called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally...

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds...

In differential geometry, an almost symplectic structure on a differentiable manifold M {\displaystyle M} is a two-form ω {\displaystyle \omega } on M...

Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in...

classical Lie groups Symplectic manifold, Symplectic matrix, Symplectic vector space, Symplectic representation Unitary group Θ10 "Symplectic group", Encyclopedia...

In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism...

the manifold, whose equivalence is the content of the Frobenius theorem. Contact geometry is in many ways an odd-dimensional counterpart of symplectic geometry...

nondegenerate 2-form ω, called the symplectic form. A symplectic manifold is an almost symplectic manifold for which the symplectic form ω is closed: dω = 0. A...

a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure....

manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in symplectic geometry...

to be done. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian...

A symplectic space may refer to: Symplectic manifold Symplectic vector space This disambiguation page lists articles associated with the title Symplectic...

infinite-dimensional manifold and a real valued function on it. In the symplectic version, this is the free loop space of a symplectic manifold with the symplectic action...

W(V). A symplectic manifold is a smooth manifold with a smoothly-varying closed symplectic form on each tangent space. Maslov index A symplectic representation...

angle. A symplectic manifold is a manifold equipped with a closed, nondegenerate 2-form. This condition forces symplectic manifolds to be even-dimensional...

algebra Symplectic integrator Symplectic manifold Symplectic matrix Symplectic representation Symplectic vector space, a vector space with a symplectic bilinear...

symplectic manifold can be used to define a Hamiltonian system. The function H is known as "the Hamiltonian" or "the energy function." The symplectic...

hyperkähler manifolds have been extensively studied using techniques from algebraic geometry, sometimes under the name holomorphically symplectic manifolds. The...

(to H {\displaystyle H} ). Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are G-structures...

a symplectic vector field is one whose flow preserves a symplectic form. That is, if ( M , ω ) {\displaystyle (M,\omega )} is a symplectic manifold with...

Topological manifold Almost complex manifold Almost symplectic manifold Calibrated manifold Complex manifold Contact manifold CR manifold Finsler manifold Hermitian...

is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's theorem...

mathematics, a Lagrangian foliation or polarization is a foliation of a symplectic manifold, whose leaves are Lagrangian submanifolds. It is one of the steps...

specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two...

In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named...

{\displaystyle n} th exterior power of the symplectic form on a symplectic manifold is a volume form. Many classes of manifolds have canonical volume forms: they...

Lagrangian submanifold L into a symplectic manifold M, and π : M ↠ B is a Lagrangian fibration of the symplectic manifold M. The caustic is a subset of...

a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum...

Kähler manifold. Any symplectic manifold admits a compatible almost complex structure making it into an almost Kähler manifold. A Kähler manifold is an...

and embeddability of manifolds in Euclidean space. Suppose a manifold X {\displaystyle X} is embedded in to a symplectic manifold ( M , ω ) {\displaystyle...