3-sphere

In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point.  Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions).  A 3-sphere is an example of a 3-manifold and an n-sphere.

Definition
In coordinates, a 3-sphere with center  and radius  is the set of all points  in real, 4-dimensional space () such that

The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted :

It is often convenient to regard  as the space with 2 complex dimensions () or the quaternions (). The unit 3-sphere is then given by

or

This description as the quaternions of norm one identifies the 3-sphere with the versors in the quaternion division ring. Just as the unit circle is important for planar polar coordinates, so the 3-sphere is important in the polar view of 4-space involved in quaternion multiplication. See polar decomposition of a quaternion for details of this development of the three-sphere.
This view of the 3-sphere is the basis for the study of elliptic space as developed by Georges Lemaître.

Properties

Elementary properties
The 3-dimensional cubic hyperarea of a 3-sphere of radius  is

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

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 that reaches its maximal size when the hyperplane cuts right through the "equator" of the 3-sphere. Then the 2-sphere shrinks again down to a single point as the 3-sphere leaves the hyperplane.

Topological properties
A 3-sphere is a compact, connected, 3-dimensional manifold without boundary. It is also simply connected. What this means, in the broad sense, is that any loop, or circular path, on the 3-sphere can be continuously shrunk to a point without leaving the 3-sphere. The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold (up to homeomorphism) with these properties.

The 3-sphere is homeomorphic to the one-point compactification of . In general, any topological space that is homeomorphic to the 3-sphere is called a topological 3-sphere.

The homology groups of the 3-sphere are as follows:  and  are both infinite cyclic, while  for all other indices . 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 , but then he himself constructed a non-homeomorphic one, now known as the Poincaré homology sphere. Infinitely many homology spheres are now known to exist. For example, a Dehn filling with slope  on any knot in the 3-sphere gives a homology sphere; typically these are not homeomorphic to the 3-sphere.

As to the homotopy groups, we have  and  is infinite cyclic. The higher-homotopy groups () are all finite abelian but otherwise follow no discernible pattern. For more discussion see homotopy groups of spheres.

Geometric properties
The 3-sphere is naturally a smooth manifold, in fact, a closed embedded submanifold of . The Euclidean metric on  induces a metric on the 3-sphere giving it the structure of a Riemannian manifold. As with all spheres, the 3-sphere has constant positive sectional curvature equal to  where  is the radius.

Much of the interesting geometry of the 3-sphere stems from the fact that the 3-sphere has a natural Lie group structure given by quaternion multiplication (see the section below on group structure). The only other spheres with such a structure are the 0-sphere and the 1-sphere (see circle group).

Unlike the 2-sphere, the 3-sphere admits nonvanishing vector fields (sections of its tangent bundle). One can even find three linearly independent and nonvanishing vector fields. These may be taken to be any left-invariant vector fields forming a basis for the Lie algebra of the 3-sphere. This implies that the 3-sphere is parallelizable. It follows that the tangent bundle of the 3-sphere is trivial. For a general discussion of the number of linear independent vector fields on a -sphere, see the article vector fields on spheres.

There is an interesting action of the circle group  on  giving the 3-sphere the structure of a principal circle bundle known as the Hopf bundle. If one thinks of  as a subset of , the action is given by
.
The orbit space of this action is homeomorphic to the two-sphere . Since  is not homeomorphic to , the Hopf bundle is nontrivial.

Topological construction
There are several well-known constructions of the three-sphere.  Here we describe gluing a pair of three-balls and then the one-point compactification.

Gluing
A 3-sphere can be constructed topologically by "gluing" 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.  In analogy with the case of the 2-sphere (see below), the gluing surface is called an equatorial sphere.

Note that the interiors of the 3-balls are not glued to each other.  One way to think of the fourth dimension is as a continuous real-valued function of the 3-dimensional coordinates of the 3-ball, perhaps considered to be "temperature".  We take the "temperature" to be zero along the gluing 2-sphere and let one of the 3-balls be "hot" and let the other 3-ball be "cold". The "hot" 3-ball could be thought of as the "upper hemisphere" and the "cold" 3-ball could be thought of as the "lower hemisphere". The temperature is highest/lowest at the centers of the two 3-balls.

This construction is analogous to a construction of a 2-sphere, performed by gluing 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 and glue corresponding points on their boundaries.  Again one may think of the third dimension as temperature.  Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres.

One-point compactification
After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean plane. In the same way, removing a single point from the 3-sphere yields three-dimensional space. 
An extremely useful way to see this is via stereographic projection. We first describe the lower-dimensional version.

Rest the south pole of a unit 2-sphere on the -plane in three-space.  We map a point  of the sphere (minus the north pole ) to the plane by sending  to the intersection of the line  with the plane.  Stereographic projection of a 3-sphere (again removing the north pole) maps to three-space in the same manner.  (Notice that, since stereographic projection is conformal, round spheres are sent to round spheres or to planes.)

A somewhat different way to think of the one-point compactification is via the exponential map.  Returning to our picture of the unit two-sphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the two-sphere of the same length, based at the south pole.  Under this map all points of the circle of radius  are sent to the north pole.  Since the open unit disk is homeomorphic to the Euclidean plane, this is again a one-point compactification.

The exponential map for 3-sphere is similarly constructed; it may also be discussed using the fact that the 3-sphere is the Lie group of unit quaternions.

Coordinate systems on the 3-sphere
The four Euclidean coordinates for  are redundant since they are subject to the condition that . As a 3-dimensional manifold one should be able to parameterize  by three coordinates, just as one can parameterize the 2-sphere using two coordinates (such as latitude and longitude). Due to the nontrivial topology of  it is impossible to find a single set of coordinates that cover the entire space. Just as on the 2-sphere, one must use at least two coordinate charts. Some different choices of coordinates are given below.

Hyperspherical coordinates
It is convenient to have some sort of hyperspherical coordinates on  in analogy to the usual spherical coordinates on . One such choice — by no means unique — is to use , where

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

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

and the volume form by

These coordinates have an elegant description in terms of quaternions. Any unit quaternion  can be written as a versor:

where  is a unit imaginary quaternion; that is, a quaternion that satisfies . This is the quaternionic analogue of Euler's formula. Now the unit imaginary quaternions all lie on the unit 2-sphere in  so any such  can be written:

With  in this form, the unit quaternion  is given by

where  are as above.

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

Hopf coordinates

For unit radius another choice of hyperspherical coordinates, , makes use of the embedding of  in . In complex coordinates  we write

This could also be expressed in  as

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

For any fixed value of  between 0 and , the coordinates  parameterize a 2-dimensional torus. Rings of constant  and  above form simple orthogonal grids on the tori. See image to right. In the degenerate cases, when  equals 0 or , these coordinates describe a circle.

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

and the volume form by

To get the interlocking circles of the Hopf fibration, make a simple substitution in the equations above

In this case , and  specify which circle, and  specifies the position along each circle. One round trip (0 to 2) of  or  equates to a round trip of the torus in the 2 respective directions.

Stereographic coordinates
Another convenient set of coordinates can be obtained via stereographic projection of  from a pole onto the corresponding equatorial  hyperplane. For example, if we project from the point  we can write a point  in  as

where  is a vector in  and . In the second equality above, we have identified  with a unit quaternion and  with a pure quaternion. (Note that the numerator and denominator commute here even though quaternionic multiplication is generally noncommutative). The inverse of this map takes  in  to

We could just as well have projected from the point , in which case the point  is given by

where  is another vector in . The inverse of this map takes  to

Note that the  coordinates are defined everywhere but  and the  coordinates everywhere but . This defines an atlas on  consisting of two coordinate charts or "patches", which together cover all of . Note that the transition function between these two charts on their overlap is given by

and vice versa.

Group structure
When considered as the set of unit quaternions,  inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication,  takes on the structure of a group. Moreover, since quaternionic multiplication is smooth,  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  is often denoted  or .

It turns out that the only spheres that admit a Lie group structure are , thought of as the set of unit complex numbers, and , the set of unit quaternions. One might think that , the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. The octonionic structure does give  one important property: parallelizability. It turns out that the only spheres that are parallelizable are , , and .

By using a matrix representation of the quaternions, , one obtains a matrix representation of . One convenient choice is given by the Pauli matrices:

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

The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the special unitary group . Thus,  as a Lie group is isomorphic to .

Using our Hopf coordinates  we can then write any element of  in the form

Another way to state this result is if we express the matrix representation of an element of  as a linear combination of the Pauli matrices. It is seen that an arbitrary element  can be written as

The condition that the determinant of  is +1 implies that the coefficients  are constrained to lie on a 3-sphere.

In literature
In Edwin Abbott Abbott's Flatland, published in 1884, and in Sphereland, a 1965 sequel to Flatland by Dionys Burger, the 3-sphere is referred to as an oversphere, and a 4-sphere is referred to as a hypersphere.

Writing in the American Journal of Physics, Mark A. Peterson describes three different ways of visualizing 3-spheres and points out language in The Divine Comedy that suggests Dante viewed the Universe in the same way.

See also
 1-sphere, 2-sphere, n-sphere
 tesseract, polychoron, simplex
 Pauli matrices
 rotation group SO(3)
 charts on SO(3)
 quaternions and spatial rotations
 Hopf bundle, Riemann sphere
 Poincaré sphere
 Reeb foliation
 Clifford torus

References

 David W. Henderson, Experiencing Geometry: In Euclidean, Spherical, and Hyperbolic Spaces, second edition, 2001,  (Chapter 20: 3-spheres and hyperbolic 3-spaces.)
 Jeffrey R. Weeks, The Shape of Space: How to Visualize Surfaces and Three-dimensional Manifolds, 1985, () (Chapter 14: The Hypersphere) (Says: A Warning on terminology: Our two-sphere is defined in three-dimensional space, where it is the boundary of a three-dimensional ball. This terminology is standard among mathematicians, but not among physicists. So don't be surprised if you find people calling the two-sphere a three-sphere.)

External links
  Note: This article uses the alternate naming scheme for spheres in which a sphere in -dimensional space is termed an -sphere.

Category:Four-dimensional geometry
Category:Algebraic topology
Category:Geometric topology
Category:Analytic geometry
Category:Quaternions
Category:Spheres