TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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Every symplectic manifold admits a compatible almost-complex structure

Statement

Assume ${A C}_{ω}$ . Every symplectic manifold $(M, ω)$ admits a smooth almost-complex structure $J$ compatible with $ω$ .

Facts & Assumptions

Given: A symplectic manifold $(M, ω)$ and the axiom of countable choice ${A C}_{ω}$ .

[F1]

Under ${A C}_{ω}$ , every smooth manifold admits a Riemannian metric. The Axiom of Countable Choice ( ${A C}_{ω}$ ), Every smooth manifold admits a riemannian metric.

[F2]

Smooth positive-definite self-adjoint bundle endomorphisms have unique smooth positive square roots. Positive-definite bundle endomorphisms have smooth positive square roots.

[F3]

An endomorphism $J$ is compatible when $J^{2} = - I$ and $ω (\cdot, J \cdot)$ is positive-definite symmetric. Compatible complex structure on a symplectic vector space.

Proof

technique · direct

1.1

Spend the assumed ${A C}_{ω}$ only through [F1] to choose a smooth Riemannian metric $k$ . Define the smooth invertible bundle map $A$ by $k (u, v) = ω (u, A v)$ . Fibrewise skew-symmetry gives $A^{*} = - A$ , so $- A^{2} = A^{*} A$ is smooth, self-adjoint, and positive definite.

F1givenalgebra

2.1

By [F2], $P = (- A^{2})^{1 / 2}$ is a smooth positive bundle endomorphism. Fibrewise, $A$ preserves the eigenspaces of $- A^{2}$ and $P$ is scalar on each of them, so $P$ commutes with $A$ . Hence $J = A P^{- 1}$ is smooth and $J^{2} = - I$ .

F2step 1.1algebra

3.1

Fibrewise, $ω (u, J v) = k (u, P^{- 1} v) = k (P^{- 1 / 2} u, P^{- 1 / 2} v),$ which is symmetric and positive definite. Thus [F3] proves compatibility. Empty and zero-dimensional manifolds carry the unique such structure.

F2F3step 1.1step 2.1∎

Depends on

Used by

Weinstein Lagrangian neighborhood theorem Theorem

Dependency tree · two levels

16 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

Eckhard Meinrenken, Symplectic Geometry (standard reference, not scraped)