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.

Riemannian hodge star

Definition

On an oriented Riemannian $n$ -manifold, for $0 \leq k \leq n$ the Hodge star is the fibrewise map $* : ⋀^{k} T^{*} M \to ⋀^{n - k} T^{*} M$ characterized by $α \land * β = ⟨ α, β ⟩_{g} {vol}_{g}$ for every pair of $k$ -covectors.

The pairing is the determinant-normalized one of Riemannian metrics induce metrics on dual tensor and exterior bundles, and the positive unit volume form is The riemannian volume form is the unique positive unit top form. This is the ordinary, orientation-dependent star, with no orientation-line twist. Existence, uniqueness and smoothness are proved in Hodge star is a smooth bundle isomorphism ↗. For $n = k = 0$ , it multiplies by the chosen orientation sign.

Source locator

Lee, Chapter 16, Problem 16-18(c–e), pp.437–438; the local construction is proved in the following theorem.

Depends on

Used by

The hodge star is defined without an orientation False statement
Riemannian inner product of compactly supported forms Proposition
Hodge star is a smooth bundle isomorphism Theorem

Dependency tree · two levels

8 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)