In geometry of higher dimensions, a hypersphere is the set of points at a constant distance from a given point called its center. It is a manifold of codimension one (i.e. with one dimension less than that of the ambient space). As the radius increases the curvature of the hypersphere decreases; in the limit a hypersphere approaches the zero curvature of a hyperplane. Hyperplanes and hyperspheres are examples of hypersurfaces.
Some spheres are not hyperspheres: suppose S is a sphere in Em where m < n and the space had n dimensions, then S is not a hypersphere. Similarly, any n-sphere in a proper flat is not a hypersphere. For example, a circle is not a hypersphere in three-dimensional space, but it is a hypersphere in the plane.
- Kazuyuki Enomoto (2013) Review of an article in International Electronic Journal of Geometry.MR 3125833
- Jemal Guven (2013) "Confining spheres in hyperspheres", Journal of Physics A 46:135201, doi:10.1088/1751-8113/46/13/135201