Tubular neighborhood

Anormal tube to a smooth curve is a manifold defined as the union of all discs such that

• 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 ${\displaystyle S\subseteq M}$  be smooth manifolds. A tubular neighborhood of ${\displaystyle S}$ in${\displaystyle M}$  is a vector 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 ${\displaystyle U\subseteq E}$  and some ${\displaystyle V\subseteq M}$  with ${\displaystyle 0_{E}[$S]\subseteq U}   and ${\displaystyle S\subseteq V}$  such that ${\displaystyle J\vert _{U}:U\to V}$  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 ${\displaystyle M.}$ 

Generalizations of smooth manifolds yield generalizations of tubular neighborhoods, such as regular neighborhoods, or spherical fibrations for Poincaré spaces.

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)
• Tube lemma – proof in topologyPages displaying wikidata descriptions as a fallback

• 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.