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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A fine Euclidean approximation lands in a prescribed tubular neighbourhood

Statement

Let $j : N ↪ R^{m}$ be a closed embedded smooth submanifold with tubular neighbourhood $U$ , and let $F : M \to N$ be continuous. Then there exists a positive continuous error function $ε$ on $M$ such that every smooth map $\tilde{H} : M \to R^{m}$ satisfying

∥ \tilde{H} (p) - j (F (p)) ∥ < ε (p)

for all $p$ has image contained in $U$ .

Facts & Assumptions

Given: A continuous map $F : M \to N$ and a tubular neighbourhood $U$ of the embedded image $j (N) \subseteq R^{m}$ .

[F1]

A positive continuous error function is a continuous map into $(0, \infty)$ (Positive continuous error functions for strong approximation).

[L1]

Euclidean embedded submanifolds admit tubular neighbourhoods (The Euclidean tubular neighbourhood theorem).

Proof

technique · direct

1.1

For each $p \in M$ , the point $j (F (p))$ lies in the open set $U$ , so its Euclidean distance to the closed complement $R^{m} ∖ U$ is positive. Define $ε (p) : = \frac{1}{2} dist ⁣ (j (F (p)), R^{m} ∖ U) .$ Because $j \circ F$ is continuous and the distance-to-a-fixed-closed-set function is continuous, [F1] shows that $ε$ is a positive continuous error function.

F1L1givenconstruct

2.1

If $∥ \tilde{H} (p) - j (F (p)) ∥ < ε (p)$ , then $\tilde{H} (p)$ lies in the open Euclidean ball of radius $ε (p)$ around $j (F (p))$ . By the definition of $ε (p)$ , that ball is contained in $U$ . Hence $\tilde{H} (p) \in U$ .

step 1.1algebra

3.1

Therefore every approximation with error bound $ε$ lands in the prescribed tubular neighbourhood $U$ .

step 2.1∎

Depends on

Used by

Dependency tree · two levels

9 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., Smooth Approximation of Maps Between Manifolds (standard reference, not scraped)