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 {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:
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[edit] 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
[edit] 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.
[edit] 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.
[edit] Stereographic coordinates
The standard coordinate charts on Sn are obtained by stereographic projection.
[edit] Hyperspherical coordinates
[edit] 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.
[edit] 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
[edit] 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.
[edit] 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
[edit] To do
- curvature tensors and traces
- isometry group
- geodesics, great circles, and spherical distance
[edit] 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).
