TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-09-01

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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.

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Exterior-power transition laws define a smooth vector bundle

Statement

For every smooth manifold $M$ and $k \geq 0$ , the bundle $⋀^{k} T^{*} M \to M$ is a smooth vector bundle.

Facts & Assumptions

Given: A smooth manifold $M$ and an integer $k \geq 0$ .

[F1]

The bundle $⋀^{k} T^{*} M$ is the fibrewise bundle of alternating $k$ -covectors (The exterior power bundle of the cotangent bundle).

[L1]

The alternating fibrewise part of the covariant tensor bundle is a smooth vector subbundle (Symmetric and alternating images are smooth subbundles).

Proof

technique · direct

1.1

The bundle $T_{k}^{0} M$ is the covariant $k$ -tensor bundle, and its alternating fibrewise image is the collection of alternating $k$ -covectors at each point.

F1given

2.1

By [L1], that alternating fibrewise image is a smooth vector subbundle of $T_{k}^{0} M$ . By step 1.1, this subbundle is exactly $⋀^{k} T^{*} M$ .

F1L1step 1.1

3.1

Therefore the exterior-power transition laws define a smooth vector bundle.

step 2.1∎

Depends on

Used by

A k-form on an n-manifold must vanish when k>n False statement

Cited to discharge well-definedness by The exterior power bundle of the cotangent bundle.

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, 2nd ed. (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)