3-sphere

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In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. A regular sphere, or 2-sphere, consists of all points equidistant from a single point in ordinary 3-dimensional Euclidean space, R3. A 3-sphere consists of all points equidistant from a single point in R4. Whereas a 2-sphere is a smooth 2-dimensional surface, a 3-sphere is an object with three dimensions, also known as 3-manifold.

In an entirely analogous manner one can define higher-dimensional spheres called hyperspheres or n-spheres. Such objects are n-dimensional manifolds.

Some people refer to a 3-sphere as a glome from the Latin word glomus meaning ball. Roughly speaking, a glome is to a sphere as a sphere is to a circle.

Contents

Definition

In coordinates, a 3-sphere with center (x0y0z0w0) and radius r is the set of all points (x,y,z,w) in R4 such that

<math>( x - x_0 )^2 + ( y - y_0 )^2 + ( z - z_0 )^2 + ( w - w_0 )^2 = r^2. \,<math>

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

<math>S^3 = \left\{(x_1,x_2,x_3,x_4)\in\mathbb{R}^4\mid x_1^2 + x_2^2 + x_3^2 + x_4^2 = 1\right\}<math>
<math>S^3 = \left\{(z_1,z_2)\in\mathbb{C}^2\mid |z_1|^2 + |z_2|^2 = 1\right\}<math>
<math>S^3 = \left\{q\in\mathbb{H}\mid |q| = 1\right\}.<math>

The last description is often the most useful. It describes the 3-sphere as the set of all unit quaternionsquaternions with absolute value equal to one. Just as the set of all unit complex numbers is important in complex geometry, the set of all unit quaternions is important to the geometry of the quaternions.

Elementary properties

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

<math>2\pi^2 r^3 \,<math>

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

<math>\begin{matrix} \frac{1}{2} \end{matrix} \pi^2 r^4.<math>

Every non-empty 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). As a 3-sphere moves through a given three-dimensional hyperplane, the intersection starts out as a point, then becomes a growing 2-sphere which reaches its maximal size when the hyperplane cuts right through the "middle" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.

Topological construction

A 3-sphere can be constructed topologically by "glueing" together the boundaries of a pair of 3-balls. The boundary of a 3-ball is a 2-sphere, and these two 2-spheres are to be identified. That is, imagine a pair of 3-balls of the same size, then superpose them so that their 2-spherical boundaries match, and let matching pairs of points on the pair of 2-spheres be identically equivalent to each other.

The interiors of the 3-balls do not match: only their boundaries. In fact, the 4th dimension can be thought of as a continuous scalar field, a function of the 3-dimensional coordinates of the 3-ball, similar to "temperature". Let this "temperature" be zero at the 2-spherical boundary, but let one of the 3-balls be "hot" (have positive values of its scalar field) and let the other 3-ball be "cold" (have negative values of its scalar field). The "hot" 3-ball could be thought of as the "hot hemi-3-sphere" and the "cold" 3-ball could be thought of as the "cold hemi-3-sphere". The temperature is highest at the hot 3-ball's very center and lowest at the cold 3-ball's center.

This construction is analogous to a construction of a 2-sphere, performed by joining the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter; superpose them so that their circular boundaries match, then let corresponding points on the circular boundaries become equivalent identically to each other. The boundaries are now glued together. Now "inflate" the disks. One disk inflates upwards and becomes the Northern hemisphere and the other inflates downwards and becomes the Southern hemisphere.

It is possible for a point travelling on the 3-sphere to move from one hemi-3-sphere to the other hemiglome by crossing the 2-spherical boundary, which could be thought of as a "3-quator" — analogous to an equator on a 2-sphere. The point would seem to be bouncing off the 3-quator and reversing direction of motion in 3-D, but also its "temperature" would become reversed, e.g. from positive on the "hot hemiglome" to zero on the 3-quator to negative on the "cold hemiglome".

Topological properties

A 3-sphere is a compact, 3-dimensional manifold without boundary. It is also simply-connected. What this means, loosely speaking, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. There is a long-standing, unproven conjecture, known as the Poincaré conjecture, stating that the 3-sphere is the only three dimensional manifold with these properties (up to homeomorphism).

The 3-sphere is also homeomorphic to the one-point compactification of R3.

The homology groups of the 3-sphere are as follows: H0(S3,Z) and H3(S3,Z) are both infinite cyclic, while Hi(S3,Z) = {0} for all other indices i. Any topological space with these homology groups is known as a homology 3-sphere. Initially Poincaré conjectured that all homology 3-spheres are homeomorphic to S3, but then he himself constructed a non-homeomorphic one, now known as the Poincaré sphere. Infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope 1/n on any knot in the three-sphere gives a homology sphere; typically these are not homeomorphic to the three-sphere.

As to the homotopy groups, we have π1(S3) = π2(S3) = {0} and π3(S3) is infinite cyclic. The higher homotopy groups (k ≥ 4) are all finite abelian but otherwise follow no discernable pattern. For more discussion see homotopy groups of spheres.

Homotopy groups of S3
k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
<math>\pi_{k}(S^3)<math> 0 0 0 Z Z2 Z2 Z12 Z2 Z2 Z3 Z15 Z2 Z2Z2 Z12Z2 Z84Z2Z2 Z2Z2 Z6

There is an interesting group action of S1 (thought of as the group of complex numbers of absolute value 1) on S3 (thought of as a subset of C2): λ.(z1,z2) = (λz1z2). The orbit space of this action is naturally homeomorphic to the two-sphere S2. The resulting map from the 3-sphere to the 2-sphere is known as the Hopf bundle. It is the generator of the homotopy group π3(S2).

Coordinates on S3

Hyperspherical coordinates

It is convenient to have some sort of hyperspherical coordinates on S3 in analogy to the usual spherical coordinates on S2. One such choice—by no means unique—is to use (ψ, θ, φ) where

<math>x_0 = \cos\psi\, <math>
<math>x_1 = \cos\phi\,\sin\theta\,\sin\psi <math>
<math>x_2 = \sin\phi\,\sin\theta\,\sin\psi <math>
<math>x_3 = \cos\theta\,\sin\psi <math>

where ψ and θ runs over the range 0 to π, and φ runs over 0 to 2π. Note that for any fixed value of ψ, θ and φ parameterize a 2-sphere of radius sin(ψ), except for the degenerate cases, when ψ equals 0 or π, in which case they describe a point.

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

<math>ds^2 = d\psi^2 + \sin^2\psi\left(d\theta^2 + \sin^2\theta\, d\phi^2\right)<math>

and the volume form by

<math>\left(\sin^2\psi\,\sin\theta\right)\;d\psi\wedge d\theta\wedge d\phi.<math>

These coordinates have a nice description in terms of quaternions. Any unit quaternion q can be written in the form:

q = eτψ = cos ψ + τ sin ψ

where τ is a unit imaginary quaternion—that is, any quaternion which satisfies τ2 = −1. This is the quaternionic analogue of Euler's formula. Now the unit imaginary quaternions all lie on the unit 2-sphere in Im H so any such τ can be written:

τ = cos φ sin θ i + sin φ sin θ j + cos θ k

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

q = eτψ = x0 + x1 i + x2 j + x3 k

where the x’s are as above.

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

Alternative hyperspherical system

Another choice of hyperspherical coordinates is (θ, ξ1, ξ2), where in terms of complex coordinates (z1, z1) ∈ C2 we have

<math>z_1 = e^{i\,\xi_1}\sin\frac{\theta}{2} <math>
<math>z_2 = e^{i\,\xi_2}\cos\frac{\theta}{2}. <math>

Here θ runs over the range 0 to π, and ξ1 and ξ2 can take any values between 0 and 2π. These coordinates are useful in the description of the 3-sphere as the Hopf bundle S1S3S2.

Note that for any fixed value of θ, (ξ1, ξ2) parameterize a 2-dimensional torus, except for the degenerate cases, when θ equals 0 or π, in which case they describe a circle.

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

<math>ds^2 = \frac{1}{4}d\theta^2 + \sin^2\left(\frac{\theta}{2}\right)d\xi_1^2 + \cos^2\left(\frac{\theta}{2}\right)d\xi_2^2<math>

and the volume form by

<math>\left(\frac{1}{4}\sin\theta\right)\;d\theta\wedge d\xi_1\wedge d\xi_2.<math>

Stereographic coordinates

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

<math>p = \left(\frac{1-\|u\|^2}{1+\|u\|^2}, \frac{2\mathbf{u}}{1+\|u\|^2}\right) = \frac{1+\mathbf{u}}{1-\mathbf{u}}<math>

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 = u1 i + u2 j + u3 k with a pure quaternion. (Note that the division here is well-defined even though quaternionic multiplication is generally noncommutative). The inverse of this map takes p = (x0, x1, x2, x3) in S3 to

<math>\mathbf{u} = \frac{1}{1+x_0}\left(x_1, x_2, x_3\right).<math>

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

<math>p = \left(\frac{-1+\|v\|^2}{1+\|v\|^2}, \frac{2\mathbf{v}}{1+\|v\|^2}\right) = \frac{-1+\mathbf{v}}{1+\mathbf{v}}<math>

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

<math>\mathbf{v} = \frac{1}{1-x_0}\left(x_1,x_2,x_3\right).<math>

Note that the u coordinates are defined everywhere but (−1, 0, 0, 0) and the v coordinates everywhere but (1, 0, 0, 0). Both patches together cover all of S3. This defines an atlas on S3 consisting of two coordinate charts. Note that the transition function between these two charts on their overlap is given by

<math>\mathbf{v} = \frac{1}{\|u\|^2}\mathbf{u}<math>

and vice-versa.

Group structure

When considered as the set of unit quaternions, S3 inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, S3 takes on the structure of a group. Moreover, since quaternionic multiplication is smooth, S3 can be regarded as a real Lie group. It is a nonabelian, compact Lie group of dimension 3. When thought of as a Lie group S3 is often denoted Sp(1) or U(1, H).

It turns out that the only spheres which admit a Lie group structure are S1, thought of 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 fails since octonion multiplication is nonassociative. The octonionic structure does give S7 one important property: parallelizability. It turns out that the only spheres which are parallelizable are S1, S3, and S7.

By using a matrix representation of the quaternions, H, one obtains a matrix representation of S3. One convenient choice is

<math>x_1+ x_2 i + x_3 j + x_4 k \mapsto \begin{pmatrix}\;\;\,x_1 + i x_2 & x_3 + i x_4 \\ -x_3 + i x_4 & x_1 - i x_2\end{pmatrix}.<math>

This map gives an injective algebra homomorphism from H to 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 matrix image of q.

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

Using our hyperspherical coordinates (θ, ξ1, ξ2) we can then write any element of SU(2) in the form

<math>\begin{pmatrix}e^{i\,\xi_1}\sin\frac{\theta}{2} & e^{i\,\xi_2}\cos\frac{\theta}{2} \\ -e^{-i\,\xi_2}\cos\frac{\theta}{2} & e^{-i\,\xi_1}\sin\frac{\theta}{2}\end{pmatrix}.<math>

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