Trivial cylinder
Encyclopedia
In geometry
and topology
, trivial cylinders are certain pseudoholomorphic curve
s appearing in certain cylindrical manifold
s.
In Floer homology
and its variants, chain complexes or differential graded algebras are generated by certain combinations of closed orbits of vector fields. In symplectic Floer homology, one considers the Hamiltonian vector field
of a Hamiltonian function on a symplectic manifold
; in symplectic field theory, contact homology, and their variants, one considers the Reeb vector field
associated to a contact form, or more generally a stable Hamiltonian structure.
The differentials all count some flavor of pseudoholomorphic curve
s in a manifold with a cylindrical almost-complex structure whose ends at negative infinity are the given collection of closed orbits. For instance, in symplectic Floer homology, one considers the product of the mapping torus of a symplectomorphism with the real numbers; in symplectic field theory, one considers the symplectization of a contact manifold.
The product of a given embedded closed orbit with R is always a pseudoholomorphic curve, and such a curve is called a trivial cylinder. Trivial cylinders do not generally contribute to the aforementioned differentials, but they may appear as components of more complicated curves which do.
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, trivial cylinders are certain pseudoholomorphic curve
Pseudoholomorphic curve
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of...
s appearing in certain cylindrical manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
s.
In Floer homology
Floer homology
Floer homology is a mathematical tool used in the study of symplectic geometry and low-dimensional topology. First introduced by Andreas Floer in his proof of the Arnold conjecture in symplectic geometry, Floer homology is a novel homology theory arising as an infinite dimensional analog of finite...
and its variants, chain complexes or differential graded algebras are generated by certain combinations of closed orbits of vector fields. In symplectic Floer homology, one considers the 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...
of a Hamiltonian function on 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...
; in symplectic field theory, contact homology, and their variants, one considers the Reeb vector field
Reeb vector field
In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry including:...
associated to a contact form, or more generally a stable Hamiltonian structure.
The differentials all count some flavor of pseudoholomorphic curve
Pseudoholomorphic curve
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of...
s in a manifold with a cylindrical almost-complex structure whose ends at negative infinity are the given collection of closed orbits. For instance, in symplectic Floer homology, one considers the product of the mapping torus of a symplectomorphism with the real numbers; in symplectic field theory, one considers the symplectization of a contact manifold.
The product of a given embedded closed orbit with R is always a pseudoholomorphic curve, and such a curve is called a trivial cylinder. Trivial cylinders do not generally contribute to the aforementioned differentials, but they may appear as components of more complicated curves which do.