Symplectization

Symplectization

In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.

Definition

Let (V,xi) be a contact manifold, and let x in V. Consider the set:S_xV = {eta in T^*_xV - {0} ,|, ker eta = xi_x} subset T^*_xVof all nonzero 1-forms at x, which have the contact plane xi_x as their kernel. The union:SV = igcup_{x in V}S_xV subset T^*Vis a symplectic submanifold of the cotangent bundle of V, and thus possesses a natural symplectic structure.

The projection pi : SV o V supplies the symplectization with the structure of a principal bundle over V with structure group R^* equiv R - {0}.

The coorientable case

When the contact structure xi is cooriented by means of a contact form alpha, there is another version of symplectization, in which only forms giving the same coorientation to xi as alpha are considered:

:S^+_xV = {eta in T^*_xV - {0} ,|, eta = lambdaalpha,,lambda > 0} subset T^*_xV,

:S^+V = igcup_{x in V}S^+_xV subset T^*V.

Note that xi is coorientable if and only if the bundle pi : SV o V is trivial. Any section of this bundle is a coorienting form for the contact structure.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Contact geometry — Contact form redirects here. For a web email form, see Form (web)#Form to email scripts. The standard contact structure on R3. Each point in R3 has a plane associated to it by the contact structure, in this case as the kernel of the one form dz − …   Wikipedia

  • Symplectic geometry — is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2 form. Symplectic geometry has its origins in the Hamiltonian formulation of classical… …   Wikipedia

  • ADE classification — The simply laced Dynkin diagrams classify diverse mathematical objects. In mathematics, the ADE classification (originally A D E classifications) is the complete list of simply laced Dynkin diagrams or other mathematical objects satisfying… …   Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Morse homology — In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be… …   Wikipedia

  • Weinstein conjecture — In mathematics, the Weinstein conjecture refers to a general existence problem for periodic orbits of Hamiltonian or Reeb vector flows. More specifically, the current understanding is that a regular compact contact type level set of a Hamiltonian …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”