PropositionStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Every Morse function admits a complete downward gradient-like field on a closed manifold

Statement

If $M$ is a closed smooth manifold and $f : M \to R$ is Morse, then $f$ admits a complete downward gradient-like vector field.

Facts & Assumptions

Given: A closed smooth manifold $M$ and a Morse function $f : M \to R$ .

[F1]

$f$ has finitely many critical points (A Morse function on a compact manifold has finitely many critical points).

[F2]

Every smooth vector field on a compact manifold is complete (Every smooth vector field on a compact manifold is complete).

[F3]

A downward gradient-like field must have strict descent off the critical set and the stated Morse-coordinate model at every critical point (Downward gradient-like vector fields for a Morse function).

Proof

technique · direct

1.1

By [F1], choose pairwise disjoint Morse-coordinate neighbourhoods of the finitely many critical points, and smaller neighbourhoods inside them. On each smaller neighbourhood prescribe the local field in [F3].

F1F3choose

2.1

On the complement of the smaller neighbourhoods, $d f$ is nowhere zero. In each coordinate patch choose a vector $Y$ with $d f (Y) < 0$ ; a partition of unity and cutoffs that equal one on the smaller neighbourhoods patch these choices with the prescribed local fields to a smooth $X$ satisfying both clauses of [F3].

F3step 1.1construct

3.1

The resulting $X$ is smooth on the compact manifold $M$ , so [F2] makes it complete. Therefore it is the required complete downward gradient-like field.

F2step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Ralph L. Cohen, Bundles, Manifolds, and Homotopy, Proposition 13.6 (standard reference, not scraped)