Symplectic geometry is a branch of differential topology/geometry which studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, a symplectic manifold is a Smooth manifold M equipped with a closed, Nondegenerate, 2-form ω called the A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, especially Vector calculus and Differential topology, a closed form is a Differential form α whose differential is In Mathematics, specifically Linear algebra, a degenerate Bilinear form f(xy on a Vector space V is one such that In the mathematical fields of Differential geometry and Tensor calculus, differential forms are an approach to Multivariable calculus which is 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. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects In Mathematics and Physics, a phase space, introduced by Willard Gibbs in 1901 is a Space in which all possible states of a System

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). Elliptic geometry is also sometimes called Riemannian geometry. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. In Mathematics, specifically Differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension at least 3 is too complicated to be described 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. Darboux's theorem is a Theorem in the mathematical field of Differential geometry and more specifically Differential forms, partially generalizing 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. In Mathematics, de Rham cohomology (after Georges de Rham) is a tool belonging both to Algebraic topology and to Differential topology, capable In Mathematics, an n -sphere is a generalization of an ordinary Sphere to arbitrary Dimension. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

Every Kähler manifold is also a symplectic manifold. In Mathematics, a Kähler manifold is a Manifold with unitary structure (a ''U''(''n''-structure) satisfying an Integrability condition 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. William Paul Thurston (born October 30, 1946) is an American Mathematician. In Mathematics, one method of defining a group is by a presentation. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

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. Mikhail Leonidovich Gromov Russian Михаил Леонидович Громов (born December 23, 1943, also known In Mathematics, an almost complex manifold is a Smooth manifold equipped with smooth Linear complex structure on each Tangent space. Holomorphic functions are the central object of study of Complex analysis; they are functions defined on an open subset of the complex number plane

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. In Mathematics, specifically in Topology and Geometry, a pseudoholomorphic curve (or J -holomorphic curve) is a smooth map from a These invariants also play a key role in string theory. String theory is a still-developing scientific approach to Theoretical physics, whose original building blocks are one-dimensional extended objects called strings

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". In Linguistics, a calque (kælk or loan translation is a Word or Phrase borrowed from another Language by Literal, word-for-word Hermann Klaus Hugo Weyl ( 9 November 1885 – 8 December 1955) was a German Mathematician. 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-. ^[1][2] This naming reflects the deep connections between complex and symplectic structures.

See also

• Symplectic flow
• Hamiltonian mechanics
• Symplectic integration
• Riemannian geometry
• Contact geometry
• Moment map

References

1. ^ etymology of symplectic, by Murray Gell-Mann. Hamiltonian mechanics is a re-formulation of Classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton. In Mathematics, a symplectic integrator (SI is a numerical integration scheme for a specific group of differential equations related to Classical mechanics Elliptic geometry is also sometimes called Riemannian geometry. In Mathematics, contact geometry is the study of a geometric structure on Smooth manifolds given by a hyperplane distribution in the Tangent bundle In Mathematics, specifically in Symplectic geometry, the moment map (or momentum map) is a tool associated with a Hamiltonian action of Murray Gell-Mann (born September 15, 1929) is an American Physicist who received the 1969 Nobel Prize in physics for his work
2. ^ [1], p. 13

• Dusa McDuff and D. Dusa McDuff FRS (born Margaret Dusa Waddington in London on 18 October 1945) is an English Mathematician at Barnard 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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