A spherical surface is a simple example for a curved surface. It is easily pictured as a surface embedded in three-dimensional space: a spherical surface is the set of all points at a certain fixed distance from a given point (the point being the centre of the sphere).

Mathematically, a spherical surface can be described without recourse to three-dimensional space – when mathematicians talk of the geometry of such a surface, they (almost) always mean the “inner geometry”: Those properties of the surface noticeable to two-dimensional beings, living and working in that surface, capable of measuring distances and angles in it.

Sphere is regularly used as a synonym for spherical surface (instead of describing a solid, three-dimensional ball). And not only for the two-dimensional spherical surface described above, but also for its analogues in lower and higher dimensions. A one-sphere, for instance, is the same as a circle, a two-sphere is the spherical surface defined above, a three-sphere its three-dimensional analogue.