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 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
which establishes a bijective correspondence between the zero section N0 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 is an open set in M and j is a homeomorphism between N and j is called a tubular neighbourhood.
Often one calls the open set T = j, rather than j itself, a tubular neighbourhood of S, it is assumed implicitly that the homeomorphism j mapping N to T exists.

Normal tube

A normal tube to a smooth curve is a manifold defined as the union of all discs such that
Let S ⊂ M be smooth manifolds. A tubular neighborhood of S in M is a vector bundle together with a smooth map such that
The normal bundle is a tubular neighborhood and because of the diffeomorphism condition in the second point, all tubular neighborhood have the same dimension, namely that of M.
E should be embedded in M.

Generalizations

Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, or spherical fibrations for Poincaré spaces.
These generalizations are used to produce analogs to the normal bundle, or rather to the stable normal bundle, which are replacements for the tangent bundle.