DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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The exterior derivative by the invariant vector-field formula

Definition

For $ω \in Ω^{k} (M)$ , define the candidate $d ω$ on smooth vector fields by $d ω (X_{0}, \dots, X_{k}) = \sum_{i} (- 1)^{i} X_{i} ω (X_{0}, {\hat{X}}_{i}, \dots, X_{k}) + \sum_{i < j} (- 1)^{i + j} ω ([X_{i}, X_{j}], X_{0}, {\hat{X}}_{i}, {\hat{X}}_{j}, \dots, X_{k}) .$ The next lemma proves that this candidate is a form.

Depends on

Used by

Finite-dimensional vector-valued forms and their exterior derivative Definition
The exterior derivative depends on a Riemannian metric False statement
Stokes formula for finite ordinary surface corners Lemma
The invariant exterior-derivative formula is C^∞-multilinear Lemma
The exterior derivative commutes with restriction Proposition
The exterior derivative is local Proposition
The exterior derivative of a function is its differential Proposition
The supplied smooth Godbillon-Vey construction does not cover merely C1 foliations Remark
Cartan's magic formula Theorem
Gaussian curvature structure equation Theorem
Maurer--Cartan structure equation Theorem
The local coordinate formula for the exterior derivative Theorem

Dependency tree · two levels

9 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)