Tubular neighborhood

Tubular neighborhood

In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.

The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve draw a line perpendicular to the curve. Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion. However, if one looks only in a narrow band around the curve, the portions of the lines in that band will not intersect, and will cover the entire band without gaps. This band is the tubular neighborhood.

In general, let "S" be a submanifold of a manifold "M", and let "N" be the normal bundle of "S" in "M". Here "S" plays the role of the curve and "M" the role of the plane containing the curve. Consider the natural
0 → "S"which establishes a bijective correspondence between the zero section "N"0 of "N" and the submanifold "S" of "M". An extension "j" of this map to the entire normal bundle "N" with values in "M" such that "j"("N") is an open set in "M" and "j" is a homeomorphism between "N" and "j"("N") is called a tubular neighbourhood.

Often one calls the open set "T"="j"("N"), rather than "j" itself, a tubular neighbourhood of "S", it is assumed implicitly that the homeomorphism "j" mapping "N" to "T" exists.

References

* cite book
author=Raoul Bott, Loring W. Tu
title=Differential forms in algebraic topology
publisher=Springer-Verlag
location=Berlin
year=1982
pages=
isbn=0-387-90613-4
oclc=
doi=

* cite book
author=Waldyr Muniz Oliva
title=Geometric Mechanics
publisher=Springer
location=Berlin
year=
pages=
isbn=3-540-44242-1
oclc=
doi=


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