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In an entirely analogous manner one can define higher-dimensional spheres called hyperspheres or n-spheres. Such objects are n-dimensional manifolds.
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2 Properties 3 Coordinates on S3 4 Group structure 5 Related topics |
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
Definition
The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted S3.
We could just as well define a 3-sphere as a subset of C2, in which case the unit 3-sphere is given by
Better still, we can identify R4 with the quaternions, H, in which case the unit 3-sphere is simply the set of all quaternions with absolute value equal to one:
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
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 (θ, ξ1, ξ2) where
In terms of complex coordinates we have
Hyperspherical coordinates are sometimes convenient for doing integrals on R4. Here one describes R4 in terms of a radial coordinate r together with the coordinates (θ, ξ1, ξ2). To do an integral it is necessary to know the Jacobian determinant for the coordinate transformation, which is given by
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 we can write a point in S3 as
We could just have well have projected onto the plane tangent to the point in which case a point in S3 is given by
Properties
while the hypervolume (the volume of the 4-dimensional region bounded by the sphere) is
Coordinates on S3
Hyperspherical coordinates
where θ runs over the range 0 to π, and ξ1 and ξ2 can taken any values between 0 and 2π.
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.Stereographic coordinates
where (u1, u2, u3) are coordinates on R3 and . The inverse of this map takes in S3 to
in R3.
where (v1, v2, v3) are coordinates on R3. The inverse of this map takes in S3 to
Note that the first map is well-defined everywhere but and the second everywhere but . By using both patches with their respective maps we can define an atlas on S3. Note that the transition function between the two coordinate patches is given by
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 (nonabelian) Lie group.
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: parallelizablity. 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
Group structure
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