User:Fropuff/Draft 3

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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 {pr, 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:

Elementary properties

Take an (n−1)-sphere of radius r inside Rn. 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 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 SnRn+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.

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 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

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).

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