# User:Fropuff/Draft 3

*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

*S*

^{n}. The unit

*n*-sphere is often referred to as

*the*

*n*-sphere. In symbols:

## Elementary properties

Take an (*n*−1)-sphere of radius *r* inside **R**^{n}. The "surface area" of this sphere is

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

## Structure

The unit *n*-sphere *S ^{n}* can naturally be regarded as a topological space with the relative topology from

**R**

^{n+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 **R**^{n+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, *S ^{n}* has a natural geometric structure. It inherits a Riemannian metric from the ambient Euclidean space. Specifically, the metric on

*S*is the pullback of the Euclidean metric by the inclusion map

^{n}*S*→

^{n}**R**

^{n+1}. This canonical metric on

*S*is often called the

^{n}**round metric**.

Together with the round metric, *S ^{n}* is a compact,

*n*-dimensional Riemannian manifold which is isometrically embedded in

**R**

^{n+1}. For specific values on

*n*,

*S*may have additional algebraic structure. This will be discussed further below.

^{n}## Coordinate charts

As an *n*-dimensional manifold, *S ^{n}* should be covered by

*n*-dimensional coordinate charts. That is, an arbirtary point on

*S*should be specifiable by

^{n}*n*coordinates. Since

*S*is not contractible it is impossible to find a single chart that covers the entire space. At least two charts are necessary.

^{n}### Stereographic coordinates

The standard coordinate charts on *S ^{n}* are obtained by stereographic projection.

### Hyperspherical coordinates

## 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

**if the context is clear). For example, any knot in**

*n*-sphere**R**

^{3}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 *S ^{n}* is the one induced from the Euclidean metric on

**R**

^{n+1}. Concretely, if one identifies the tangent space to a point

*p*on

*S*with the orthogonal complement of the vector

^{n}*p*in

**R**

^{n+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

### 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,
**R**P^{1}. Parallizable. SO(2) = U(1). - 2-sphere
- Complex structure; see Riemann sphere. Equivalent to the complex projective line,
**C**P^{1}. 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,
**H**P^{1}. SO(5)/SO(4). - 5-sphere
- Principal U(1)-bundle over
**C**P^{2}. 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) =
*G*_{2}/SU(3). - 7-sphere
- Topological quasigroup structure as the set of unit octonions. Principal Sp(1)-bundle over
*S*^{4}. Parallizable. SO(8)/SO(7) = SU(4)/SU(3) = Sp(2)/Sp(1) = Spin(7)/*G*_{2}= Spin(6)/SU(3).