False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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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.
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The orthogonal normal bundle of a submanifold is defined without a metric

Statement

The orthogonal normal bundle of an embedded submanifold is defined without a metric.

Facts & Assumptions

Given: The displayed claim.

[L1]

The quotient normal bundle is intrinsic, but the orthogonal normal bundle is obtained only after choosing an ambient metric (Normal and conormal bundles of an embedded submanifold, Assuming countable choice, an ambient metric identifies the two normal bundles).

Refutation

technique · direct

1.1

Let $S = R \times {0} \subseteq R^{2}$ . For the Euclidean metric, the orthogonal complement of $T S$ is spanned by $\partial_{y}$ .

L1given

2.1

For the metric $g = d x^{2} + 2 d x d y + 2 d y^{2}$ , a vector $a \partial_{x} + b \partial_{y}$ is orthogonal to $\partial_{x}$ exactly when $a + b = 0$ , so the orthogonal complement is spanned by $\partial_{y} - \partial_{x}$ . The orthogonal normal line therefore depends on the chosen metric, and only the quotient normal bundle is intrinsic.

L1step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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 (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)