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

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The flow definition of tensor Lie derivative is local and well-defined

Statement

The local-flow definition of $L_{X} T$ is independent of the chosen local flow and depends only on $X$ , $T$ , and the point.

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

Two local flows of $X$ have, for each starting point, integral curves with the same initial condition.

F1given

2.1

Local uniqueness makes the flows equal near $(0, p)$ ; their pullback curves therefore have the same derivative at zero.

step 1.1∎

Depends on

Used by

A tensor field is flow-invariant exactly when its Lie derivative vanishes Proposition

Cited to discharge well-definedness by The Lie derivative of a tensor field.

Dependency tree · two levels

7 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)