A 2-sphere in is the set of all points having a fixed distance (termed the radius) from a fixed point (termed the center). The 2-sphere is a particular case of the more general notion of sphere, which makes sense in all dimensions.
The radius is denoted as .
The 2-sphere with center and radius has the following Cartesian equation:
since the center can be translated to the origin, we can consider the sphere centered at the origin, whose equation is:
All 2-spheres are the same upto similarity, and the 2-sphere is thus uniquely determined topologically, differentially, and upto isometry (upto scaling factor by a constant).
The abstract 2-sphere is denoted as .
Structure and symmetry
The 2-sphere is a quotient space, viz it occurs as the quotient of a Lie group by a subgroup. Thus, the group of isometries is transitive on all points.
Every point on the 2-sphere is an umbilic point, viz the curvature in all directions at each point is equal. In fact, the curvature in any direction is , hence we have:
Given any two points on the 2-sphere, there is an isometry taking one to the other.
This provides an instance of application of the following theorem: Liebmann's theorem
We saw that the mean curvature of the 2-sphere is where is the radius. Liebmann's theorem establishes a converse of sorts: any differentiable, closed and convex surface whose mean curvature is constant, must be a 2-sphere.