Symplectic geometry

Symplectic geometry is a special area of mathematics that helps us understand the shapes and movements of objects in space. It is part of two bigger fields: differential geometry and differential topology. These fields study how smooth surfaces and spaces can be bent, twisted, and changed in various ways.

In symplectic geometry, mathematicians look at something called "symplectic manifolds." These are special kinds of spaces that have certain rules about how they can change. These rules come from a mathematical object called a "2-form," which helps describe relationships between points in the space.

This type of geometry began with the study of classical mechanics, which looks at how things move and change according to physical laws. In particular, it grew from the Hamiltonian formulation, a way of describing motion that uses something called "phase space." This phase space often has the special properties that symplectic geometry studies. This connection makes symplectic geometry very important in physics and other areas of science.

Etymology

The word "symplectic" was introduced into mathematics by Hermann Weyl. It comes from a Greek phrase meaning "twining or plaiting together," similar to the word "complex." Both words share an ancient root that means "folding or weaving."

Thanks to an important math rule called Darboux's theorem, we know that symplectic shapes look the same up close, no matter where you look. This means the only differences between them are in how they are put together overall, which is a topic in topology. Sometimes, people use the words "symplectic topology" and "symplectic geometry" as if they mean the same thing.

Main article: Symplectic group
Main articles: Symplectic vector space, Symplectic topology

Overview

Symplectic geometry is a special kind of math used on smooth, even-dimensional spaces. These spaces are like surfaces that can be measured and bent in certain ways. They have something called a "symplectic 2-form." This helps measure the size of two-dimensional objects inside them, like how we might measure area with a ruler.

This idea comes from studying how things move in the real world. For example, to know where a moving object will go, we need to know both its position and its momentum. Together, these form a point in a two-dimensional space. The symplectic form helps us measure the area covered by the object's motion. This area stays the same even as the object moves. In higher dimensions, symplectic geometry works by measuring areas in pairs of directions, keeping the total area unchanged as things move.

Comparison with Riemannian geometry

Riemannian geometry studies special shapes called differentiable manifolds that have something called metric tensors. Symplectic geometry, which looks at symplectic manifolds, is similar but has some important differences.

One big difference is that symplectic manifolds do not have local features like curvature, thanks to Darboux's theorem. This theorem says that near any point, a symplectic manifold looks like a simple standard shape. Also, not every manifold can have a symplectic form — there are rules, like the manifold must have an even number of dimensions and be orientable.

There is also an interesting link between the two geometries: geodesics in Riemannian geometry, which are the shortest paths, are like pseudoholomorphic curves in symplectic geometry, which are surfaces with the smallest area. Both ideas are very important in their fields.

Main article: Riemannian geometry

Examples and structures

Every Kähler manifold is also a symplectic manifold. For a long time, experts were not sure if there were compact non-Kähler symplectic manifolds, but many examples have since been found.

Most symplectic manifolds do not have a special kind of structure called an integrable complex structure, but they do have something called almost complex structures. This idea helped mathematicians develop new tools and theories.