Symplectic geometry

"Complex" comes from the Latin com-plexus, meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek sym-plektikos (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures.

A symplectic geometry is defined on a smooth even-dimensional space that is a differentiable manifold.

To specify the trajectory of the object, one requires both the position q and the momentum p, which form a point (p,q) in the Euclidean plane

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] 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.

Another difference with Riemannian geometry is that not every differentiable manifold need admit a symplectic form; there are certain topological restrictions.

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 maps be holomorphic.