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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(N) is an open set in M and j is a homeomorphism between N and j(N) is called a tubular neighbourhood.
Often one calls the open set 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.
- 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 S ⊂ M be smooth manifolds. A tubular neighborhood of S in M is a vector bundle together with a smooth map such that
- where i is the embedding and the zero section
- 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 M.
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).
- Parallel curve (aka offset curve)
- Raoul Bott, Loring W. Tu (1982). Differential forms in algebraic topology. Berlin: Springer-Verlag. ISBN 0-387-90613-4.
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