PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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Time-t flow maps are diffeomorphisms between open domains

Statement

Let $Φ : D \to M$ be the maximal flow of a smooth vector field $X$ . For each $t \in R$ , the time- $t$ map

$Φ_{t} : D_{t} \to D_{- t}, Φ_{t} (p) : = Φ (t, p),$

where $D_{t} : = {p : (t, p) \in D}$ , is a diffeomorphism with inverse $Φ_{- t}$ .

Facts & Assumptions

Given: The maximal flow $Φ : D \to M$ of a smooth vector field $X$ and a time $t \in R$ .

[L1]

The maximal flow has open domain and satisfies the local group law (The fundamental theorem on flows).

Proof

technique · direct

1.1

By [L1], the slices $D_{t}$ and $D_{- t}$ are open in $M$ because $D$ is open in $R \times M$ . The map $Φ_{t}$ is smooth as a restriction of the smooth flow map.

L1given

1.2

Whenever $(t, p) \in D$ , the local group law from [L1] gives $Φ_{- t} (Φ_{t} (p)) = Φ (- t, Φ (t, p)) = Φ (0, p) = p .$ The same argument with $- t$ in place of $t$ shows $Φ_{t} (Φ_{- t} (q)) = q$ for $q \in D_{- t}$ .

2.1

Thus $Φ_{t}$ and $Φ_{- t}$ are inverse smooth maps between the open sets $D_{t}$ and $D_{- t}$ , so $Φ_{t}$ is a diffeomorphism.

step 1.1step 1.2∎

Depends on

The fundamental theorem on flows

Used by

The Lie derivative of a vector field Definition
The generating vector field is invariant under its own flow Proposition
The flow-box theorem Theorem
The Lie derivative of a vector field equals the Lie bracket Theorem
Two vector fields commute if and only if their local flows commute Theorem

Dependency tree · two levels

4 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)