tubular neighborhood

In mathematics, a tubular neighborhoodMathworldPlanetmath of a submanifoldMathworldPlanetmath of a smooth manifoldMathworldPlanetmath 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 perpendicularPlanetmathPlanetmath 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 the tubular neighborhood.

In general, let M be a submanifold of a manifold M, and let N be the normal bundle of M in M (M will play the role of the curve, and M will be like the plane containing the curve). Consider the map


which establishes a bijective correspondence between the zero sectionMathworldPlanetmath N0 of N and the submanifold M of M. The mapping i maps the curve (blue in the following diagram) at the bottom to the blue curve on top, and each of the infinite lines on the bottom, to each of the finite lines (they can also be curves) on top.

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 homeomorphismMathworldPlanetmath between N and j(N) is called a tubular neighbourhood.

Often times one calls the open set T=j(N), rather than j itself, a tubular neighbourhood of M, it is assumed implicitly that the homeomorphism j mapping N to T exists.

The following schematic illustration of the normal bundle N, with the zero section N0 in blue. The transformationMathworldPlanetmath j maps N0 to the curve M, and N to the tubular neighborhood of M.


  • 1 Raoul Bott & Loring W. Tu Differential forms in algebraic topology. Berlin: Springer-Verlag. (1982)
  • 2 Waldyr Muniz Oliva Geometric Mechanics. Berlin: Springer. (1982)

This entry was adapted from the Wikipedia article http://en.wikipedia.org/wiki/Tubular_neighborhoodTubular neighborhood as of June 10, 2007.

These diagrams were created by Oleg Alexandrov and released to the public domain by him.

Title tubular neighborhood
Canonical name TubularNeighborhood
Date of creation 2013-03-22 17:13:53
Last modified on 2013-03-22 17:13:53
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 8
Author PrimeFan (13766)
Entry type Definition
Classification msc 54A05
Synonym tubular neighbourhood