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

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Downward gradient-like vector fields for a Morse function

Definition

Let $f : M \to R$ be Morse. A smooth vector field $X$ is downward gradient-like for $f$ if both conditions hold:

$d f_{x} (X_{x}) < 0$ at every $x \notin Crit (f)$ ; and
for every $p \in Crit (f)$ there are Morse coordinates $(u, v)$ centred at $p$ , with $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ , in which $X = 2 \sum_{i} u_{i} \partial_{u_{i}} - 2 \sum_{j} v_{j} \partial_{v_{j}} .$

Thus the field is the negative Euclidean gradient in the required local Morse model, not merely a strictly descending field off the critical set.

Depends on

Used by

Dependency tree · two levels

7 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

Liviu I. Nicolaescu, An Invitation to Morse Theory, §2.4 (standard reference, not scraped)