DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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.

Compatible almost-Kähler metric

Definition

Let $(M, ω)$ be symplectic and let $J$ be an $ω$ -compatible smooth almost-complex structure. The Riemannian metric

$g_{J} (X, Y) = ω (X, J Y)$

is the compatible almost-Kähler metric, and $(M, ω, J, g_{J})$ is an almost-Kähler manifold. No integrability of $J$ is included in this term.

Once any two of $ω$ , $J$ , and $g_{J}$ are fixed subject to compatibility, the displayed identity determines the third.

Depends on

Compatible complex structure on a symplectic vector space

Used by

Compatible almost-complex structures and Kähler geometry Remark

Dependency tree · two levels

2 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)