TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Through each point there is a unique maximal integral curve

Statement

For every point $p \in M$ and every smooth vector field $X$ on $M$ , there is a unique maximal integral curve $γ_{p} : I_{p} \to M$ of $X$ with $γ_{p} (0) = p$ .

Facts & Assumptions

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

[L1]

Through each point there is a unique integral curve on some open interval about $0$ , depending smoothly on the initial value (Local existence, uniqueness, and smooth dependence for manifold integral curves).

Proof

technique · direct

1.1

By [L1], there exists at least one integral curve of $X$ through $p$ on some open interval about $0$ . If two such curves are defined on overlapping intervals, [L1] forces them to agree on the overlap because they solve the same initial-value problem at any common time.

L1given

2.1

Let $I_{p}$ be the union of all intervals carrying an integral curve through $p$ , and define $γ_{p} (t)$ by any one of those curves. Step 1.1 shows this is well defined on $I_{p}$ . Because all the intervals contain $0$ and pairwise overlap along the common trajectory, their union is again an interval.

step 1.1construct

3.1

The map $γ_{p} : I_{p} \to M$ is an integral curve, since near each $t \in I_{p}$ it coincides with one of the local curves from which it was assembled. If it extended to a larger interval, that larger curve would belong to the family defining $I_{p}$ , contradicting the definition of the union.

step 2.1

4.1

Therefore $γ_{p}$ is the unique maximal integral curve of $X$ through $p$ .

step 1.1step 3.1∎

Depends on

Used by

Complete vector fields Definition
Local and global flows generated by a vector field Definition
The fundamental theorem on flows Theorem

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