# Symplectic geometry

﻿
Symplectic geometry

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 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 tensors). 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 2"n"-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 compact 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 curves, 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.

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 of "complex", by Hermann Weyl; 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-plektos" (συμπλεκτικός); in both cases the suffix comes from the Indo-European root *plek-. [ [http://www.santafe.edu/~mgm/plectics.html etymology of symplectic] , by Murray Gell-Mann.] [ [http://www.math.hawaii.edu/~gotay/Symplectization.pdf] , p. 13] This naming reflects the deep connections between complex and symplectic structures.

ee also

* Symplectic flow
* Hamiltonian mechanics
* Symplectic integration
* Riemannian geometry
* Contact geometry
* Moment map

References

* Dusa McDuff and D. Salamon, "Introduction to Symplectic Topology", Oxford University Press, 1998. ISBN 0-19-850451-9.
* A. T. Fomenko, "Symplectic Geometry (2nd edition)" (1995) Gordon and Breach Publishers, ISBN 2-88124-901-9. "(An undergraduate level introduction.)"

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Geometry — (Greek γεωμετρία ; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences. Initially a body of… …   Wikipedia

• Geometry and topology — In mathematics, geometry and topology is an umbrella term for geometry and topology, as the line between these two is often blurred, most visibly in local to global theorems in Riemannian geometry, and results like the Gauss–Bonnet theorem and… …   Wikipedia

• 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.… …   Wikipedia

• Symplectic sum — In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a… …   Wikipedia

• Symplectic group — For finite groups with all characteristc abelian subgroups cyclic, see group of symplectic type. Group theory …   Wikipedia

• Symplectic cut — In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that… …   Wikipedia

• Symplectic integrator — In mathematics, a symplectic integrator (SI) 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… …   Wikipedia

• Symplectic filling — In mathematics, a filling of a manifold X is a cobordism W between X and the empty set. More to the point, the n dimensional topological manifold X is the boundary of an n+1 dimensional manifold W . Perhaps the most active area of current… …   Wikipedia

• symplectic — adjective Describing the geometry of differentiable manifolds equipped with a closed, nondegenerate 2 form …   Wiktionary

• Differential geometry — A triangle immersed in a saddle shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines. Differential geometry is a mathematical discipline that uses the techniques of differential and integral calculus, as well as… …   Wikipedia