PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passaudited 2026-09-06

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Tensor Lie derivative agrees with $X$ on functions and bracket on vector fields

Statement

For a function $f$ and vector field $Y$ , $L_{X} f = X f and L_{X} Y = [X, Y] .$

Facts & Assumptions

Given: The manifolds, forms, vector fields, maps, and coordinates explicitly named in the statement.

[F1]

The preceding result states that If $X$ has local flow $Φ_{t}$ and $T$ is a smooth tensor field, its Lie derivative is $L_{X} T = {\frac{d}{d t} ∣}_{t = 0} Φ_{t}^{*} T,$ on every local flow domain where this derivative is defined. (The Lie derivative of a tensor field).

Proof

technique · direct

1.1

For functions, $Φ_{t}^{*} f = f \circ Φ_{t}$ , whose derivative at zero is $X f$ .

F1given

2.1

For vector fields, differentiating the pullback uses the inverse-time pushforward convention and yields the established bracket $[X, Y]$ .

step 1.1∎

Depends on

Used by

The coordinate formula for the Lie derivative of a contravariant tensor Proposition
The coordinate formula for the Lie derivative of a covariant tensor Proposition

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

John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)

Tensor Lie derivative agrees with XX on functions and bracket on vector fields