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SYMPLECTIZATION

Contact geometry

Branch of geometry

There is a precise relation: the lift of a Legendrian submanifold in a symplectization of a contact manifold is a Lagrangian submanifold. The simplest example

Contact geometry

Contact_geometry

Symplectization

In mathematics, the symplectization (or symplectification) of a contact manifold is a symplectic manifold which naturally corresponds to it. Let ( V

Symplectization

Weinstein conjecture

α) can be embedded into a canonical symplectic manifold, called the symplectization of M, such that M is a contact type level set (of a canonically defined

Weinstein conjecture

Weinstein_conjecture

ADE classification

Mathematical classification

systems. He tried to introduce informal concepts of Complexification and Symplectization based on analogies between Picard–Lefschetz theory which he interprets

ADE classification

ADE_classification

Trivial cylinder

with the real numbers; in symplectic field theory, one considers the symplectization of a contact manifold. The product of a given embedded closed orbit

Trivial cylinder

Trivial_cylinder

Floer homology

Symplectic topology tool

Lagrangian, and its differential counts certain holomorphic strips in the symplectization of the contact manifold whose ends are asymptotic to given Reeb chords

Floer homology

Floer_homology

Bitangents of a quartic

28 lines which touch a general quartic plane curve in two places

1997, p. 13 – Arnold, Vladimir, 1997, Toronto Lectures, Lecture 2: Symplectization, Complexification and Mathematical Trinities, June 1997 (last updated

Bitangents of a quartic

Bitangents_of_a_quartic

Morse homology

Theory in differential topology

between the critical submanifolds are pseudoholomorphic curves in the symplectization of a contact manifold asymptotic to Reeb orbits in the relevant critical

Morse homology

Morse_homology

Helmut Hofer

German-American mathematician

S2CID 121426158. Hofer, Helmut (1993). "Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three".

Helmut Hofer

Helmut_Hofer

Seifert conjecture

arXiv:dg-ga/9708011. Hofer, H. (1993). "Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three".

Seifert conjecture

Seifert_conjecture