False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-09-01

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

Statement

False claim: on an $n$ -manifold, a $k$ -form can be nonzero even when $k > n$ .

Facts & Assumptions

Given: An $n$ -manifold $M$ and an integer $k > n$ .

[L1]

If $\dim V = n$ and $k > n$ , then ${Alt}^{k} (V) = 0$ (Dimension of the $k$ th exterior power is binomial).

[L2]

The bundle $⋀^{k} T^{*} M$ has fibre ${Alt}^{k} (T_{p} M)$ at each point (Exterior-power transition laws define a smooth vector bundle).

Refutation

technique · direct

1.1

For each $p \in M$ , the tangent space $T_{p} M$ has dimension $n$ . Hence [L1] gives ${Alt}^{k} (T_{p} M) = 0$ .

L1given

2.1

By [L2], every fibre of $⋀^{k} T^{*} M$ is zero. Therefore every section of that bundle is the zero form.

L2step 1.1

3.1

So no nonzero $k$ -form exists when $k > n$ , and the claim is false.

step 2.1∎

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

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

A kk-form on an nn-manifold must vanish when k>nk>n