Symplectic geometry

Phase portrait of the Van der Pol oscillator, a one-dimensional system. Phase space was the original object of study in symplectic geometry.

Symplectic geometry is a branch of differential geometry and differential topology that 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 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 maps 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", introduced by Weyl (1939, footnote, p.165); 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-.^[1] This naming reflects the deep connections between complex and symplectic structures.

See also

Look up symplectic in Wiktionary, the free dictionary.

• Contact geometry
• Hamiltonian mechanics
• Geometric Mechanics
• Moment map
• Symplectic flow
• Symplectic frame bundle
• Symplectic integration
• Symplectic manifold

Notes

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.)
• Maurice A. de Gosson: Symplectic Geometry and Quantum Mechanics (2006) Birkhäuser Verlag, Basel ISBN 978-3-7643-7574-4.
• Alan Weinstein, Symplectic geometry
• Weyl, Hermann (1939), The Classical Groups. Their Invariants and Representations, Princeton University Press, ISBN 978-0-691-05756-9, MR 0000255 

External links

• Hazewinkel, Michiel, ed. (2001), "Symplectic structure", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4