Symplectic geometry

equation in classical mechanics

bilinear differential operation on scalar fields on a symplectic (or, more generally, Poisson) manifold

conjectured relation between pairs of Calabi–Yau manifolds; situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory

branch of differential geometry and differential topology

in differential geometry, a smooth manifold equipped with a closed, nondegenerate differential 2-form

group of matrices preserving a non-degenerate alternating quadratic form

vector space equipped with an alternating nondegenerate bilinear form

sets of coordinates which can be used to describe a physical system at any given point in time

smooth manifold carrying compatible complex, Riemannian, and symplectic structures

mathematical concept

formulation of quantum mechanics in phase space

Mathematical structure in differential geometry

isomorphism between the tangent and cotangent bundles on a smooth manifold; induced by either a RIemannian or symplectic structure

foundational result in symplectic geometry

Kähler–Einstein metric on a complex projective space

vector field associated to a hamiltonian on a symplectic manifold

scientific theory

associative algebra together with a Lie bracket that also satisfies Leibniz's law

canonical differential form defined on the cotangent bundle of a smooth manifold

quantization method for constrained Hamiltonian systems with second-class constraints

tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action, generalizing the classical notions of linear and angular momentum

vector space equipped with both a Lie bracket and a Lie cobracket, with certain compatibility conditions between the two