False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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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.
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Every boundary vector field has a local two-sided flow inside the manifold

Statement

False. On $[0, \infty)$ , the constant field $- \partial_{x}$ at $0$ has integral curve $t \mapsto - t$ , which immediately leaves the half-line for $t > 0$ .

Facts & Assumptions

Given: The manifold with boundary $M = [0, \infty)$ , the smooth constant vector field $X = - \partial_{x}$ , and the boundary point $0$ .

[L1]

Boundary-tangent vector fields have local two-sided flows preserving the boundary (Boundary-tangent fields have boundary-preserving local two-sided flows).

[L2]

Inward-pointing vector fields are guaranteed only a local forward flow at the boundary (Inward-pointing fields have local forward semiflows at the boundary).

Refutation

technique · direct

1.1

The integral curve through $0$ satisfies $γ^{'} (t) = - 1$ and $γ (0) = 0$ , hence $γ (t) = - t$ . For every $t > 0$ it lies outside $M$ , so even a local two-sided ambient solution need not restrict to a flow inside the manifold.

givenalgebra

2.1

This does not contradict [L1], because $X$ is not tangent at $0$ , or [L2], because $X$ points outward rather than inward there. The explicit trajectory in step 1.1 therefore refutes the unrestricted two-sided claim.

L1L2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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

Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)