LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

Disk bundles over compact bases are compact manifolds with boundary

Statement

Let $E \to M$ be a smooth rank- $q$ vector bundle over a boundaryless smooth manifold, with a supplied smooth bundle metric $h$ . The closed disk bundle $D_{h} (E) = {∥ v ∥_{h} \leq 1}$ is a smooth manifold with boundary of dimension $\dim M + q$ , with boundary $S_{h} (E) = {∥ v ∥_{h} = 1}$ and interior ${∥ v ∥_{h} < 1}$ . The projection and zero section are smooth. If $M$ is compact, the disk bundle and its boundary are compact. For $q = 0$ the disk bundle is $M$ and the boundary is empty; when $\dim M + q \geq 1$ the boundary is a closed embedded smooth manifold of dimension $\dim M + q - 1$ .

Facts & Assumptions

Given: A smooth vector bundle $E \to M$ over a boundaryless smooth manifold, with smooth metric $h$ and rank $q \geq 0$ .

[F1]

In a bundle chart the metric squared is $H (u, v) = v^{T} A (u) v$ , with $A$ smooth positive definite (Smooth vector bundles, rank, fibres, and trivial bundles); disk and sphere bundles have their indicated inequalities (Disk, sphere, and Thom spaces of a metric vector bundle).

[L1]

At a regular level a smooth real-valued function has coordinate normal form, giving half-space charts for its sublevel (Local normal form for submersions, Regular sublevels are compact manifolds with boundary). Manifold boundaries are closed embedded submanifolds (The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold).

[L2]

Small coordinate balls have compact closures; compact subsets admit finite ambient subcovers (Coordinate balls form a basis of a topological manifold, A subspace is compact exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it). Continuous positive functions on nonempty compact Euclidean sets have a positive minimum, and closed bounded Euclidean sets are compact (A continuous real-valued function on a nonempty compact metric space is bounded and attains a greatest and a least value, Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

Proof

technique · direct

1.1F1L1givenalgebra

The smooth function $H : E \to R$ , $H (v) = ∥ v ∥_{h}^{2}$ , has vertical derivative $w \mapsto 2 h (v, w)$ . At $H (v) = 1$ , evaluating on $w = v$ gives $2$ , so $1$ is regular. The local normal-form argument of [L1] supplies the subspace smooth half-space charts on $H \leq 1$ , without needing compactness. Its boundary is $H = 1$ and its interior is $H < 1$ . For $q = 0$ , $H = 0$ and the disk bundle is simply $M$ .

2.1L1step 1.1

These charts are restrictions of smooth ambient charts, so projection and zero section remain smooth. The boundary is closed and embedded by [L1], with the asserted dimension when the total dimension is positive.

3.1F1L2step 1.1algebra∎

If $M$ is compact, cover it by finitely many compact coordinate pieces $K_{i}$ lying inside bundle-trivialization domains. For $q > 0$ , on the compact set $K_{i} \times S^{q - 1}$ the function $(u, z) \mapsto z^{T} A (u) z$ has a positive minimum $c_{i}$ by [L2]. Thus $v^{T} A (u) v \leq 1$ implies $∣ v ∣ \leq c_{i}^{- 1 / 2}$ . The disk bundle over $K_{i}$ is a closed bounded subset of a Euclidean coordinate product, hence compact by [L2]. Their finite union is $D_{h} (E)$ , which is therefore compact; its closed boundary is compact too. Empty pieces are omitted. For $q = 0$ compactness is just compactness of $M$ .

Depends on

Used by

Positive-codimension thickening reduces closed sources to the open case Lemma

Dependency tree · two levels

77 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.