- Symplectic geometry
Symplectic geometry is a branch of differential topology/geometry which studies
symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanicswhere the phase spaceof 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 theoremwhich 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 cohomologygroup "H2(M)" is nontrivial; this implies, for example, that the only n-spherethat admits a symplectic form is the 2-sphere.
Kähler manifoldis 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 Gompfhas shown that every finitely presented groupoccurs as the fundamental groupof 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.
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
calqueof "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.
* Symplectic integration
Dusa McDuffand 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.)"
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