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In mathematics, an n-sphere or hypersphere is an -dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional, and the sphere 2-dimensional, because the surfaces themselves are 1- and 2-dimensional respectively, not because they ...
The n-sphere may be defined geometrically as the set of points in a Euclidean space of dimension n + 1 located at a unit distance from the origin. The i-th homotopy group π i (S n) summarizes the different ways in which the i-dimensional sphere S i can be mapped continuously into the n-dimensional sphere S n.
In an isotropic chart (on a static spherically symmetric spacetime), the metric (aka line element) takes the form = + (+ (+ ())), < <, < <, < <, < < Depending on context, it may be appropriate to regard , as undetermined functions of the radial coordinate (for example, in deriving an exact static spherically symmetric solution of the Einstein field equation).
For example, the isometry group of the n-sphere is the orthogonal group O(n + 1). Given any finite subgroup G thereof in which only the identity matrix possesses 1 as an eigenvalue , the natural group action of the orthogonal group on the n -sphere restricts to a group action of G , with the quotient manifold S n / G inheriting a geodesically ...
Special cases are right triangles (p q 2). Uniform solutions are constructed by a single generator point with 7 positions within the fundamental triangle, the 3 corners, along the 3 edges, and the triangle interior. All vertices exist at the generator, or a reflected copy of it. Edges exist between a generator point and its image across a mirror.
The n-dimensional model is the celestial sphere of the (n + 2)-dimensional Lorentzian space R n+1,1. Here the model is a Klein geometry : a homogeneous space G / H where G = SO( n + 1, 1) acting on the ( n + 2) -dimensional Lorentzian space R n +1,1 and H is the isotropy group of a fixed null ray in the light cone .
The rotation group generalizes quite naturally to n-dimensional Euclidean space, with its standard Euclidean structure. The group of all proper and improper rotations in n dimensions is called the orthogonal group O( n ), and the subgroup of proper rotations is called the special orthogonal group SO( n ), which is a Lie group of dimension n ( n ...
A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation + + + = defines an algebraic hypersurface of dimension n − 1 in the Euclidean space of dimension n.