This article may be confusing or unclear to readers. (August 2014)
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 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 is an open set in M and j is a homeomorphism between N and is called a tubular neighbourhood.
Often one calls the open set rather than j itself, a tubular neighbourhood of S, it is assumed implicitly that the homeomorphism j mapping N to T exists.
- all the discs have the same fixed radius;
- the center of each disc lies on the curve; and
- each disc lies in a plane normal to the curve where the curve passes through that disc's center.
Let be smooth manifolds. A tubular neighborhood of in is a vector bundle together with a smooth map such that
- where is the embedding and the zero section
- there exists some and some with and such that is a diffeomorphism.
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
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 (which does not admit a direct description for these spaces).
- Raoul Bott, Loring W. Tu (1982). Differential forms in algebraic topology. Berlin: Springer-Verlag. ISBN 0-387-90613-4.
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