Muso

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 a 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 map

i
:

N

0

→
S

{displaystyle i:N_{0}to S}
which establishes a bijective correspondence between the zero section

N

0

{displaystyle 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
)

{displaystyle j(N)}
is an open set in M and j is a homeomorphism between N and

j
(
N
)

{displaystyle j(N)}
is called a tubular neighbourhood.
Often one calls the open set

T
=
j
(
N
)
,

{displaystyle 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.