n-sphere

From Mathematics wiki

An n-sphere, denoted $\mathbb{S}^n\,$ , also known as a hypersphere, is the $n\,$ -dimensional generalization of the sphere, and may be defined by the locus of points equidistant from some point (the center) in $\mathbb{R}^{n+1}\,$ , i.e.,

$\mathbb{S}^n = \left\{x \in \mathbb{R}^{n+1} : \left \Vert x\right \Vert = r\right\}\,$ ,

where $r\,$ is the radius of the sphere. Thus

$\mathbb{S}^0\,$	is two points on $\mathbb{R}^1\,$
$\mathbb{S}^1\,$	is the circle on $\mathbb{R}^2\,$ , not the disc
$\mathbb{S}^2\,$	is the sphere in $\mathbb{R}^3\,$ , not the ball
	and so forth...

If $r = 1\,$ then we speak of the unit n-sphere or unit hypersphere.

1 Symmetry
2 Metric
3 Curvature
4 Surface area
5 Enclosed volume
6 Coordinates

Symmetry

The isometry group of an n-sphere $\mathbb{S}^n\,$ is $SO(n+1)\,$ , the special orthogonal group.

Metric

An n-sphere of radius $r\,$ can be embedded in Euclidean $\mathbb{R}^{n+1}\,$ , which induces the following metric (see spherical coordinates):

$ds^2 = r^2 d\Omega^2_n = r^2\begin{pmatrix} 1 & 0 & 0& \cdots \\ 0 & \sin^2\!\theta_{n-1} & 0 & \cdots \\ 0 & 0 & \sin^2\!\theta_{n-1} \sin^2\!\theta_{n-2}& \cdots \\ \vdots & \vdots & \vdots & \ddots \end{pmatrix}\,$ ,