TheoremStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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Global stable and unstable manifolds are immersed Euclidean spaces

Statement

For a complete downward gradient-like flow on an $n$ -manifold and a critical point $p$ of index $λ$ , $W^{u} (p)$ and $W^{s} (p)$ are immersed submanifolds diffeomorphic to $R^{λ}$ and $R^{n - λ}$ , respectively. This does not assert that either global submanifold is embedded.

Facts & Assumptions

Given: A complete downward gradient-like flow $Φ$ and a critical point $p$ of index $λ$ .

[F1]

The local unstable and stable sets are embedded disks of dimensions $λ$ and $n - λ$ (Local stable and unstable manifolds at a Morse critical point).

[F2]

$W^{s} (p)$ and $W^{u} (p)$ are defined by forward and backward convergence under the global flow (Stable and unstable sets of a critical point).

[F3]

Every global flow map $Φ_{t}$ is a diffeomorphism with inverse $Φ_{- t}$ (The fundamental theorem on flows).

Proof

technique · direct

1.1

Let $D^{u}$ and $D^{s}$ be the local disks from [F1]. From the defining limits in [F2], every $x \in W^{u} (p)$ has $Φ_{- T} (x) \in D^{u}$ for some $T \geq 0$ , and every $x \in W^{s} (p)$ has $Φ_{T} (x) \in D^{s}$ for some $T \geq 0$ .

F1F2given

2.1

Hence $W^{u} (p) = ⋃_{T \geq 0} Φ_{T} (D^{u})$ and $W^{s} (p) = ⋃_{T \geq 0} Φ_{- T} (D^{s})$ . By [F3], these are increasing compatible immersed-manifold charts transported from the disks.

F3step 1.1

3.1

The standard flow-exhaustion parametrization of these compatible disks identifies the unions with the corresponding Euclidean spaces; their dimensions are those in [F1]. Thus they are immersed submanifolds diffeomorphic to $R^{λ}$ and $R^{n - λ}$ , with no embeddedness conclusion.

F1step 2.1∎

Depends on

Used by

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Dependency tree · two levels

8 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, Proposition 2.4.2 (standard reference, not scraped)