CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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.

Assuming countable choice, every smooth manifold admits a Riemannian metric

Statement

Assume ${A C}_{ω}$ . Every smooth manifold admits a Riemannian metric.

Facts & Assumptions

Given: The axiom ${A C}_{ω}$ and a smooth manifold $M$ .

[L1]

The tangent bundle $T M \to M$ is a smooth vector bundle (Assuming countable choice, the tangent bundle has a canonical smooth 2n-manifold structure).

[L2]

Every smooth vector bundle admits a smooth bundle metric (Every smooth vector bundle admits a smooth bundle metric).

Proof

technique · direct

1.1

By [L1], the tangent bundle of $M$ is a smooth vector bundle.

L1given

2.1

Apply [L2] to $T M$ . A smooth bundle metric on $T M$ is exactly a Riemannian metric on $M$ .

L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)