2-sphere

Definition

A 2-sphere in ${\mathbb{R}}^3$ 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 $r$.

Equational descriptions

Cartesian equation

The 2-sphere with center $(x_0,y_0,z_0)$ and radius $r$ has the following Cartesian equation:

$(x-x_0{)}^2+(y-y_0{)}^2+(z-z_0{)}^2=r^2$

since the center can be translated to the origin, we can consider the sphere centered at the origin, whose equation is:

$x^2+y^2+z^2=r^2$

Abstract structure

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 $S^2$.

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.

Curvatures

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 $1/r$, hence we have:

• The Gaussian curvature is $1/r^2$
• The mean curvature is $2/r$

Properties

The metric on the 2-sphere has constant Gaussian curvature (The two-dimensional version of sectional curvature) everywhere.

Given any two points on the 2-sphere, there is an isometry taking one to the other.

Related theorems

Liebmann's theorem

This provides an instance of application of the following theorem: Liebmann's theorem

We saw that the mean curvature of the 2-sphere is $2/r$ where $r$ 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.