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 Rs,t, is a real vector space with a nondegenerate quadratic form with signature (s, t). The quadratic form is permitted to be definite (where s = 0 or t = 0), making this a generalization of a Euclidean vector space.[lower-alpha 1]
  • A pseudo-Euclidean space, denoted Es,t, is a real affine space in which displacement vectors are the elements of the space Rs,t. It is distinguished from the vector space.
  • The quadratic form Q acting on a vector x ∈ Rs,t is denoted Q(x), called the quadrance[lower-alpha 2] of x (a generalization of the square of distance in a Euclidean space).
  • The symmetric bilinear form B acting on two vectors x, y ∈ Rs,t is denoted B(x, y) or x ⋅ y.[lower-alpha 3] This is associated with the quadratic form Q.[lower-alpha 4]
  • Two vectors x, y ∈ Rs,t are orthogonal if x ⋅ y = 0.
  • 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 Es,t consisting of the points u for which the displacement vector x = u − o from a reference point o satisfies the equation

a x ⋅ x + b ⋅ x + c = 0,

where a, c ∈ R and b, x ∈ Rs,t.[1][lower-alpha 5]

Since a = 0 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.[2]

A quasi-sphere P = {x ∈ X : Q(x) = k} in a quadratic space (X, Q) has a counter-sphere N = {x ∈ X : Q(x) = −k}.[lower-alpha 6] Furthermore, if k ≠ 0 and L is an isotropic line in X through x = 0, then L ∩ (P ∪ N) = ∅, 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 quadrance

The centre of a quasi-sphere is a point that has equal quadrance from every point of the quasi-sphere – i.e., the quadratic form applied to the displacement vector from the centre to a point of the quasi-sphere (the radius vector) yields a constant, called the radial quadrance, or equivalently, 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 a ≠ 0, the displacement vector p of the centre from the reference point and the radial quadrance r may be found as follows. We put Q(x − p) = r, and comparing to the defining equation above for a quasi-sphere, we get

The case of a = 0 may be interpreted as the centre p being a well-defined point at infinity with either infinite or zero radial quadrance (the latter for the case of a null hyperplane). Knowing p (and r) in this case does not determine the hyperplane's position, though, only its orientation in space.

The radial quadrance may take on a positive, zero or negative value. When the quadratic form is definite, even though p and r may be determined from the above expressions, the set of vectors x satisfying the defining equation may be empty, as is the case in a Euclidean space for a negative radial quadrance.

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. Q(x − p)) as the radial quadrance, in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radial quadrance, those with negative radial quadrance, those with zero radial quadrance.[lower-alpha 7]

In a space with a positive-definite quadratic form (i.e. a Euclidean space), a quasi-sphere with negative radial quadrance is the empty set, one with zero radial quadrance consists of a single point, one with positive radial quadrance is a standard n-sphere, and one with zero curvature is a hyperplane that is partitioned with the n-spheres.

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See also

Notes
  1. Some authors exclude the definite cases, but in the context of this article, the qualifier indefinite will be used where this exclusion is intended.
  2. Élie Cartan uses the term scalar square for the quandrance.
  3. The symmetric bilinear form applied to the two vectors is also called their scalar product.
  4. The associated symmetric bilinear form of a (real) quadratic form Q is defined such that Q(x) = B(x, x), and may be determined as B(x, y) = 1/4(Q(x + y) − Q(x − y)). See Polarization identity for variations of this identity.
  5. Though not mentioned in the source, we must exclude the combination b = 0 and a = 0.
  6. There are caveats when Q is definite. Also, when k = 0, it follows that N = P.
  7. A hyperplane (a quasi-sphere with infinite radial quadrance or zero curvature) is partitioned with quasi-spheres to which it is tangent. The three sets may be defined according to whether the quadratic form applied to a vector that is a normal of the tangent hypersurface is positive, zero or negative. The three sets of objects are preserved under conformal transformations of the space.

References
  1. Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. p. 140. ISBN 9780191085789.
  2. Ian R. Porteous (1995), Clifford Algebras and the Classical Groups, Cambridge University Press
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