TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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The Lie derivative of a vector field equals the Lie bracket

Statement

For smooth vector fields $X$ and $Y$ on $M$ ,

$L_{X} Y = [X, Y] .$

Facts & Assumptions

Given: Smooth vector fields $X, Y$ on $M$ , the maximal flow $Φ$ of $X$ , a point $p \in M$ , and a smooth function $f$ .

[L1]

The Lie derivative of a vector field is defined by the inverse-time pushforward difference quotient (The Lie derivative of a vector field).

[L2]

The Lie bracket acts on functions by $[X, Y] f = X (Y f) - Y (X f)$ (The Lie bracket of smooth vector fields).

[L3]

The flow of $X$ satisfies ${\frac{d}{d t} ∣}_{t = 0} (f \circ Φ_{t}) = X f, {\frac{d}{d t} ∣}_{t = 0} (f \circ Φ_{- t}) = - X f .$ (The fundamental theorem on flows).

Proof

technique · direct

1.1L1given

By [L1], evaluating $L_{X} Y$ on $f$ gives $(L_{X} Y)_{p} (f) = {\frac{d}{d t} ∣}_{t = 0} ((Φ_{- t})_{*} Y)_{Φ_{t} (p)} (f) = {\frac{d}{d t} ∣}_{t = 0} Y_{Φ_{t} (p)} (f \circ Φ_{- t}) .$

2.1L3step 1.1

The expression in step 1.1 is ${\frac{d}{d t} ∣}_{t = 0} (Y (f \circ Φ_{- t})) (Φ_{t} (p)) .$ Differentiating the outer evaluation along the $X$ -flow contributes $X (Y f) (p)$ , while differentiating the inner function $f \circ Φ_{- t}$ contributes $- Y (X f) (p)$ by [L3]. Therefore $(L_{X} Y)_{p} (f) = X (Y f) (p) - Y (X f) (p) .$

3.1L2step 2.1

By [L2], the right-hand side of step 2.1 is exactly $[X, Y]_{p} (f)$ . Since this holds for every smooth $f$ , the tangent vectors $(L_{X} Y)_{p}$ and $[X, Y]_{p}$ agree.

4.1step 3.1∎

Therefore $L_{X} Y = [X, Y]$ .

Depends on

Used by

A vector field is flow-invariant if and only if its Lie derivative vanishes Proposition
Cartan commutator identities Proposition
Tensor Lie derivative agrees with X on functions and bracket on vector fields Proposition
Fundamental vector fields form a Lie-algebra homomorphism Theorem
The differential of Ad is ad Theorem
Two vector fields commute if and only if their local flows commute Theorem

Dependency tree · two levels

11 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)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)