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A negative-gradient trajectory on a compact Morse manifold has single critical alpha and omega limits

Statement

Let $M$ be compact, let $f : M \to R$ be Morse, and let $γ$ be a negative-gradient trajectory. Then $γ$ is full and there are critical points $α (γ)$ and $ω (γ)$ such that

$\lim_{t \to - \infty} γ (t) = α (γ), \lim_{t \to \infty} γ (t) = ω (γ) .$

Facts & Assumptions

Given: A compact smooth manifold $M$ , a Morse function $f$ , and a negative-gradient trajectory $γ$ .

[F1]

A smooth vector field on a compact manifold is complete (Every smooth vector field on a compact manifold is complete).

[F2]

Precompact full tails have nonempty compact connected invariant limit sets (Precompact trajectory tails have nonempty compact connected flow-invariant limit sets).

[F3]

Such limit sets for a negative-gradient trajectory consist of critical points (Every precompact end-limit point of a negative-gradient trajectory is critical).

[F4]

A Morse function on a compact manifold has finitely many critical points (A Morse function on a compact manifold has finitely many critical points).

Proof

technique · direct

1.1

By [F1], the negative-gradient field is complete, so $γ$ has domain $R$ . Both tails have compact closure because they lie in $M$ .

F1given

2.1

By [F2] each of $α (γ)$ and $ω (γ)$ is nonempty and connected, and [F3] places it in $Crit (f)$ .

F2F3step 1.1

3.1

By [F4], $Crit (f)$ is finite and hence discrete. A connected subset of a discrete finite set is one point, so both limit sets are single critical points.

F4step 2.1

4.1

A trajectory with singleton tail-limit set converges to that point: otherwise a sequence of tail times outside a fixed neighbourhood would have a limit point in the same tail-limit set. Thus the two displayed limits hold.

step 2.1step 3.1∎

Depends on

Used by

A Morse trajectory from one critical point to another Definition

Dependency tree · two levels

15 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, Theorem 13.2 (standard reference, not scraped)