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The term 3-sphere has a different meaning depending on whether it is being used in a geometrical or a
topological context. This article mostly follows the topological meaning. However, it is clearly important to use the correct
definition in the context.
Geometrical definition
In geometry, following Coxeter, a 3-sphere means a regular 3-dimensional sphere in 3-dimensional Euclidean space,
R3. To learn about this sphere, visit 2-sphere.
Topological definition
In topology, a 3-sphere is a higher-dimensional analogue of a
sphere. A regular sphere, or 2-sphere, consists of all points equidistant away from a
single point in ordinary 3-dimensional Euclidean space,
R3. A 3-sphere consists of all points equidistant away 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.
Definition
In coordinates, a 3-sphere with center
(x0, y0, z0, w0) and radius
r is the set of all points (x,y,z,w) in R4 such that
-
The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted
S3. It can be desribed as a subset of either R4, C2,
or H (the quaternions):
-
-
-
The last description is often the most useful. It describes the 3-sphere as the set of all unit
quaternions—quaternions 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.
Properties
In topological terms 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).
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).
The volume (or hyperarea) of a 3-sphere of radius r is
-
while the hypervolume (the volume of the 4-dimensional region bounded by the sphere) is
-
Construction
A 3-sphere can be constructed topologically by "glueing" the boundaries of a pair of 3-balls. The boundary of a 3-ball is a
2-sphere: so bind the 2-spherical boundaries of a pair of 3-balls together. 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-sphere 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".
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 to negative; "hot" to "cold".
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
-
-
-
-
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
-
and the volume form by
-
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 desribes 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
-
-
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 S1 → S3 → S2.
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
-
and the volume form by
-
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
-
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
-
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
-
where v = (v1, v2, v3) is a vector in the second
R3. The inverse of this map takes p to
-
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 consiting of two coordinate charts. 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, 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
-
This map gives an 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 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
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Related topics
External link
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