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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every smooth vector bundle admits a smooth bundle metric

Statement

Every smooth vector bundle admits a smooth bundle metric.

Facts & Assumptions

Given: A smooth vector bundle $E \to M$ .

[L1]

The base manifold admits smooth partitions of unity subordinate to open covers (Smooth partitions of unity exist on manifolds).

[L2]

Local frames are equivalent to local trivializations (Local frames and local trivializations are equivalent data).

Proof

technique · direct

1.1

Choose a trivializing open cover $(U_{α})$ of $M$ and, by [L2], a local frame on each $U_{α}$ . Pull back the Euclidean inner product on $R^{r}$ through that trivialization to obtain a smooth local bundle metric $h_{α}$ on $E ∣_{U_{α}}$ .

L2givenchoose

2.1

By [L1], choose a smooth partition of unity $(ρ_{α})$ subordinate to $(U_{α})$ and define $h : = \sum_{α} ρ_{α} h_{α} .$ The sum is locally finite, so $h$ is smooth. At each point $p$ , some $ρ_{α} (p) > 0$ , all weights are nonnegative, and $\sum_{α} ρ_{α} (p) = 1$ , so $h_{p}$ is a positive-definite inner product on $E_{p}$ . Therefore $h$ is a smooth bundle metric on $E$ .

L1step 1.1constructalgebra∎

Depends on

Used by

Assuming countable choice, every smooth manifold admits a Riemannian metric Corollary
Every vector subbundle has a smooth complement Corollary
Orthogonal complements of subbundles are smooth subbundles Proposition

Dependency tree · two levels

31 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)