False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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The hodge star is defined without an orientation

Statement

The ordinary Hodge star is determined by the Riemannian metric without any orientation.

Facts & Assumptions

Given: The same Euclidean metric $d x^{2}$ on $R$ , with the two possible orientations.

[F1]

Riemannian hodge star: 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 prop-riemannian-metrics-induce-metrics-on-dual-tensor-and-exterior-bundles, and the positive unit volume form is prop-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 thm-hodge-star-is-a-smooth-bundle-isomorphism. For $n = k = 0$ , it multiplies by the chosen orientation sign.

[F2]

Hodge star is a smooth bundle isomorphism: The Hodge star exists uniquely and is a smooth bundle isomorphism in every degree $0 \leq k \leq n$ .

Refutation

technique · direct

1.1

With positive orientation the volume form is $d x$ ; with negative orientation it is $- d x$ . In degree zero, $⟨ 1, 1 ⟩ = 1$ , so the defining identity $1 \land * 1 = ⟨ 1, 1 ⟩ vol$ gives $*_{+} 1 = d x$ and $*_{-} 1 = - d x$ . Both are the stars for their respective oriented metrics.

F1F2given

2.1

The one-form $d x$ is nonzero, so the two results differ, although the underlying metric is identical. Hence that metric alone cannot specify the ordinary Hodge star; the claimed orientation independence fails already in dimension one.

step 1.1∎

Source locator

Lee, Problem 16-18(a–c), pp. 437–438, Hodge star on oriented inner-product spaces; the one-dimensional orientation reversal is calculated above.

Depends on

Used by

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Dependency tree · two levels

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Sources

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)