Difference between revisions of "Tubular neighbourhood"
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| + | $#C+1 = 26 : ~/encyclopedia/old_files/data/T094/T.0904420 Tubular neighbourhood | ||
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| − | Suppose that in | + | 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|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|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|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> | ||
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| − | |||
====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
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