How are the two bottom left examples considered disks?
Does this only apply to manifolds in 3D? How would we classify a manifold in other geometries? Or other structures? Say we are in an Orthogonal Group (SO2) or Euclidean group SE2 which are differentiable manifolds. Is there a test as well?
@RCD-Y I believe we're applying topological rules here. In topology, you can stretch and morph surfaces, but you can't tear or glue them (i.e. a coffee mug and a donut are the same). The two on the bottom left can be stretched into a disk, as they are still both single planar surfaces. Compare that to the bottom right, which is two planar surfaces connected to a point, which can not be simply morphed into a disk without "gluing", which is not allowed.
Another definition of manifold is "any 2D surface (including a plane) that doesn't self-intersect is also a 2D manifold". The simplest one is a sphere. You can imagine each infinitesimal patch of the sphere locally resembles a 2D Euclidean plane.