TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedaudited 2026-09-07

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The unparametrized trajectory space is a smooth manifold

Statement

For a Morse--Smale pair and distinct critical points $p, q$ , $M (p, q)$ is a smooth manifold of dimension $λ (p) - λ (q) - 1$ .

Facts & Assumptions

Given: A Morse--Smale pair and distinct $p, q$ with nonempty $\tilde{M} (p, q)$ .

[F1]

The parametrized space is a smooth manifold of dimension $λ (p) - λ (q)$ (A parametrized Morse trajectory space is a manifold).

[F2]

A regular intermediate level gives one representative of each time orbit (A regular level identifies unparametrized trajectories).

Proof

technique · direct

1.1

Choose a regular $c$ strictly between $f (q)$ and $f (p)$ . The map $f$ restricted to $\tilde{M} (p, q)$ has nonzero derivative along the flow direction, since $d f (X) < 0$ away from critical points. Hence its $c$ -level is a codimension-one smooth submanifold.

F1givenalgebra

2.1

By [F2], that level is bijective to $M (p, q)$ ; transport its smooth structure across this bijection. Its dimension is $λ (p) - λ (q) - 1$ by [F1] and step 1.1.

F1F2step 1.1algebra∎

Depends on

Used by

No Morse--Smale trajectories for nonpositive index drop Corollary
A Morse--Smale flow on the circle Example
Regular-level slices for unparametrized trajectories Example
Index-one trajectory spaces are zero-dimensional Proposition
Index-two trajectory spaces are one-dimensional Proposition
Ambient orientability is not required for Morse--Smale transversality Remark

Dependency tree · two levels

16 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

Michèle Audin and Mihai Damian, Morse Theory and Floer Homology, §2.2.b (standard reference, not scraped)