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

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Musical isomorphisms

Definition

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.

Source locator

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

Depends on

Riemannian metric and riemannian manifold

Used by

The euclidean metric and its musical maps Example
The musical maps are smooth inverse bundle isomorphisms Theorem

Dependency tree · two levels

3 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, second edition (standard reference, not scraped)