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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

A noncompact continuation datum can lose its trajectories at infinity Counterexample
An index-one moduli locus can be infinite without compactness Counterexample
Critical levels connected by a trajectory cannot always be interchanged Counterexample
A regular continuation datum between Morse--Smale pairs Definition
Geometric convergence to a broken trajectory Definition
Morse function adapted to a cobordism Definition
Morse--Smale pairs Definition
Parametrized Morse trajectory space Definition
Stable and unstable sets of a critical point Definition
The mod-two Morse chain group Definition
Broken trajectories in an index-two torus moduli space Example
The Morse complex of the circle Example
The Morse complex of the two-sphere Example
A regular level identifies unparametrized trajectories Lemma
Boundary orientation of the compactified one-dimensional Morse moduli space Lemma
Broken Morse trajectories have strictly decreasing critical values and indices Lemma
Gluing handle Morse models along collars Lemma
Gradient-like perturbation separates adjacent critical levels Lemma
Standard handle admits an adapted Morse function Lemma
Time translation acts freely on nonconstant trajectories Lemma
Unstable orientations induce orientations of the trajectory moduli spaces Lemma
Every Morse function admits a complete downward gradient-like field on a closed manifold Proposition
Morse cancellation criterion via a unique connecting orbit Proposition
Adapted excellent Morse functions exist on compact cobordisms Theorem
Compactness up to breaking of Morse trajectory spaces Theorem
Local stable and unstable manifolds at a Morse critical point Theorem
Relative Morse--Smale perturbation of a gradient-like field Theorem
Self-indexing Morse functions exist Theorem
The unparametrized trajectory space is a smooth manifold Theorem

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)