2-sphere

From Diffgeom
Jump to: navigation, search

Definition

A 2-sphere in \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:

Properties

Template:Constant-curvature metric

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

Template:Homogeneous metric

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.