# 2-sphere

## Contents

## Definition

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 .

## Equational descriptions

### Cartesian equation

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:

## 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 .

## 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 , hence we have:

- The Gaussian curvature is
- The mean curvature is

## Properties

Template:Constant-curvature metric

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 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.