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

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Time-dependent vector fields have local smooth evolution operators

Statement

Let $I \subseteq R$ be an open interval, and let $X_{t}$ be a smooth time-dependent vector field on $M$ over $I$ . For every $(s, p) \in I \times M$ there exist an open interval $J \subseteq I$ containing $s$ , open neighbourhoods $U_{r}$ of the evolving points, and a smooth map

$Ψ : {(t, s, q) : t, s \in J, q \in U_{s}} \to M$

such that $t \mapsto Ψ_{t, s} (q)$ is the unique solution of $\dot{γ} (t) = X_{t} (γ (t))$ with $Ψ_{s, s} (q) = q$ .

Facts & Assumptions

Given: An open interval $I \subseteq R$ , a smooth time-dependent vector field $X_{t}$ over $I$ , and a base point $(s, p) \in I \times M$ .

[L1]

Chart maps identify manifold neighbourhoods with Euclidean open sets (Chart maps are diffeomorphisms onto Euclidean open sets).

[L2]

Smooth nonautonomous ODEs on Euclidean open sets have unique local smooth evolution operators (The fundamental theorem for nonautonomous smooth ODEs).

Proof

technique · direct

1.1

Choose a chart $(U, x)$ around $p$ . By [L1], the chart identifies $U$ with an open subset of $R^{n}$ , and the field $X_{t}$ becomes a smooth time-dependent Euclidean vector field ${\tilde{X}}_{t}$ there.

L1given

2.1

Apply [L2] to ${\tilde{X}}_{t}$ at $(s, x (p))$ . Because $I$ is open, the Euclidean field is defined on the open set $I \times x (U)$ . The theorem therefore yields an open interval $J \subseteq I$ containing $s$ , an open set $W \subseteq x (U)$ around $x (p)$ , and a smooth Euclidean evolution map ${\tilde{Ψ}}_{t, s} (y)$ .

L2step 1.1choose

3.1

Transport back by the chart: $Ψ_{t, s} (q) : = x^{- 1} ({\tilde{Ψ}}_{t, s} (x (q))) .$ This map is smooth and its time slices solve the manifold differential equation because the chart intertwines derivatives with the coordinate vector field.

L1step 2.1construct

4.1

Any other local manifold solution would push forward under $x$ to a Euclidean solution of the same nonautonomous ODE with the same initial data, so [L2] gives uniqueness.

L2step 3.1

5.1

Therefore smooth time-dependent vector fields have unique local smooth evolution operators.

step 3.1step 4.1∎

Depends on

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Sources

Marius Crainic, Rui Loja Fernandes, and Ioan Marcut, Lectures on Poisson Geometry (standard reference, not scraped)
John M. Lee, Introduction to Smooth Manifolds, 2nd ed. (standard reference, not scraped)