CorollaryStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 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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No Morse--Smale trajectories for nonpositive index drop

Statement

If $(f, X)$ is Morse--Smale, $p \neq q$ , and $λ (p) \leq λ (q)$ , then $M (p, q) = \emptyset$ .

Facts & Assumptions

Given: Distinct critical points $p, q$ in a Morse--Smale pair with $λ (p) \leq λ (q)$ .

[F1]

The unparametrized trajectory space, when nonempty, is a smooth manifold of dimension $λ (p) - λ (q) - 1$ (The unparametrized trajectory space is a smooth manifold).

Proof

technique · direct

1.1F1givenassume-contraalgebra

If $M (p, q)$ were nonempty, [F1] would give it dimension $λ (p) - λ (q) - 1 \leq - 1$ .

2.1step 1.1discharge-contradiction∎

A nonempty smooth manifold has a nonnegative integer dimension, contradicting step 1.1. Thus the moduli space is empty.

Depends on

The unparametrized trajectory space is a smooth manifold

Used by

Broken continuation trajectories and geometric convergence Definition
Broken Morse trajectories Definition
The mod-two Morse differential Definition
The Morse complex of the two-sphere Example
Breaking length is bounded by the index drop Lemma
Broken Morse trajectories have strictly decreasing critical values and indices Lemma
The index-two compactification is a compact one-manifold with boundary Theorem
The mod-two Morse differential squares to zero Theorem

Dependency tree · two levels

14 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

Alexander F. Ritter, Part III Morse Homology, Lecture 9 (standard reference, not scraped)