DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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A Morse trajectory from one critical point to another

Definition

Fix a Riemannian metric $g$ on $M$ . For critical points $p, q$ of a Morse function $f : M \to R$ , a Morse trajectory from $p$ to $q$ is a nonconstant full trajectory of $- {grad}_{g} f$ with

$\lim_{t \to - \infty} γ (t) = p and \lim_{t \to \infty} γ (t) = q .$

On a compact manifold the preceding endpoint lemma supplies these limits. On a noncompact manifold, their existence is part of this definition's hypothesis.

Depends on

Used by

Morse trajectories have a positive energy drop Lemma

Dependency tree · two levels

9 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, §13.1 (standard reference, not scraped)