Symplectic geometry

Symplectic topology (also called symplectic geometry) is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with closed, nondegenerate, 2-forms. Symplectic topology 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 topology 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 the Riemannian case, symplectic manifolds have no local invariants such as curvature. This is a consequence of Darboux's theorem which states that every pair of symplectic manifolds are locally isomorphic. Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions. For starters, the manifold must be even-dimensional. Much work in symplectic topology has centered around investigating which manifolds admit symplectic structures.

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 made, however, the important observation that symplectic manifolds do admit an abundance of compatible almost complex structures, so that they satisfy all the axioms for a complex 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.

See also

• symplectic manifold
• Hamiltonian mechanics
• symplectic integration
• Riemannian geometry
• contact geometry

References

• Dusa McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford University Press, 1998. ISBN 0-198-50451-9.
• A. T. Fomenko, Symplectic Geometry (2nd edition) (1995) Gordon and Breach Publishers, ISBN 2-88124-901-9. (Provides an undergrad level introduction.)