Symplectic geometry
Symplectic geometry is a branch of
The term "symplectic", introduced by
By
The name "complex group" formerly advocated by me in allusion to line complexes, as these are defined by the vanishing of antisymmetric bilinear forms, has become more and more embarrassing through collision with the word "complex" in the connotation of complex number. I therefore propose to replace it by the corresponding Greek adjective "symplectic". Dickson called the group the "Abelian linear group" in homage to Abel who first studied it.
Weyl (1939, p. 165)
Introduction
A symplectic geometry is defined on a smooth even-dimensional space that is a differentiable manifold. On this space is defined a geometric object, the symplectic 2-form, that allows for the measurement of sizes of two-dimensional objects in the space. The symplectic form in symplectic geometry plays a role analogous to that of the metric tensor in Riemannian geometry. Where the metric tensor measures lengths and angles, the symplectic form measures oriented areas.[3]
Symplectic geometry arose from the study of
and is an area form that measures the area A of a region S in the plane through integration:
The area is important because as conservative dynamical systems evolve in time, this area is invariant.[3]
Higher dimensional symplectic geometries are defined analogously. A 2n-dimensional symplectic geometry is formed of pairs of directions
in a 2n-dimensional manifold along with a symplectic form
This symplectic form yields the size of a 2n-dimensional region V in the space as the sum of the areas of the projections of V onto each of the planes formed by the pairs of directions[3]
Comparison with Riemannian geometry
Symplectic geometry has a number of similarities with and differences from 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 . 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
Examples and structures
Every
Most symplectic manifolds, one can say, are not Kähler; and so do not have an integrable
Gromov used the existence of almost complex structures on symplectic manifolds to develop a theory of pseudoholomorphic curves,[4] which has led to a number of advancements in symplectic topology, including a class of symplectic invariants now known as Gromov–Witten invariants. Later, using the pseudoholomorphic curve technique Andreas Floer invented another important tool in symplectic geometry known as the Floer homology.[5]
See also
- Contact geometry
- Geometric mechanics
- Moment map
- Poisson geometry
- Symplectic integration
- Symplectic vector space
Notes
- ^ Hartnett, Kevin (February 9, 2017). "A Fight to Fix Geometry's Foundations". Quanta Magazine.
- ^ Weyl, Hermann (1939). The Classical Groups. Their Invariants and Representations. Reprinted by Princeton University Press (1997). ISBN 0-691-05756-7. MR0000255
- ^ ISBN 9789814277686
- ^ Gromov, Mikhael. "Pseudo holomorphic curves in symplectic manifolds." Inventiones mathematicae 82.2 (1985): 307–347.
- ^ Floer, Andreas. "Morse theory for Lagrangian intersections." Journal of differential geometry 28.3 (1988): 513–547.
References
- ISBN 978-0-8053-0102-1.
- Arnol'd, V. I. (1986). "Первые шаги симплектической топологии" [First steps in symplectic topology]. Успехи математических наук (in Russian). 41 (6(252)): 3–18. S2CID 250908036 – via Russian Mathematical Surveys, 1986, 41:6, 1–21.
- ISBN 978-0-19-850451-1.
- Fomenko, A. T. (1995). Symplectic Geometry (2nd ed.). Gordon and Breach. ISBN 978-2-88124-901-3. (An undergraduate level introduction.)
- ISBN 978-3-7643-7574-4.
- .
- MR0000255.
External links
- Media related to Symplectic geometry at Wikimedia Commons
- "Symplectic structure", Encyclopedia of Mathematics, EMS Press, 2001 [1994]