False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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FALSE: an inner product determines an orientation

Statement

An inner product on a finite-dimensional real vector space determines an orientation of that space.

Facts & Assumptions

Given: The standard inner product on $R^{3}$ and the ordered bases $(e_{1}, e_{2}, e_{3})$ and $(e_{2}, e_{1}, e_{3})$ .

[L1]

With the standard metric fixed, $R^{3}$ carries the two opposite orientations represented by $(e_{1}, e_{2}, e_{3})$ and $(e_{2}, e_{1}, e_{3})$ , and reversing the orientation negates the Hodge star (Reversing orientation negates the Hodge star while keeping the metric fixed).

[L2]

The sign of a change-of-basis determinant records which orientation class a basis lies in (A real linear isomorphism preserves or reverses orientation according to the sign of its determinant).

Refutation

technique · direct

1.1

By [L1], the same standard inner product is compatible with both the orientation represented by $(e_{1}, e_{2}, e_{3})$ and the opposite one represented by $(e_{2}, e_{1}, e_{3})$ ; neither is preferred by the metric.

1.2

By [L2], these are genuinely different orientations: the change-of-basis determinant is $- 1$ , so the two bases lie in opposite classes.

2.1

Since one inner product is compatible with two different orientations, the inner product does not determine an orientation.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, §8.2 (standard reference, not scraped)