Quasi-sphere
In mathematics and theoretical physics, a quasi-sphere is a generalization of the hypersphere and the hyperplane to the context of a pseudo-Euclidean space. It may be described as the set of points for which the quadratic form for the space applied to the displacement vector from a centre point is a constant value, with the inclusion of hyperplanes as a limiting case.
Notation and terminology
This article uses the following notation and terminology:
- A pseudo-Euclidean vector space, denoted Template:Math, is a real vector space with a nondegenerate quadratic form with signature Template:Math. The quadratic form is permitted to be definite (where Template:Math or Template:Math), making this a generalization of a Euclidean vector space.Template:Efn
- A pseudo-Euclidean space, denoted Template:Math, is a real affine space in which displacement vectors are the elements of the space Template:Math. It is distinguished from the vector space.
- The quadratic form Template:Math acting on a vector Template:Math, denoted Template:Math, is a generalization of the squared Euclidean distance in a Euclidean space. Élie Cartan calls Template:Math the scalar square of Template:Math.Template:Refn
- The symmetric bilinear form Template:Math acting on two vectors Template:Math is denoted Template:Math or Template:Math.Template:Efn This is associated with the quadratic form Template:Math.Template:Efn
- Two vectors Template:Math are orthogonal if Template:Math.
- A normal vector at a point of a quasi-sphere is a nonzero vector that is orthogonal to each vector in the tangent space at that point.
Definition
A quasi-sphere is a submanifold of a pseudo-Euclidean space Template:Math consisting of the points Template:Math for which the displacement vector Template:Math from a reference point Template:Math satisfies the equation
where Template:Math and Template:Math.Template:RefnTemplate:Efn
Since Template:Math in permitted, this definition includes hyperplanes; it is thus a generalization of generalized circles and their analogues in any number of dimensions. This inclusion provides a more regular structure under conformal transformations than if they are omitted.
This definition has been generalized to affine spaces over complex numbers and quaternions by replacing the quadratic form with a Hermitian form.Template:Refn
A quasi-sphere Template:Math in a quadratic space Template:Math has a counter-sphere Template:Math.Template:Efn Furthermore, if Template:Math and Template:Math is an isotropic line in Template:Math through Template:Math, then Template:Math, puncturing the union of quasi-sphere and counter-sphere. One example is the unit hyperbola that forms a quasi-sphere of the hyperbolic plane, and its conjugate hyperbola, which is its counter-sphere.
Geometric characterizations
Centre and radial scalar square
The centre of a quasi-sphere is a point that has equal scalar square from every point of the quasi-sphere, the point at which the pencil of lines normal to the tangent hyperplanes meet. If the quasi-sphere is a hyperplane, the centre is the point at infinity defined by this pencil.
When Template:Math, the displacement vector Template:Math of the centre from the reference point and the radial scalar square Template:Math may be found as follows. We put Template:Math, and comparing to the defining equation above for a quasi-sphere, we get
The case of Template:Math may be interpreted as the centre Template:Math being a well-defined point at infinity with either infinite or zero radial scalar square (the latter for the case of a null hyperplane). Knowing Template:Math (and Template:Math) in this case does not determine the hyperplane's position, though, only its orientation in space.
The radial scalar square may take on a positive, zero or negative value. When the quadratic form is definite, even though Template:Math and Template:Math may be determined from the above expressions, the set of vectors Template:Math satisfying the defining equation may be empty, as is the case in a Euclidean space for a negative radial scalar square.
Diameter and radius
Any pair of points, which need not be distinct, (including the option of up to one of these being a point at infinity) defines a diameter of a quasi-sphere. The quasi-sphere is the set of points for which the two displacement vectors from these two points are orthogonal.
Any point may be selected as a centre (including a point at infinity), and any other point on the quasi-sphere (other than a point at infinity) define a radius of a quasi-sphere, and thus specifies the quasi-sphere.
Partitioning
Referring to the quadratic form applied to the displacement vector of a point on the quasi-sphere from the centre (i.e. Template:Math) as the radial scalar square, in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radial scalar square, those with negative radial scalar square, those with zero radial scalar square.Template:Efn
In a space with a positive-definite quadratic form (i.e. a Euclidean space), a quasi-sphere with negative radial scalar square is the empty set, one with zero radial scalar square consists of a single point, one with positive radial scalar square is a standard Template:Math-sphere, and one with zero curvature is a hyperplane that is partitioned with the Template:Math-spheres.