False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-01

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FALSE: every noncompact submanifold has a uniform-radius tubular neighbourhood

Statement

False claim: every noncompact embedded submanifold of Euclidean space has a tubular neighbourhood of one fixed radius.

Facts & Assumptions

Given: A smooth embedded curve obtained by joining, for each integer $n \geq 1$ , a long horizontal segment to a smoothed hairpin whose two parallel strands are distance $2^{- n}$ apart.

[L1]

The Euclidean tubular neighbourhood theorem only guarantees a positive radius function along the submanifold (The Euclidean tubular neighbourhood theorem).

Refutation

technique · direct

1.1

The described curve is a smooth embedding of $R$ into $R^{2}$ : each hairpin lives far to the right of the previous ones, and the smoothing keeps successive pieces joined with nonvanishing tangent.

givenconstruct

2.1

Let $r > 0$ . Choose $n$ with $2^{- n} < 2 r$ . In the $n$ th hairpin the two nearly parallel strands are closer than $2 r$ , so the normal discs of radius $r$ centered on opposite strands meet before reaching the turning cap. Hence the normal addition map is not injective on the radius- $r$ neighbourhood of that part of the curve.

step 1.1algebra

3.1

Since this happens for every fixed $r > 0$ , no uniform tubular radius works. Thus [L1] is sharp: in the noncompact case one generally needs a variable radius.

L1step 2.1∎

Depends on

The Euclidean tubular neighbourhood theorem

Used by

A noncompact embedded curve with no uniform tubular radius Example

Dependency tree · two levels

5 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)