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

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A diffeomorphism pulls back tensor fields and forms isomorphically

Statement

If $F : M \to N$ is a diffeomorphism, then pullback by $F$ is an isomorphism on covariant tensor fields and on differential forms. Its inverse is pullback by $F^{- 1}$ .

Facts & Assumptions

Given: A diffeomorphism $F : M \to N$ with inverse $F^{- 1}$ .

[F1]

A diffeomorphism has a smooth inverse (Diffeomorphisms and local diffeomorphisms of manifolds).

[L1]

Covariant tensor pullback and form pullback are functorial (Pullback of covariant tensors is smooth and functorial, Pullback of forms is smooth functorial and preserves wedges).

Proof

technique · direct

1.1

By [F1], both $F$ and $F^{- 1}$ are smooth, so [L1] gives pullback maps in both directions on covariant tensor fields and on forms.

F1L1given

2.1

Functoriality from [L1] yields $(F^{- 1})^{*} \circ F^{*} = (F \circ F^{- 1})^{*} = {id}^{*}, F^{*} \circ (F^{- 1})^{*} = ((F^{- 1}) \circ F)^{*} = {id}^{*} .$ The identity pullback is the identity map by the same functoriality statements.

L1step 1.1algebra

3.1

Hence $F^{*}$ is an isomorphism with inverse $(F^{- 1})^{*}$ on covariant tensor fields and on differential forms.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

12 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)