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.

Unless the curve is straight, these lines will intersect among themselves in a rather complicated fashion.

Consider the natural map which establishes a bijective correspondence between the zero section

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

rather than j itself, a tubular neighbourhood of S, it is assumed implicitly that the homeomorphism j mapping N to T exists.

A normal tube to a smooth curve is a manifold defined as the union of all discs such that Let

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 (the dimension of the vector bundle considered as a manifold is) that of

Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, or spherical fibrations for Poincaré spaces.

A curve, in blue, and some lines perpendicular to it, in green. Small portions of those lines around the curve are in red.
A close up of the figure above. The curve is in blue, and its tubular neighborhood T is in red. With the notation in the article, the curve is S , the space containing the curve is M , and
A schematic illustration of the normal bundle N , with the zero section in blue. The transformation j maps N 0 to the curve S in the figure above, and N to the tubular neighbourhood of S .