DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Pushforwards and pullbacks of vector fields by a diffeomorphism

Definition

Assume ${A C}_{ω}$ , so that $T M$ and $T N$ carry their canonical smooth structures. Let $F : M \to N$ be a diffeomorphism.

For a smooth vector field $X$ on $M$ , the pushforward $F_{*} X$ is the unique vector field on $N$ that is $F$ -related to $X$ . Explicitly,

$(F_{*} X)_{F (p)} : = d F_{p} (X_{p}) .$

For a smooth vector field $Y$ on $N$ , the pullback $F^{*} Y$ is the vector field on $M$ defined by

$(F^{*} Y)_{p} : = d (F^{- 1})_{F (p)} (Y_{F (p)}) .$

Because $F$ and $F^{- 1}$ are smooth and their global differentials are smooth, both constructions yield smooth vector fields.

Depends on

Used by

Diffeomorphism pushforward preserves Lie brackets Corollary
The Lie derivative of a vector field Definition
FALSE: every smooth vector field can be pushed forward by every smooth map False statement
Flow reparametrization realizes a level isotopy Lemma
The generating vector field is invariant under its own flow Proposition
Fundamental vector fields form a Lie-algebra homomorphism Theorem
The differential of Ad is ad Theorem

Dependency tree · two levels

10 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)
Will J. Merry, Differential Geometry (standard reference, not scraped)