S^{1})}
 
. Then the cone on 
 
 
 
 K
 
 
 {\displaystyle K}
 
 from the center 
 
 
 
 
 
 0
 _
 
 
 
 
 {\displaystyle {\underline {0}}}
 
 of 
 
 
 
 
 D
 
 4
 
 
 
 
 {\displaystyle D^{4}}
 
 is a submanifold of 
 
 
 
 
 D
 
 4
 
 
 
 
 {\displaystyle D^{4}}
 
, but it is not locally flat at 
 
 
 
 
 
 0
 _
 
 
 
 
 {\displaystyle {\underline {0}}}
 
.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a></li>
<li><a href="/facts/Neat_submanifold/57IUarVn">Neat submanifold</a></li></ul>

<ul><li>Brown, Morton (1962), Locally flat imbeddings [<a href="/facts/Sic/qB9MUUWF">sic</a>] of topological manifolds. Annals of Mathematics, Second series, Vol. 75 (1962), pp. 331–341.</li>
<li>Mazur, Barry. On embeddings of spheres. Bulletin of the American Mathematical Society, Vol. 65 (1959), no. 2, pp. 59–65. <a href="http://projecteuclid.org/euclid.bams/1183523034">http://projecteuclid.org/euclid.bams/1183523034</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">András Juhász, Differential and Low-Dimensional Topology, p. 3 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
</ol>

Local flatness open-in-new

Local flatness