TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Morse--Smale metrics are residual for a fixed Morse function

Statement

Assume dependent choice. Let $M$ be closed and $f : M \to R$ Morse. The set of smooth Riemannian metrics $g$ for which $(f, g)$ is Morse--Smale is residual in the $C^{\infty}$ metric space.

Facts & Assumptions

Given: Dependent choice, a closed manifold $M$ , and a fixed Morse function $f$ .

[F1]

A fixed pair of critical points is transverse for a residual set of smooth metrics (Baire diagonal passage from finite regularity to smooth metrics).

[F2]

A Morse function on a compact manifold has finitely many critical points.

Proof

technique · direct

1.1

By [F2], there are only finitely many ordered pairs $(p, q)$ of critical points. For each pair, take the residual set of metrics supplied by [F1].

F1F2given

2.1

Their finite intersection is residual and consists exactly of metrics for which every $W^{u} (p)$ is transverse to every $W^{s} (q)$ , namely the Morse--Smale metrics.

F1step 1.1

3.1

Thus the conclusion is residual in the smooth metric space. No openness or simultaneous statement about varying functions or continuation families has been used.

step 2.1∎

Depends on

Used by

Morse--Smale residuality does not assert simultaneous genericity for all data Remark

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

Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex, §2.12 (standard reference, not scraped)