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2008-05-18T19:33:15Z

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Vipul: /* Properties */

2007-07-31T07:35:08Z

Properties

← Older revision		Revision as of 07:35, 31 July 2007
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	Given any two points on the 2-sphere, there is an isometry taking one to the other.		Given any two points on the 2-sphere, there is an isometry taking one to the other.

			==Related theorems==

			{{applicationinstance\|Liebmann's theorem}}

			We saw that the mean curvature of the 2-sphere is <math>2/r</math> where <math>r</math> 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.

Vipul at 07:26, 31 July 2007

2007-07-31T07:26:48Z

New page

==Definition==

A '''2-sphere''' in <math>\R^3</math> 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 <math>r</math>.
==Equational descriptions==

===Cartesian equation===

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

<math>(x-x_0)^2 + (y-y_0)^2 + (z - z_0)^2 = r^2</math>

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

<math>x^2 + y^2 + z^2 = r^2</math>

==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 <math>S^2</math>.

==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 <math>1/r</math>, hence we have:

* The [[Gaussian curvature]] is <math>1/r^2</math>
* The [[mean curvature]] is <math>2/r</math>

==Properties==

{{constant-curvature metric}}

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

{{homogeneous metric}}

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

2-sphere - Revision history

Vipul: 2 revisions

Vipul: /* Properties */

Vipul at 07:26, 31 July 2007