DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-09-01

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Tubular neighbourhoods of embedded submanifolds

Definition

Let $i : S ↪ M$ be a smooth embedding, and suppose its normal bundle $ν (S)$ (Normal and conormal bundles of an embedded submanifold) has been equipped with its standard smooth-vector-bundle structure. Under ${A C}_{ω}$ the existence of this structure is supplied by Assuming countable choice, normal and conormal bundles are smooth vector bundles.

A tubular neighbourhood of $S$ in $M$ consists of:

an open neighbourhood $Ω \subseteq ν (S)$ of the zero section, and
a smooth embedding $Φ : Ω \to M$

such that:

$Φ (0_{p}) = i (p)$ for every $p \in S$ , and
$Φ (Ω)$ is an open neighbourhood of $i (S)$ in $M$ .

Equivalently, $Φ$ is a diffeomorphism from $Ω$ onto an ambient open neighbourhood of $S$ , and its restriction to the zero section is the original inclusion.

Depends on

Used by

The normal addition map for a Euclidean submanifold Definition
Two tubular neighbourhood germs are isomorphic near the zero section Proposition
The tubular neighbourhood theorem in a smooth ambient manifold Theorem

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Marco Gualtieri, Topology I: Smooth Manifolds, Part 11, Definition 3.53 (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Tubular Neighborhoods (standard reference, not scraped)