PropositionStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-31

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A fibrewise bijective smooth bundle map over a diffeomorphism is a bundle isomorphism

Statement

Let $Φ : E \to F$ be a smooth vector bundle map over a diffeomorphism $f : M \to N$ . If each fibre map $Φ_{p} : E_{p} \to F_{f (p)}$ is bijective, then $Φ$ is a vector bundle isomorphism.

Facts & Assumptions

Given: A smooth bundle map $Φ : E \to F$ over a diffeomorphism $f : M \to N$ , with each $Φ_{p}$ bijective.

[L1]

In local frames, smooth bundle maps are given by smooth matrix-valued functions (Smoothness of a bundle map is equivalent to smooth local matrices).

[L2]

A real square matrix is invertible exactly when its determinant is nonzero (A finite square real matrix is invertible if and only if its determinant is nonzero).

[L3]

A smooth matrix-valued map has smooth inverse matrix entries wherever its determinant never vanishes (Matrix inversion preserves $C^{k}$ regularity where the determinant is nonzero).

Proof

technique · direct

1.1

If the common fibre rank is $0$ , then every fibre is the zero vector space, so $Φ$ is already the unique smooth bundle map between zero bundles over $f$ and hence a bundle isomorphism. Otherwise choose local frames so that on one trivializing neighborhood, $Φ (p, v) = (f (p), A (p) v)$ . Fibrewise bijectivity means that each matrix $A (p)$ is invertible, so [L2] gives $\det A (p) \neq 0$ for every $p$ .

L1L2given

2.1

In the positive-rank case, [L3] makes the entries of $A^{- 1}$ smooth on the same neighborhood. Thus the local inverse is $(q, w) \mapsto (f^{- 1} (q), A (f^{- 1} (q))^{- 1} w)$ , which is smooth because $f^{- 1}$ is smooth. These local inverses agree on overlaps, so $Φ$ is a smooth bundle isomorphism. Together with the rank- $0$ branch of step 1.1, this proves the proposition.

L3step 1.1algebra∎

Depends on

Used by

Dependency tree · two levels

26 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)