Tubular neighborhood

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A curve, in blue, and some lines perpendicular to it, in green. Small portions of those lines around the curve are in red.
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 T=j(N).
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 T=j(N).

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. 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 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

i:N0 → S

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.

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A schematic illustration of the normal bundle N, with the zero section N0 in blue. The transformation j maps N0 to the curve S in the figure above, and N to the tubular neighbourhood of S.
A schematic illustration of the normal bundle N, with the zero section N0 in blue. The transformation j maps N0 to the curve S in the figure above, and N to the tubular neighbourhood of S.
  • Raoul Bott, Loring W. Tu (1982). Differential forms in algebraic topology. Berlin: Springer-Verlag. ISBN 0-387-90613-4.