Tubular neighborhood

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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

${\displaystyle i:N_{0}\to S}$

which establishes a bijective correspondence between the zero section ${\displaystyle N_{0}}$ 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 ${\displaystyle j(N)}$ is an open set in M and j is a homeomorphism between N and ${\displaystyle j(N)}$ is called a tubular neighbourhood.

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