Lee Hwa Chung theorem
Encyclopedia
The Lee Hwa Chung theorem is a theorem
in symplectic topology
.
The statement is as follows. Let M be a symplectic manifold
with symplectic form ω. Let
be a differential k-form
on M which is invariant for all Hamiltonian vector field
s. Then:
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
in symplectic topology
Symplectic topology
Symplectic geometry is a branch of differential geometry and differential topology which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form...
.
The statement is as follows. Let M be 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...
with symplectic form ω. Let
be a differential k-formDifferential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
on M which is invariant for all Hamiltonian vector field
Hamiltonian vector field
In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field, defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is a geometric manifestation of Hamilton's equations...
s. Then:
- If k is odd,
- If k is even,
, where 