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A neighbourhood of a smooth submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t0944201.png" /> in a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t0944202.png" /> that is fibred over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t0944203.png" /> with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t0944204.png" />, where
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t0944205.png" /></td> </tr></table>
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Suppose that in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t0944206.png" /> a Riemannian metric is chosen and consider segments of geodesics that are normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t0944207.png" /> and start in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t0944208.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t0944209.png" /> is compact, then there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442010.png" /> such that no two segments of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442011.png" /> and starting at different points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442012.png" /> intersect. The union of all such segments of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442013.png" /> is an open neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442015.png" />, and is called a tubular neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442016.png" />. It is possible to construct for a non-compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442017.png" /> a tubular neighbourhood by covering <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442018.png" /> with a countable family of compacta and by decreasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442019.png" /> as the number of elements of the covering increases. There is a [[Deformation retract|deformation retract]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442020.png" /> associating with each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442021.png" /> the beginning of a geodesic containing this point. This retract determines a vector bundle with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442022.png" /> that is isomorphic to the [[Normal bundle|normal bundle]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442023.png" /> of the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442024.png" />. In this way, the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442025.png" /> is homeomorphic to the [[Thom space|Thom space]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t094/t094420/t09442026.png" />.
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A neighbourhood of a smooth submanifold  $  N $
 +
in a smooth manifold  $  M $
 +
that is fibred over  $  N $
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with fibre  $  \mathbf R  ^ {d} $,
 +
where
 +
 
 +
$$
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d  =  \mathop{\rm dim}  M -  \mathop{\rm dim}  N.
 +
$$
 +
 
 +
Suppose that in $  M $
 +
a Riemannian metric is chosen and consider segments of geodesics that are normal to $  N $
 +
and start in $  N $.  
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If $  N $
 +
is compact, then there exists an $  \epsilon > 0 $
 +
such that no two segments of length $  \leq  \epsilon $
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and starting at different points of $  N $
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intersect. The union of all such segments of length < \epsilon $
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is an open neighbourhood $  U $
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of $  N $,  
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and is called a tubular neighbourhood of $  N $.  
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It is possible to construct for a non-compact $  N $
 +
a tubular neighbourhood by covering $  N $
 +
with a countable family of compacta and by decreasing $  \epsilon $
 +
as the number of elements of the covering increases. There is a [[Deformation retract|deformation retract]] $  r: U \rightarrow N $
 +
associating with each point of $  U $
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the beginning of a geodesic containing this point. This retract determines a vector bundle with fibre $  \mathbf R  ^ {d} $
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that is isomorphic to the [[Normal bundle|normal bundle]] $  \nu $
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of the imbedding $  N \rightarrow M $.  
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In this way, the quotient space $  \overline{U}\; / \partial  \overline{U}\; $
 +
is homeomorphic to the [[Thom space|Thom space]] of $  \nu $.
  
 
An analogue of the notion of a tubular neighbourhood can also be introduced for topological manifolds (where one has to consider locally flat imbeddings, [[#References|[2]]]).
 
An analogue of the notion of a tubular neighbourhood can also be introduced for topological manifolds (where one has to consider locally flat imbeddings, [[#References|[2]]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Thom,  "Quelques propriétés globales des variétés différentiables"  ''Comm. Math. Helv.'' , '''28'''  (1954)  pp. 17–86</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Kirby,  L.C. Siebenmann,  "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press  (1977)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Thom,  "Quelques propriétés globales des variétés différentiables"  ''Comm. Math. Helv.'' , '''28'''  (1954)  pp. 17–86</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.C. Kirby,  L.C. Siebenmann,  "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press  (1977)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:26, 6 June 2020


A neighbourhood of a smooth submanifold $ N $ in a smooth manifold $ M $ that is fibred over $ N $ with fibre $ \mathbf R ^ {d} $, where

$$ d = \mathop{\rm dim} M - \mathop{\rm dim} N. $$

Suppose that in $ M $ a Riemannian metric is chosen and consider segments of geodesics that are normal to $ N $ and start in $ N $. If $ N $ is compact, then there exists an $ \epsilon > 0 $ such that no two segments of length $ \leq \epsilon $ and starting at different points of $ N $ intersect. The union of all such segments of length $ < \epsilon $ is an open neighbourhood $ U $ of $ N $, and is called a tubular neighbourhood of $ N $. It is possible to construct for a non-compact $ N $ a tubular neighbourhood by covering $ N $ with a countable family of compacta and by decreasing $ \epsilon $ as the number of elements of the covering increases. There is a deformation retract $ r: U \rightarrow N $ associating with each point of $ U $ the beginning of a geodesic containing this point. This retract determines a vector bundle with fibre $ \mathbf R ^ {d} $ that is isomorphic to the normal bundle $ \nu $ of the imbedding $ N \rightarrow M $. In this way, the quotient space $ \overline{U}\; / \partial \overline{U}\; $ is homeomorphic to the Thom space of $ \nu $.

An analogue of the notion of a tubular neighbourhood can also be introduced for topological manifolds (where one has to consider locally flat imbeddings, [2]).

References

[1] R. Thom, "Quelques propriétés globales des variétés différentiables" Comm. Math. Helv. , 28 (1954) pp. 17–86
[2] R.C. Kirby, L.C. Siebenmann, "Foundational essays on topological manifolds, smoothings, and triangulations" , Princeton Univ. Press (1977)

Comments

Tubular neighbourhoods were introduced by H. Whitney in his treatment of differentiable manifolds (see [a2] for some history).

References

[a1] M.W. Hirsch, "Differential topology" , Springer (1976) pp. Chapt. 5, Sect. 3
[a2] J. Dieudonné, "A history of algebraic and differential topology: 1900–1960" , Birkhäuser (1989) pp. Chapt. III
How to Cite This Entry:
Tubular neighbourhood. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tubular_neighbourhood&oldid=15283
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article