TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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.

Local existence, uniqueness, and smooth dependence for manifold integral curves

Statement

Let $X$ be a smooth vector field on $M$ and let $p \in M$ . Then there exist $h > 0$ , an open neighbourhood $U$ of $p$ , and a smooth map

$Φ : (- h, h) \times U \to M$

such that for every $q \in U$ , the curve $t \mapsto Φ (t, q)$ is the unique integral curve of $X$ on $(- h, h)$ with initial value $Φ (0, q) = q$ .

Facts & Assumptions

Given: A smooth vector field $X$ on $M$ and a point $p \in M$ .

[L1]

Chart maps are diffeomorphisms onto open subsets of Euclidean space (Chart maps are diffeomorphisms onto Euclidean open sets).

[L2]

In a chart, a smooth vector field has smooth coordinate components (Smoothness of a vector field is equivalent to smooth coordinate components).

[L3]

A smooth autonomous vector field on an open subset of $R^{n}$ has a local smooth flow depending smoothly on the initial point (The fundamental theorem for autonomous smooth ODEs).

Proof

technique · direct

1.1

Choose a chart $(V, x)$ around $p$ . By [L1], $x : V \to x (V) \subseteq R^{n}$ is a diffeomorphism onto an open set, and by [L2] the vector field $X ∣_{V}$ corresponds to a smooth Euclidean vector field $\tilde{X}$ on $x (V)$ .

L1L2given

2.1

Apply [L3] to $\tilde{X}$ at the point $x (p)$ . This gives $h > 0$ , an open neighbourhood $W \subseteq x (V)$ of $x (p)$ , and a smooth map $\tilde{Φ} : (- h, h) \times W \to x (V)$ whose time slices are the unique integral curves of $\tilde{X}$ .

L3step 1.1choose

3.1

Set $U : = x^{- 1} (W)$ and define $Φ (t, q) : = x^{- 1} (\tilde{Φ} (t, x (q)))$ . Because $x$ and $x^{- 1}$ are smooth by [L1], $Φ$ is smooth. Each curve $t \mapsto Φ (t, q)$ is an integral curve of $X$ and satisfies $Φ (0, q) = q$ .

L1step 2.1construct

4.1

If another curve in $M$ through $q \in U$ solved the same initial-value problem, its coordinate expression under $x$ would solve the Euclidean problem for $\tilde{X}$ with the same initial value. Uniqueness in [L3] then forces the two curves to agree.

L3step 3.1

5.1

Therefore $X$ has unique local integral curves depending smoothly on the initial point.

step 3.1step 4.1∎

Depends on

Used by

The fundamental theorem on flows Theorem
Through each point there is a unique maximal integral curve Theorem

Dependency tree · two levels

19 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)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)