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

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

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

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

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...

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

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...

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

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

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

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...

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...

symplectic manifold can be made to look locally like the linear symplectic space C n {\displaystyle \mathbb {C} ^{n}} with its canonical symplectic form...

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...

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

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

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

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

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

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

study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups...

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

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

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...

{\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...

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

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

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

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

complex manifold Calabi–Yau manifold Hyperkähler manifold K3 surface hypercomplex manifold Quaternion-Kähler manifold Symplectic topology Symplectic space...