en 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 $R s, t$ , 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.^[a]
A pseudo-Euclidean space, denoted $E s, t$ , is a real affine space in which displacement vectors are the elements of the space $R s, t$ . It is distinguished from the vector space.
The quadratic form $Q$ acting on a vector $x \in R s, t$ , denoted $Q (x)$ , is a generalization of the squared Euclidean distance in a Euclidean space. Élie Cartan calls $Q (x)$ the scalar square of $x$ .^[1]
The symmetric bilinear form $B$ acting on two vectors $x, y \in R s, t$ is denoted $B (x, y)$ or $x \cdot y$ .^[b] This is associated with the quadratic form $Q$ .^[c]
Two vectors $x, y \in R s, t$ are orthogonal if $x \cdot y = 0$ .
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 $E s, t$ consisting of the points $u$ for which the displacement vector $x = u - o$ from a reference point $o$ satisfies the equation

a x \cdot x + b \cdot x + c = 0

,

where $a, c \in R$ and $b, x \in R s, t$ .^[2]^[d]

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.^[3]

A quasi-sphere $P = {x \in X : Q (x) = k}$ in a quadratic space $(X, Q)$ has a counter-sphere $N = {x \in X : Q (x) = - k}$ .^[e] Furthermore, if $k \neq 0$ and $L$ is an isotropic line in $X$ through $x = 0$ , then $L \cap (P \cup N) = \emptyset$ , 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 $a \neq 0$ , the displacement vector $p$ of the centre from the reference point and the radial scalar square $r$ may be found as follows. We put $Q (x - p) = r$ , and comparing to the defining equation above for a quasi-sphere, we get

p=-{\frac {b}{2a}},

r=p\cdot p-{\frac {c}{a}}.

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 scalar square (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 scalar square 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 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. $Q (x - p)$ ) 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.^[f]

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 $n$ -sphere, and one with zero curvature is a hyperplane that is partitioned with the $n$ -spheres.

Notes

^ Some authors exclude the definite cases, but in the context of this article, the qualifier indefinite will be used where this exclusion is intended.
^ The symmetric bilinear form applied to the two vectors is also called their scalar product.
^ 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 $B(x, y) = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/4⁠(Q(x + y) − Q(x − y))$ . See Polarization identity for variations of this identity.
^ Though not mentioned in the source, we must exclude the combination $b = 0$ and $a = 0$ .
^ There are caveats when $Q$ is definite. Also, when $k = 0$ , it follows that $N = P$ .
^ A hyperplane (a quasi-sphere with infinite radial scalar square 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

^ Élie Cartan (1981) [First published in 1937 in French, and in 1966 in English], The Theory of Spinors, Dover Publications, p. 3, ISBN 0486640701
^ Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. p. 140. ISBN 9780191085789.
^ Ian R. Porteous (1995), Clifford Algebras and the Classical Groups, Cambridge University Press

[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.

[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 $B(x, y) = .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}⁠1/4⁠(Q(x + y) − Q(x − y))$ . See Polarization identity for variations of this identity.

[6] Though not mentioned in the source, we must exclude the combination $b = 0$ and $a = 0$ .

[8] There are caveats when $Q$ is definite. Also, when $k = 0$ , it follows that $N = P$ .

[9] A hyperplane (a quasi-sphere with infinite radial scalar square 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.

[2] Élie Cartan (1981) [First published in 1937 in French, and in 1966 in English], The Theory of Spinors, Dover Publications, p. 3, ISBN 0486640701

[5] Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. p. 140. ISBN 9780191085789.

[7] Ian R. Porteous (1995), Clifford Algebras and the Classical Groups, Cambridge University Press

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Quasi-sphere