Gromov's Non-Squeezing Theorem states that a ball in symplectic space cannot be mapped, by a transformation preserving the symplectic structure, into a cylinder of smaller radius, even though such a map is easy to achieve using an ordinary volume-preserving transformation. Named for Mikhail Gromov, who proved it in 1985, it revealed that symplectic geometry carries a genuine rigidity invisible to volume alone, and it launched the modern theory of symplectic capacities.
Facts
StatementOne cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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Source Non-squeezing theorem (Wikipedia)
Sources
1. Gromov's non-squeezing theorem (Wikipedia)
Background and statement section
one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder.
Background and statement section, second sentence
It was first proven in 1985 by Mikhail Gromov.
View the SourceNon-squeezing theorem (Wikipedia)
In Branch: Geometry, Lead sentenceQuote, In Branch: Geometry, Lead sentence
one of the most important theorems in symplectic geometry.
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