by PWrong » Fri Jun 26, 2015 5:59 am
I haven't really encountered them, but I can try to translate the Wikipedia article to some extent.
A hyperkahler manifold is like a Kahler manifold except with a quaternionic structure instead of a complex structure. So I'll talk about Kahler manifolds first, and if you understand the difference between complex numbers and quaternions then you can fill in the rest.
First I'll talk about real manifolds. First of all a manifold is a set with some nice topological properties. The surface of the Earth is curved, but when you're very small it looks flat. A manifold is basically a set that has that property. All of the toratopes are real manifolds, and the closed toratopes are smooth manifolds (no nasty corners or breaks). There's no requirement that a manifold has to be embedded in any higher space, or that it has to be finite. Note that Euclidean space itself is a manifold. The precise definition involves creating maps or functions from subsets of the (n-dimensional) manifold to subsets of R^n. For examples you can look up various projections like the Mercator projection.
Now for a complex manifold, you need to be able to map subsets of the manifold to subsets of C^n, where C is the set of complex numbers. Other than that, the idea is similar.
You can put all sorts of structures on a manifold. For example, lines of longitude and latitude are a structure on the Earth. They let you talk about things like position. The Riemannian structure lets you talk about inner products of vectors and vector fields. That gives you a notion of the distance between two points on the manifold, and lets you talk about geodesic curves (the shortest curve between two points). It's a very useful structure. Then there's symplectic structure, which I don't really know anything about, but apparently it's used in mathematical physics.
Finally, putting it all together, a hyperkahler manifold is a quaternionic manifold with both a Riemannian structure and a symplectic structure. I don't think they would relate directly to toratopes because toratopes are reals manifolds and hyperkahler manifolds are quaternionic.