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 the tubular neighborhood.
In general, let be a submanifold of a manifold , and let be the normal bundle of in ( will play the role of the curve, and will be like the plane containing the curve). Consider the map
which establishes a bijective correspondence between the zero section of and the submanifold of . The mapping 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 of this map to the entire normal bundle with values in such that is an open set in and is a homeomorphism between and is called a tubular neighbourhood.
Often times one calls the open set , rather than itself, a tubular neighbourhood of , it is assumed implicitly that the homeomorphism mapping to exists.
The following schematic illustration of the normal bundle , with the zero section in blue. The transformation maps to the curve , and to the tubular neighborhood of .
- 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.
|Date of creation||2013-03-22 17:13:53|
|Last modified on||2013-03-22 17:13:53|
|Last modified by||PrimeFan (13766)|