PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-01

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Two tubular neighbourhood germs are isomorphic near the zero section

Statement

Let $Φ_{1} : Ω_{1} \to M$ and $Φ_{2} : Ω_{2} \to M$ be two tubular neighbourhoods of the same closed embedded submanifold $S \subseteq M$ built on the same normal bundle $ν (S)$ . Then, after shrinking $Ω_{1}$ and $Ω_{2}$ around the zero section, there is a diffeomorphism $Ψ : Ω_{1}^{'} \to Ω_{2}^{'}$ such that $Φ_{2} \circ Ψ = Φ_{1}$ and $Ψ$ restricts to the identity on the zero section.

Facts & Assumptions

Given: Two tubular neighbourhood charts $Φ_{1} : Ω_{1} \to M$ and $Φ_{2} : Ω_{2} \to M$ for the same closed embedded submanifold $S \subseteq M$ .

[F1]

A tubular neighbourhood chart is a diffeomorphism from an open normal-bundle neighbourhood of the zero section onto an ambient open neighbourhood of $S$ (Tubular neighbourhoods of embedded submanifolds).

[L1]

Tubular neighbourhoods exist in smooth ambient manifolds (The tubular neighbourhood theorem in a smooth ambient manifold).

Proof

technique · direct

1.1F1givenconstruct

By [F1], both $Φ_{1}$ and $Φ_{2}$ are diffeomorphisms onto open neighbourhoods of $S$ . Shrink the domains so that their images lie in the common overlap. Then $Ψ : = Φ_{2}^{- 1} \circ Φ_{1}$ is a diffeomorphism between the shrunken domains.

2.1F1step 1.1

On the zero section both tubular charts agree with the inclusion of $S$ into $M$ , so $Ψ (p, 0) = (p, 0)$ for every $p \in S$ . Thus $Ψ$ restricts to the identity on the zero section.

3.1L1step 1.1step 2.1∎

The relation $Φ_{2} \circ Ψ = Φ_{1}$ is built into the definition of $Ψ$ , and [L1] guarantees that these tubular charts are honest smooth objects rather than formal placeholders. Hence the two tubular neighbourhoods define the same germ near the zero section.

Depends on

Used by

Dependency tree · two levels

20 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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Tubular Neighborhoods (standard reference, not scraped)