- Symplectic geometry
**Symplectic geometry**is a branch of differential topology/geometry which studiessymplectic manifold s; that is,differentiable manifold s equipped with a closed,nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation ofclassical mechanics where thephase 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 (calledmetric tensor s). Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. This is a consequence ofDarboux'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**R**^{2n}. 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 2ndde Rham cohomology group "H^{2}(M)" is nontrivial; this implies, for example, that the onlyn-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 toWilliam Thurston ); in particular,Robert Gompf has shown that everyfinitely presented group occurs as thefundamental 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 compatiblealmost complex structure s, so that they satisfy all the axioms for a Kähler manifold "except" the requirement that the transition functions beholomorphic .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 asGromov-Witten invariant s. These invariants also play a key role instring 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", byHermann 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*] , byMurray Gell-Mann .*[*] This naming reflects the deep connections between complex and symplectic structures.*http://www.math.hawaii.edu/~gotay/Symplectization.pdf*] , p. 13**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.)"

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