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

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The flow of a vector field tangent to a closed embedded submanifold preserves it

Statement

Let $S ↪ M$ be a closed embedded submanifold, and let $X$ be a smooth vector field on $M$ tangent to $S$ . Then for every $(t, p)$ in the domain of the flow of $X$ with $p \in S$ , one has $Φ_{t} (p) \in S$ .

Facts & Assumptions

Given: A closed embedded submanifold $S ↪ M$ , a smooth vector field $X$ tangent to $S$ , and the maximal flow $Φ$ of $X$ .

[L1]

A tangent vector field restricts to a smooth vector field on the embedded submanifold (A vector field tangent to an embedded submanifold restricts to a vector field on it).

[L2]

The maximal flow time slices are exactly the maximal integral curves (The fundamental theorem on flows).

Proof

technique · direct

1.1L1given

By [L1], the restriction $X ∣_{S}$ is a smooth vector field on $S$ . Let $p \in S$ , and let $η$ be the maximal integral curve of $X ∣_{S}$ through $p$ . Then $η$ is also an integral curve of the ambient field $X$ .

2.1L2step 1.1

By [L2], the ambient curve $t \mapsto Φ_{t} (p)$ is the maximal integral curve of $X$ through $p$ . Since step 1.1 gives another integral curve of $X$ through the same initial point, uniqueness forces $η (t) = Φ_{t} (p)$ wherever both are defined.

3.1step 2.1∎

Because the image of $η$ lies in $S$ , step 2.1 shows that $Φ_{t} (p) \in S$ for every time for which the flow is defined.

Depends on

Used by

Boundary-tangent fields have boundary-preserving local two-sided flows Theorem

Dependency tree · two levels

11 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)