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. 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. 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, 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. The symmetric bilinear form B acting on two vectors x, y ∈ Rs,t is denoted B(x, y) or x ⋅ y. This is associated with the quadratic form Q. 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. 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. 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.