Symplectic topology
Encyclopedia
Symplectic geometry is a branch of differential geometry and differential topology
which studies symplectic manifold
s; that is, differentiable manifold
s equipped with a closed, nondegenerate 2-form
. Symplectic geometry has its origins in the Hamiltonian formulation
of classical mechanics
where the phase space
of certain classical systems takes on the structure of a symplectic manifold.
Symplectic geometry has a number of similarities and differences with Riemannian geometry
, which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors (called metric tensor
s). Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature
. This is a consequence of Darboux's theorem
which states that a neighborhood of any point of a 2n-dimensional symplectic manifold is isomorphic to the standard symplectic structure on an open set of R2n. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For example, every symplectic manifold is even-dimensional and orientable. Additionally, if M is a closed symplectic manifold, then the 2nd de Rham cohomology
group H2(M) is nontrivial; this implies, for example, that the only n-sphere that admits a symplectic form is the 2-sphere
.
Every Kähler manifold
is also a symplectic manifold. Well into the 1970s, symplectic experts were unsure whether any compact non-Kähler symplectic manifolds existed, but since then many examples have been constructed (the first was due to William Thurston
); in particular, Robert Gompf has shown that every finitely presented group occurs as the fundamental group
of some symplectic 4-manifold, in marked contrast with the Kähler case.
Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable complex structure compatible with the symplectic form. Mikhail Gromov, however, made the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a Kähler manifold except the requirement that the transition functions be holomorphic.
Gromov used the existence of almost complex structures on symplectic manifolds to develop a theory of pseudoholomorphic curve
s, which has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as Gromov–Witten invariants. These invariants also play a key role in string theory
.
The term "symplectic" is a calque
of "complex", introduced by ; previously, the "symplectic group" had been called the "line complex group".
Complex comes from the Latin com-plexus, meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek sym-plektikos (συμπλεκτικός); in both cases the suffix comes from the Indo-European root *plek-. This naming reflects the deep connections between complex and symplectic structures.
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
which studies 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...
s; that is, differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
s equipped with a closed, nondegenerate 2-form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
. Symplectic geometry has its origins in the Hamiltonian formulation
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
of classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
where the phase space
Phase space
In mathematics and physics, a phase space, introduced by Willard Gibbs in 1901, is a space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the phase space...
of certain classical systems takes on the structure of a symplectic manifold.
Symplectic geometry has a number of similarities and differences with Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
, which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors (called metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
s). Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature
Curvature of Riemannian manifolds
In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define it, now known as the curvature tensor...
. This is a consequence of Darboux's theorem
Darboux's theorem
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among them being symplectic geometry...
which states that a neighborhood of any point of a 2n-dimensional symplectic manifold is isomorphic to the standard symplectic structure on an open set of R2n. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For example, every symplectic manifold is even-dimensional and orientable. Additionally, if M is a closed symplectic manifold, then the 2nd de Rham cohomology
De Rham cohomology
In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes...
group H2(M) is nontrivial; this implies, for example, that the only n-sphere that admits a symplectic form is the 2-sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
.
Every Kähler manifold
Kähler manifold
In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
is also a symplectic manifold. Well into the 1970s, symplectic experts were unsure whether any compact non-Kähler symplectic manifolds existed, but since then many examples have been constructed (the first was due to William Thurston
William Thurston
William Paul Thurston is an American mathematician. He is a pioneer in the field of low-dimensional topology. In 1982, he was awarded the Fields Medal for his contributions to the study of 3-manifolds...
); in particular, Robert Gompf has shown that every finitely presented group occurs as the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of some symplectic 4-manifold, in marked contrast with the Kähler case.
Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable complex structure compatible with the symplectic form. Mikhail Gromov, however, made the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a Kähler manifold except the requirement that the transition functions be holomorphic.
Gromov used the existence of almost complex structures on symplectic manifolds to develop a theory 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, which has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as Gromov–Witten invariants. These invariants also play a key role in string theory
String theory
String theory is an active research framework in particle physics that attempts to reconcile quantum mechanics and general relativity. It is a contender for a theory of everything , a manner of describing the known fundamental forces and matter in a mathematically complete system...
.
Name
Symplectic geometry is also called symplectic topology although the latter is really a subfield concerned with important global questions in symplectic geometry.The term "symplectic" is a calque
Calque
In linguistics, a calque or loan translation is a word or phrase borrowed from another language by literal, word-for-word or root-for-root translation.-Calque:...
of "complex", introduced by ; previously, the "symplectic group" had been called the "line complex group".
Complex comes from the Latin com-plexus, meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek sym-plektikos (συμπλεκτικός); in both cases the suffix comes from the Indo-European root *plek-. This naming reflects the deep connections between complex and symplectic structures.
See also
- Contact geometryContact geometryIn mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle and specified by a one-form, both of which satisfy a 'maximum non-degeneracy' condition called 'complete non-integrability'...
- Hamiltonian mechanicsHamiltonian mechanicsHamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
- Moment mapMoment mapIn mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The moment map generalizes the classical notions of linear and angular momentum...
- Symplectic flow
- Symplectic frame bundle
- Symplectic integrationSymplectic integratorIn mathematics, a symplectic integrator is a numerical integration scheme for a specific group of differential equations related to classical mechanics and symplectic geometry. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations...
- Symplectic manifoldSymplectic manifoldIn 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...