Riemannian geometry is a geometry on differentiable structures (manifolds) equipped with a "metric tensor" that generalizes the idea of a length to curved shapes.
Symplectic geometry is, similarly, a geometry on differentiable manifolds equipped with a "symplectic form" that generalizes the oriented area spanned by two vectors known in flat structures as the determinant.
Now here's the fun part: Riemannian geometry may exist in 1, 2, 3... dimensions; but symplectic geometry exists in even dimensions only. For example: if you tried to think of a 3D sphere of radius 1, you couldn't pass a 4D sphere through its largest section (a bit like if you couldn't pass a 3D sphere through a circle-shaped hole of same radius.
> a bit like if you couldn't pass a 3D sphere through a circle-shaped hole of same radius.
So a funny thing happened. When my eyes first skimmed over that part, I spotted the word "donut". When I read it a second later, it wasn't there.
Something in my brain literally substituted "donut" for "circle-shaped hole" to the extent that it appeared in my vision, for a single processing cycle.
Symplectic geometry is, similarly, a geometry on differentiable manifolds equipped with a "symplectic form" that generalizes the oriented area spanned by two vectors known in flat structures as the determinant.
Now here's the fun part: Riemannian geometry may exist in 1, 2, 3... dimensions; but symplectic geometry exists in even dimensions only. For example: if you tried to think of a 3D sphere of radius 1, you couldn't pass a 4D sphere through its largest section (a bit like if you couldn't pass a 3D sphere through a circle-shaped hole of same radius.
I forget what was my point initially.