# Extensive Definition

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
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 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-. This
naming reflects the deep connections between complex and symplectic
structures.

## See also

## 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.)

symplectic in French: Géométrie
symplectique

symplectic in Dutch: Symplectische
meetkunde

symplectic in Japanese: シンプレクティック幾何学

symplectic in Chinese: 辛拓扑