Lagrangian foliation
Encyclopedia
In mathematics
, a Lagrangian foliation or polarization is a foliation
of a symplectic manifold
, whose leaves are Lagrangian submanifolds. It is one of the steps involved in the geometric quantization
of a square-integrable functions on a symplectic manifold.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Lagrangian foliation or polarization is a foliation
Foliation
In mathematics, a foliation is a geometric device used to study manifolds, consisting of an integrable subbundle of the tangent bundle. A foliation looks locally like a decomposition of the manifold as a union of parallel submanifolds of smaller dimension....
of a symplectic manifold
Symplectic manifold
In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...
, whose leaves are Lagrangian submanifolds. It is one of the steps involved in the geometric quantization
Geometric quantization
In mathematical physics, geometric quantization is a mathematical approach to defining a quantum theory corresponding to a given classical theory. It attempts to carry out quantization, for which there is in general no exact recipe, in such a way that certain analogies between the classical theory...
of a square-integrable functions on a symplectic manifold.