DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Musical isomorphisms

The musical maps for $g$ are $v^{♭} = g (v, \cdot)$ and its pointwise inverse $α^{♯}$ , characterized by $g (α^{♯}, v) = α (v)$ for all $v$ .

For the metric in Riemannian metric and riemannian manifold, $v^{♭} = 0$ forces $g (v, v) = 0$ and hence $v = 0$ . Thus $♭$ is injective between equal-dimensional fibres and bijective; this gives the pointwise inverse. Smooth inverse bundle maps are proved in The musical maps are smooth inverse bundle isomorphisms ↗. On a zero fibre both are the unique map.

Lee, Introduction to Smooth Manifolds, 2nd ed., Chapter 13, pp.328–332 and 341–342.

Depends on