LemmaStatement: Literature-sourcedProof: Literature-sourcedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Morse--Smale transversality and surjectivity of the linearized flow operator

Statement

Let $f$ be Morse and let $γ$ connect critical points $p, q$ for $X = - {grad}_{g} f$ . Let $E_{γ} = C_{0}^{1} (R, γ^{*} T M)$ be the Banach space of $C^{1}$ sections for which both $ξ$ and $\nabla_{t} ξ$ tend to zero at $\pm \infty$ , with the supremum $C^{1}$ norm, and let $F_{γ} = C_{0}^{0} (R, γ^{*} T M)$ have the supremum norm. For the tangent Levi--Civita connection metric-dual to the cotangent connection, put

$D_{γ} ξ = \nabla_{t} ξ + \nabla_{ξ} {grad}_{g} f .$

Then $D_{γ} : E_{γ} \to F_{γ}$ is bounded Fredholm of index $λ (p) - λ (q)$ , and it is surjective if and only if $W^{u} (p)$ and $W^{s} (q)$ are transverse at $γ (0)$ .

Facts & Assumptions

Given: A Morse function $f$ , a connecting orbit $γ$ between its critical points $p, q$ , and the displayed $C_{0}^{1}$ and $C_{0}^{0}$ Banach spaces.

[F1]

Covariant Hessians define the linearization of the gradient equation (Riemannian metrics, symmetric cotangent-bundle connections, and covariant Hessians).

[F2]

The point-marked trajectory space is the stable--unstable intersection (Parametrized Morse trajectory space).

Proof

technique · direct

1.1

Differentiating $\dot{γ} + {grad}_{g} f (γ) = 0$ in a decaying variation gives the displayed operator, by [F1]. Its kernel is the tangent space of the point-marked solution set.

F1given

2.1

The hyperbolic Hessians at $p$ and $q$ give exponential dichotomies at the two ends. The standard first-order Fredholm theorem therefore gives index $λ (p) - λ (q)$ . Its adjoint solvability condition identifies the dual cokernel with the annihilator of $T_{γ (0)} W^{u} (p) + T_{γ (0)} W^{s} (q)$ , rather than canonically identifying the cokernel itself with a tangent-space quotient.

F2step 1.1

3.1

That annihilator vanishes exactly when the two tangent spaces span $T_{γ (0)} M$ , which is transversality. Since a Fredholm operator is onto exactly when its cokernel vanishes, this proves the biconditional.

step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

13 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

Alberto Abbondandolo and Pietro Majer, Lectures on the Morse Complex, Lemma 2.21(ii) (standard reference, not scraped)