False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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FALSE: Hodge star needs only the vector-space structure

Statement

The Hodge star is determined by the underlying real vector-space structure alone: no metric or orientation data is needed.

Facts & Assumptions

Given: The real vector space $R^{3}$ , the standard inner product, and the two opposite orientations.

[L1]

The Hodge star is built from the Gram pairing (the metric) and the oriented unit volume form (the orientation) (The Hodge star on an oriented finite-dimensional real inner-product space).

[L2]

The Hodge star is the unique operator satisfying its characterizing relation for those data (The Hodge star exists uniquely and is given by the complementary-basis formula in an oriented orthonormal basis).

[L3]

With the standard metric fixed, reversing the orientation negates the Hodge star: $⋆_{-} = - ⋆_{+}$ (Reversing orientation negates the Hodge star while keeping the metric fixed).

Refutation

technique · direct

1.1

By [L1], the definition of the Hodge star consumes a metric and an orientation as input, so the star is data attached to the pair, not to the bare vector space.

1.2

By [L3], the same underlying real vector space $R^{3}$ with the same metric but the two opposite orientations has two different stars, $⋆_{+} = - ⋆_{-}$ .

2.1

By the uniqueness clause of [L2], the operator is genuinely tied to the chosen data: the characterizing relation forces the sign change of step 1.2, so no single star is attached to the vector-space structure alone.

L2step 1.2

3.1

Steps 1.1 through 2.1 refute the claimed sufficiency of the vector-space structure.

step 1.1step 1.2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

Reyer Sjamaar, Manifolds and Differential Forms, §2.4 (standard reference, not scraped)