3-sphere


In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior of a 3-sphere is a 4-ball.
It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. For example, when traveling on a 3-sphere, you can go north and south, east and west, or along a 3rd set of cardinal directions. This means that a 3-sphere is an example of a 3-manifold.

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 surface volume of a 3-sphere of radius is
while the 4-dimensional hypervolume is
Every non-empty intersection of a 3-sphere with a three-dimensional hyperplane is a 2-sphere. 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.
In a given three-dimensional hyperplane, a 3-sphere can rotate about an "equatorial plane", in which case it appears to be a 2-sphere whose size is constant.

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 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.
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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. The only other spheres with such a structure are the 0-sphere and the 1-sphere.
Unlike the 2-sphere, the 3-sphere admits nonvanishing vector fields. 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, 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. 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 to the plane by sending to the intersection of the line with the plane. Stereographic projection of a 3-sphere maps to three-space in the same manner.
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. 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, it describes a rotation about through an angle of.