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Local stable and unstable manifolds at a Morse critical point

Statement

Let $p$ be a critical point of index $λ$ of a Morse function on an $n$ -manifold, and let $X$ be downward gradient-like. In the Morse coordinates of its definition, the local unstable and stable manifolds are respectively

${v = 0} ≅ R^{λ}, {u = 0} ≅ R^{n - λ} .$

After restricting to sufficiently small balls, they are embedded disks tangent at $p$ to the negative and positive Hessian eigenspaces, respectively.

Facts & Assumptions

Given: A Morse critical point $p$ of index $λ$ and a downward gradient-like field $X$ .

[F1]

In Morse coordinates $f = f (p) - ∣ u ∣^{2} + ∣ v ∣^{2}$ and $X = 2 u \partial_{u} - 2 v \partial_{v}$ (Downward gradient-like vector fields for a Morse function).

[F2]

The index and coindex are the dimensions of the negative and positive Hessian directions (Nondegenerate critical points, nullity, index, and coindex).

Proof

technique · direct

1.1

By [F1], the coordinate flow solves $\dot{u} = 2 u$ and $\dot{v} = - 2 v$ , hence $u (t) = e^{2 t} u (0)$ and $v (t) = e^{- 2 t} v (0)$ .

F1givenalgebra

2.1

A point remains near $p$ and converges to it in forward time exactly when $u (0) = 0$ ; in backward time exactly when $v (0) = 0$ . Thus the local stable disk is ${u = 0}$ and the local unstable disk is ${v = 0}$ .

step 1.1

3.1

By [F2], the $u$ -space has dimension $λ$ and is the negative Hessian space, while the $v$ -space has dimension $n - λ$ and is the positive one. This gives the claimed disk dimensions and tangent spaces.

F2step 2.1∎

Depends on

Used by

Global stable and unstable manifolds are immersed Euclidean spaces Theorem

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

Ralph L. Cohen, Bundles, Manifolds, and Homotopy, Proposition 13.8 and Theorem 13.9 (standard reference, not scraped)