User:Fropuff/Drafts/n-sphere

Draft in progress.

---

In mathematics, an n-sphere is a generalization of a ordinary sphere to arbitrary dimensions. For any natural number n, an n-sphere of radius r is defined the set of points in (n+1)-dimensional Euclidean space which are at distance r from a fixed point. The radius r may be any positive real number.

• a 0-sphere is a pair of points {p − r, p + r}.
• a 1-sphere is a circle of radius r.
• a 2-sphere is an ordinary sphere.
• a 3-sphere is a sphere in 4-dimensional Euclidean space.
• and so on...

Spheres for n > 2 are sometimes called hyperspheres.

The n-sphere of unit radius centered at the origin is called the unit n-sphere, denoted Sⁿ. The unit n-sphere is often referred to as the n-sphere. In symbols:

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

Elementary properties

Take an (n−1)-sphere of radius r inside Rⁿ. The "surface area" of this sphere is

$A={\frac {2\pi ^{n/2}}{\Gamma (n/2)}}r^{n-1}$

where Γ is the gamma function. The "volume" of the n+1 dimensional region it encloses is

$V={\frac {\pi ^{n/2}}{\Gamma (n/2+1)}}r^{n}$

Structure

The unit n-sphere Sⁿ can naturally be regarded as a topological space with the relative topology from Rⁿ⁺¹. As a subspace of Euclidean space it is both Hausdorff and second-countable. Moreover, since it is a closed, bounded subset of Euclidean space it is compact according to the Heine-Borel theorem.

The n-sphere can also be regarded as a smooth manifold of dimension n. In fact, it is a closed, embedded submanifold of Rⁿ⁺¹. This follows from the fact that it is the regular level set of a smooth function (namely the function Rⁿ⁺¹ → R that sends a vector to its norm squared). Since Sⁿ has codimension 1 in Rⁿ⁺¹ it is a hypersurface.

In addition to its topological and smooth structure, Sⁿ has a natural geometric structure. It inherits a Riemannian metric from the ambient Euclidean space. Specifically, the metric on Sⁿ is the pullback of the Euclidean metric by the inclusion map Sⁿ → Rⁿ⁺¹. This canonical metric on Sⁿ is often called the round metric.

Together with the round metric, Sⁿ is a compact, n-dimensional Riemannian manifold which is isometrically embedded in Rⁿ⁺¹. For specific values on n, Sⁿ may have additional algebraic structure. This will be discussed further below.

Coordinate charts

As an n-dimensional manifold, Sⁿ should be covered by n-dimensional coordinate charts. That is, an arbirtary point on Sⁿ should be specifiable by n coordinates. Since Sⁿ is not contractible it is impossible to find a single chart that covers the entire space. At least two charts are necessary.

Stereographic coordinates

The standard coordinate charts on Sⁿ are obtained by stereographic projection.

Topology

In topology, any space which is homeomorphic to the unit n-sphere in Euclidean space is called a topological n-sphere (or just an n-sphere if the context is clear). For example, any knot in R³ is a topological 1-sphere, as is the boundary of any polygon. Topological 2-spheres include spheroids and the boundaries of polyhedra.

To do

• topological constructions (e.g. one-point compactification, balls glued together, ball with boundary identified)
• homology groups and cell decomposition
• homotopy groups and connectivity
• exotic spheres and differential structures

Geometry

Spheres occupy a special place in Riemannian geometry. For n ≥ 2, the n-sphere can be characterized as the unique complete, simply connected, n-dimensional Riemannian manifold with constant sectional curvature +1. The n-sphere serves as the model space for elliptic geometry.

The round metric

The round metric on Sⁿ is the one induced from the Euclidean metric on Rⁿ⁺¹. Concretely, if one identifies the tangent space to a point p on Sⁿ with the orthogonal complement of the vector p in Rⁿ⁺¹ then the round metric at p is just the Euclidean metric restricted to the tangent space.

In stereographic coordinates, the round metric may be written

$ds^{2}=\left({\frac {2}{1+\|u\|^{2}}}\right)^{2}\left((du^{1})^{2}+\cdots +(du^{n})^{2}\right)$

To do

• curvature tensors and traces
• isometry group
• geodesics, great circles, and spherical distance

Specific spheres

0-sphere

The pair of points {±1} with the discrete topology. The only sphere which is disconnected. Has a natural Lie group structure; isomorphic to O(1). Parallizable.

1-sphere

Also known as the unit circle. Has a nontrivial fundamental group. Abelian Lie group structure U(1); the circle group. Topologically equivalent to the real projective line, RP¹. Parallizable. SO(2) = U(1).

2-sphere

Complex structure; see Riemann sphere. Equivalent to the complex projective line, CP¹. SO(3)/SO(2).

3-sphere

Lie group structure Sp(1). Principal U(1)-bundle over the 2-sphere. Parallizable. SO(4)/SO(3) = SU(2) = Sp(1) = Spin(3).

4-sphere

Equivalent to the quaternionic projective line, HP¹. SO(5)/SO(4).

5-sphere

Principal U(1)-bundle over CP². SO(6)/SO(5) = SU(3)/SU(2).

6-sphere

Almost complex structure coming from the set of pure unit octonions. SO(7)/SO(6) = G₂/SU(3).

7-sphere

Topological quasigroup structure as the set of unit octonions. Principal Sp(1)-bundle over S⁴. Parallizable. SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/G₂ = Spin(6)/SU(3).

Related topics

• hyperbolic space