This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(August 2014) (Learn how and when to remove this message)

Inmathematics, atubular neighborhood of asubmanifold of asmooth manifold is anopen set around it resembling thenormal bundle.

The idea behind a tubular neighborhood can be explained in a simple example. Consider asmooth curve in the plane without self-intersections. On each point on thecurve draw a lineperpendicular 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, letS be asubmanifold of amanifoldM, and letN be thenormal bundle ofS inM. HereS plays the role of the curve andM the role of the plane containing the curve. Consider the natural map

${\displaystyle i:N_0\to S}$

which establishes abijective correspondence between thezero section${\displaystyle N_0}$ ofN and the submanifoldS ofM. An extensionj of this map to the entire normal bundleN with values inM such that${\displaystyle j(N)}$ is an open set inM andj is ahomeomorphism betweenN and${\displaystyle j(N)}$ is called a tubular neighbourhood.

Often one calls the open set${\displaystyle T=j(N),}$ rather thanj itself, a tubular neighbourhood ofS, it is assumed implicitly that the homeomorphismj mappingN toT exists.

Anormal tube to asmooth curve is amanifold defined as theunion of all discs such that

• all the discs have the same fixed radius;
• the center of each disc lies on thecurve; and
• each disc lies in a planenormal to the curve where the curve passes through that disc's center.

Let${\displaystyle S\subseteq M}$ be smooth manifolds. A tubular neighborhood of${\displaystyle S}$ in${\displaystyle M}$ is avector bundle${\displaystyle \pi :E\to S}$ together with a smooth map${\displaystyle J:E\to M}$ such that

• ${\displaystyle J\circ 0_E=i}$ where${\displaystyle i}$ is the embedding${\displaystyle S\hookrightarrow M}$ and${\displaystyle 0_E}$ the zero section
• there exists some open sets${\displaystyle U\subseteq E}$ and${\displaystyle V\subseteq M}$ with${\displaystyle 0_E[S]\subseteq U}$ and${\displaystyle S\subseteq V}$ such that${\displaystyle J{|}_U:U\to V}$ is adiffeomorphism.

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${\displaystyle M.}$

Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, orspherical fibrations forPoincaré spaces.

These generalizations are used to produce analogs to the normal bundle, or rather to thestable normal bundle, which are replacements for the tangent bundle (which does not admit a direct description for these spaces).

• Parallel curve – Generalization of the concept of parallel lines (aka offset curve)
• Tube lemma – Lemma in topology

• Raoul Bott, Loring W. Tu (1982).Differential forms in algebraic topology. Berlin: Springer-Verlag.ISBN 0-387-90613-4.

• Morris W. Hirsch (1976).Differential Topology. Berlin: Springer-Verlag.ISBN 0-387-90148-5.

• Waldyr Muniz Oliva (2002).Geometric Mechanics. Berlin: Springer-Verlag.ISBN 3-540-44242-1.

Wikimedia Commons has media related toTubular neighborhood.

• Manifolds
• Geometric topology
• Smooth manifolds

• Articles with short description
• Short description is different from Wikidata
• Articles lacking in-text citations from August 2014
• All articles lacking in-text citations
• Commons category link is on Wikidata