The Banach-Tarski paradox involves cutting up one of these and rearranging it to two copies of itself.
If the curvature of a manifold is always greater than one fourth of its max value, it's isomorphic to one of these objects. This object is the double cover of a space whose homology group is Z over Z two for odd p less than the total dimension. Identifying certain points on one of these objects forms real projective space. Every vector field tangent to one of these in even dimensions will have at least one (*) singular point. This is the only compact 2-manifold with genus zero. The Banach-Tarski paradox involves cutting up one of these and rearranging it to two copies of itself. Every simply connected closed three-manifold is homeomorphic to one of these according to the Poincare conjecture. For 10 points, name these objects whose points are all the same distance from the center
If the curvature of a manifold is always greater than one fourth of its max value, it's isomorphic to one of these objects.
sphere