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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.88 Symplectic Manifolds

Gabriel P. Paternain

Symplectic geometry is the geometry that governs classical physics, and more generally plays an important role in helping us to understand the actions of groups on manifolds. It shares some features with Riemannian geometry and complex geometry, and there is an important special class of manifolds, the Kähler manifolds, in which all three geometric structures are unified.

1 Symplectic Linear Algebra

Just as RIEMANNIAN GEOMETRY [I.3 §6.10] is based on EUCLIDEAN GEOMETRY [I.3 §6.2], symplectic geometry is based on the geometry of the so-called linear symplectic space (Image2n, ω0).

Given two vectors ...

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