3-sphere

In topology, a 3-sphere or hypersphere (also called a glome) is analogous to a sphere in a space of greater number of dimensions. An ordinary sphere, or 2-sphere, consists of all points equidistant from a given point in ordinary three-dimensional Euclidean space, R³. A 3-sphere consists of all points equidistant from a given point on R⁴. While a 2-sphere is a "smooth" of two dimensions, a 3-sphere is an example of a 3-manifold.

In an entirely analogous way, it is possible to define spheres of a greater number of dimensions, called hyperspheres or n-spheres. Such objects are n-dimensional varieties.

Some literature refers to the 3-sphere as glomo, from the Latin glomus, ball. Informally, a glomo is to a sphere what a sphere is to a circle.

In geometry, 3-Sphere is the surface of a sphere, while in topology they refer to it as a 2-sphere and indicate it as ${displaystyle S^{2};}$. Callably, geometry and topologists adopt incompatible agreements for the meaning of "n-sphere".

Definition

In coordinates, a 3-sphere with center (x₀, y₀, z₀, w₀) and radius r is the set of all points (< i>x, y, z, w) in R⁴ such that

${displaystyle (x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}+(w-w_{0})^{2}=r^{2},}$

The 3-sphere centered at the origin and with radius 1 is called 3-unit sphere or 3-unit sphere, and is usually denoted S³. It can be described as a subset of R⁴, C², or H b> (the quaternions):

${displaystyle S^{3}=left{(x_{1},x_{2},x_{3},x_{4})in mathbb {R} ^{4}mid x_{1^}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}{2}}{2}}{2}}{2}}{2}}{2}}}{2}{2}{2}}}{2}{2}{2}}{2}}{2}}{2}{2}{2}{2}}}{2}{2}}{2}{2}}}{2}}{2}{2}{2}}}}{2}{2}}}}{2}}{2}{2}{2}}}{2}}{2}{2}}{2}}{2}}}{2}}}{2}}{2}}}}}} = ==$

${displaystyle S^{3}=left{(z_{1},z_{2})in mathbb {C} ^{2}{2}{2}{2}{2}}{2}}{2}{2}{2}{2}{2}{2}}{1right}}}}}}$

${displaystyle S^{3}=left{qin mathbb {H} mid Δq.$

The last description is usually the most useful. Describes the 3-sphere as the set of all unit quaternions, that is, quaternions with absolute value equal to 1. Just as the set of all unit complex numbers is important in complex geometry, the set of all unit quaternions is important for the geometry of quaternions.

Properties

The three-dimensional volume (or hyperarea) of a 3-sphere of radius r is

${displaystyle 2pi ^{2}r^{3},}$

while the four-dimensional hypervolume (the volume of the 4-dimensional region bounded by the 3-sphere) is

${displaystyle {tfrac {1}{2}}{2}{2}r^{4}$

Every nonempty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere, unless the hyperplane is tangent to the 3-sphere, in which case the intersection is a single point. When the 3-sphere moves through a given three-dimensional hyperplane, the intersection begins as a point, then becomes a growing 2-sphere that reaches its maximum size when the hyperplane cuts directly through the "middle" 3. 4; of the 3-sphere, and finally the 2-sphere "shrinks" again to a single point as the 3-sphere leaves the hyperplane.

Topological properties

A 3-sphere is a compact variety without delimitation. It is also simply connected. What this means, informally, is that any circular path, or any loop, in the 3-sphere can continually shrink to a point without leaving the 3-sphere. There is a conjecture, called the Poincaré conjecture, which maintains that the 3-sphere is the only three-dimensional variety with these properties (except homeomorphism). This conjecture has been proven by Grigori Perelmán in a series of works produced since November 2002.

The 3-sphere is also homeomorphic with compaction at a R³ point.

The non-trivial homology groups of the 3-sphere are the following: H₀(S³,Z) and H< sub>3(S³,Z) are both infinite cyclics, while H_i(S^{3< /sup>,Z) = {0} for every other index i. Any topological space with these homology groups is known as a homological 3-sphere. Initially Poincaré conjectured that all 3-spheres of homology were homomorphic to S3, but he later managed to construct a non-homomorphic one, now known as the Poincaré sphere. The existence of an infinite number of spheres of homology is known. For example, a Dehn filling with slope 1/n over any knot in the 3-sphere gives a homology sphere; Typically, these are not 3-sphere homomorphs.}

Regarding the homotopy groups, we have π₁(S³) = π₂(S³) = {0} and π₃(S³) is infinitely cyclic. The largest homotopy groups (k ≥ 4) are all finite Abelian, but otherwise they do not follow any discernible pattern. For more details, see homotopy groups of spheres.

There is an interesting group action of S¹ (imagined as the group of complex numbers of absolute value 1) on S³ (imagined as a subset of < b>C²): λ·(z₁,z_{2< /sub>) = (λz1,λz2). The orbital space of this action is naturally homomorphic with the S² 2-sphere. The resulting map of the 3-sphere on the 2-sphere is known as the Hopf beam. It is the generator of the homotopy group π3(S²).}

Group structure

When considered as the set of unit quaternions, S³ inherits the structure of quaternionic multiplication. Since the set of unit quaternions is closed under the multiplication S³ it has the structure of a group. Furthermore, since quaternionic multiplication is regular (infinitely differentiable), S³ can be seen as a Lie group. It is a compact, non-abelian Lie group of dimension 3. When imagined as a Lie group it is usually denoted Sp(1) or U(1, H).

It turns out that the only spheres that admit the structure of a Lie group are the unit circle, S¹, imagined as the set of unit complex numbers, and S³, the set of unit quaternions. One might think that S⁷, the set of unit octonions, would form a Lie group, but this is not the case because the multiplication of octonions is not associative. The octonionic structure gives S⁷ an important property: parallelizability. The only parallelizable spheres are S¹, S³ and S ⁷.

Using a matrix representation of the quaternions, H, a matrix representation of S³ is obtained. A convenient choice is

${displaystyle x_{1}+x_{2}i+x_{3}j+x_{4}kquad mapsto quad {begin{pmatrix}x_{1}+ix_{2}{3}{3} +ix_{4}{4}-x_{3}{4}{4}{4}{4}{x_{1}{1}{1⁄2}{1}{1}{1}{1}{xx1}}{1}{1}{1}{1}{1}{1}{1}{xx1}{1}{1}{1}}}{x1}}{xxxxxx1}}}{1}{1}{1}}{1}{2}}}}}}{xxxx1}{1}{1}{1}{xxxx1}}}}{xxxxxxx1}}{1}}}{xxxxxxxxxxx$

This map gives an injective algebraic homomorphism of H in the set of 2×2 complex matrices. It has the property that the absolute value of a quaternion q is equal to the square root of the determinant of the image matrix of q.

The set of unit quaternions is therefore given by matrices of the form indicated above with unit determinant. It turns out that this group is precisely the unitary special group SU(2). Therefore S³ as a Lie group is isomorphic to SU(2).

Using our hyperspherical coordinates (η, ξ₁, ξ₂) we can write any element of SU(2) as

${displaystyle {begin{pmatrix}e^{i,xi _{1}operatorname {sen} eta &e^{i,xi _{2}}}cos eta \-e^{-i}{2}}{2}}cos eta &e^{-i,xi _{1p}operarname {sen} eta} eta$

Coordinate systems on the 3-sphere

Hyperspherical coordinates

It is convenient to have some type of hyperspherical coordinates in S³, analogous to the usual spherical coordinates in S². One choice (by no means the only possible one) is to use (ψ, θ, φ) where

${displaystyle x_{0}=cos psi ,}$

${displaystyle x_{1}=cos phi ,operatorname {sen} theta ,operatorname {sen} psi }$

${displaystyle x_{2}=operatorname {sen} phi ,operatorname {sen} theta ,operatorname {sen} psi }$

${displaystyle x_{3}=cos theta ,operatorname {sen} psi }$

where ψ and θ shift in the range (0,π), and φ shifts in (0,2π), where as in the case of the 2-sphere, it is not possible to parameterize the entire space with a single choice of coordinates (in the 2-sphere, at least one meridian that goes from the north pole to the south pole remains unparameterized), for this other ranges would have to be taken that cover the unparameterized parts. Note also that for any fixed value of ψ, θ and φ parameterize a 2-sphere of radius sin(ψ).

The metric tensor on the 3-sphere at these coordinates is determined by

${displaystyle ds^{2}=dpsi ^{2}+operatorname {sen} ^{2}psi left(dtheta ^{2}+operatorname {sen} ^{2}theta ,dphi ^{2}right)}}$

and the volume shape by

${displaystyle dV=left(operatorname {sen} ^{2}psi ,operatorname {sen} theta right),dpsi wedge dtheta wedge dphi }$

These coordinates can be described in terms of quaternions. Any unit quaternion q can be written in the form:

${displaystyle q=e^{tau psi }=cos psi +tau operatorname {sen} psi }$

where τ is a unit imaginary quaternion (i.e., any quaternion that satisfies τ² = −1). This is the quaternionic analogue of Euler's formula. Now the imaginary unit quaternions all lie on the 2-unit sphere in ImH, so that any τ can be written as:

${displaystyle tau =cos varphi operatorname {sen} theta ;{boldsymbol {i}+operatorname {sen} varphi operatorname {sen} theta ;{boldsymbol {j}}}+cos theta ;{boldsymbol {k}}}}}}$

With τ in this form, the unit quaternion q is given by

${displaystyle q=e^{tau psi }=x_{0}+x_{1};{boldsymbol {i}}+x_{2};{boldsymbol {j}};{boldsymbol {k}}}}$

where the x are as indicated above.

When q is used to describe spatial rotations (cf. quaternions and spatial rotation) it describes a rotation about τ through an angle of 2ψ.

Hopf Coordinates

Another choice of hyperspherical coordinates, (η, ξ₁, ξ₂), uses the fit of S^{3< /sup> in C2. In complex coordinates (z₁, z₂) ∈ C 2 we will write}

${displaystyle z_{1}=e^{i,xi _{1}}operatorname {sen} eta}$

${displaystyle z_{2}=e^{i,xi _{2}}cos eta }$

Here η shifts in the range 0 to π/2, and ξ₁ and ξ₂ can take any value between 0 and 2π. These coordinates are useful in describing the 3-sphere as a Hopf beam

${displaystyle S^{1}to S^{3}to S^{2}}$

For any value of η between 0 and π/2, the coordinates (ξ₁, ξ₂) parameterize a two-dimensional torus. In the degenerate cases, when η is equal to 0 or π/2, these coordinates describe a circle.

The metric tensor on the 3-sphere in these coordinates is given by

${displaystyle ds^{2}=deta ^{2}+operatorname {sen} ^{2}eta ,dxi _{1}^{2}+cos ^{2}eta ,dxi _{2}{2}{2}}}$

and the volume shape by

${displaystyle dV=operatorname {sen} eta cos eta eta ,deta wedge dxi _{1wedge dxi _{2}}}}$

Stereographic coordinates

Another convenient set of coordinates can be obtained by stereographic projection of S³ onto a hyperplane of R³ tangent. For example, if we project onto the tangent plane to the point (1, 0, 0, 0) a point p could be written in S³ as

${displaystyle p=left({frac {1-associatedu}{2}{2}{1+associatedu^{2}}}}},{frac {2mathbf {u}}} }{1+associatedu^{2}}}}{right)={{1+mathbf {u}{1}{1-math}}{1}{1}{1}{1}}{1}{1⁄4}}{1⁄4}{b$

where u = (u₁, u₂, u₃) is a vector in R³ and ||u||² = u₁² + u₂² + u₃². In the second equality above we have identified p with a unit quaternion and u = u₁i + u₂ j + u₃ k with a pure quaternion. (Note that the division here is well defined, even though quaternionic multiplication is generally non-commutative.) The inverse of this map transforms p = (x₀, x₁, x₂, x₃) in S³ in

${displaystyle mathbf {u} ={frac {1}{1+x_{0}}}}(x_{1},x_{2},x_{3})}}}}$

We could well have projected onto the tangent plane to the point (−1, 0, 0, 0), in which case the point p would be given by

${displaystyle p=left({frac {1+organizations}{2}{2}{1+organization}}}},{frac {2mathbf {v}} }{1+expend}{2}}}}{right}={frac {1+mathbf {v}}{1+math$

where v = (v₁, v₂, v₃) is a vector in the second R³. The inverse of this map transforms p into

${displaystyle mathbf {v} ={frac {1}{1-x_{0}}}}(x_{1},x_{2},x_{3})}}}}$

Note that the coordinates u are defined everywhere except (−1, 0, 0, 0) and coordinates v everywhere except (1, 0, 0, 0, 0). Both patches together cover the whole S³. This defines an atlas over S³ consisting of two coordinated letters. Note also that the transition function between these two letters in overlap is given by

${displaystyle mathbf {v} ={frac {1}{associatedu}{2}}}}{mathbf {u} }$

and vice versa.

Tangents

A 3-unit sphere embedded in 4-space has a 3-space tangent vector, T_pS ³, at any point p. If (x₀,x₁,x_{2,x3) are the coordinates of p, then the vector with coordinates (−x1,x0,−x3,x 2) is in TpS³, and the collection of all those vectors forms a continuous field of unit vectors in S³. (This is a section of the tangent bundle, TS³.) Such a construction is clearly possible for spheres in all spaces with an even number of dimensions, S²ⁿ⁻¹; but an implication of the Atiyah-Singer index theorem is that it is impossible for S²ⁿ (for n positive).}

N-spheres

The notion of sphere can be generalized in vector spaces of dimensions greater than three. From the fourth dimension it is no longer graphically representable, but the definition remains that the sphere is the set of points equidistant from a fixed point. In a Euclidean space of n+1 dimensions, using Cartesian coordinates, the equation of an n-sphere:

${displaystyle x_{1⁄2}{2} +x_{2}{2} +x_{3⁄2}{2}{2}{2}{2}{2}{2} =1}$

And for a sphere of radius R, and center (c₁, c₂,..., c_n ):

${displaystyle (x_{1}-c_{1})^{2}+(x_{2}-c_{2})^{2}+(x_{3}-c_{3})^{2}+cdots +(x_{n+1}-c_{n+1}{2}{2}=R^{2}{2}}}$

The volume of the ball contained on the previous closed surface, in dimension n calculated by induction n. Here are the top ten values of V_n(r) and corresponding surfaces:

The volume of the ball reaches its maximum in dimension 5, while the surface of the sphere reaches its maximum in dimension 7.

There is the possibility of representing an n-sphere or hypersphere of n dimensions as a bundle of another hypersphere of lower dimension. This only happens in three cases:

• ${displaystyle S^{3};}$, can be represented as non-trivial fiber with base space ${displaystyle S^{2};}$ and fiber ${displaystyle S^{1};}$, this construction can be obtained from a geometric-algebraic construction using complex numbers.
• ${displaystyle S^{7};}$, can be represented as non-trivial fiber with base space ${displaystyle S^{4};}$ and fiber ${displaystyle S^{3};}$, this construction can be obtained from a geometric-algebraic construction using quaternionic numbers.
• ${displaystyle S^{15};}$, can be represented as non-trivial fiber with base space ${displaystyle S^{8};}$ and fiber ${displaystyle S^{7};}$, this construction can be obtained from a geometric-algebraic construction using octonic numbers.

For a higher dimension there are no other cases in which this is possible.

In literature

Stephen Baxter used the 3-sphere in his short story Dante and the 3-Sphere, a story in which an apparently mad scientist and theologian "realizes" that Dante in the "Divine Comedy" refers to a traversal through multiple 3-spheres. The main character is taken by the scientist on a journey through multiple 3-spheres.

In Edwin Abbott Abbott's Flatland, published in 1884, the 3-spheres are referred to as superspheres.