nsphere

The n-sphere, usually denoted S^n, is the set of all points in (n+1)-dimensional Euclidean space R^{n+1} at a fixed distance r from a chosen center. When r = 1, it is called the unit n-sphere. Equivalently, S^n is the boundary of the closed (n+1)-dimensional ball B^{n+1}(r). As a manifold, S^n has dimension n.

Common examples include S^0, a two-point set; S^1, the circle; and S^2, the ordinary sphere. The standard round metric is obtained by restricting the Euclidean metric to the surface, and the sphere is a homogeneous space under the action of the orthogonal group O(n+1).

Topologically, S^n has nontrivial reduced homology only in degrees 0 and n, with H_0 ≅ Z and H_n ≅ Z, and H_k(S^n) = 0 for 0 < k < n. The Euler characteristic χ(S^n) equals 2 for n even and 0 for n odd; more generally χ(S^n) = 1 + (-1)^n for n > 0. The cohomology ring is generated by a single class in degree n.

Formulas for measures include the surface area of the unit n-sphere: A_n = 2π^{(n+1)/2} / Γ((n+1)/2). For a sphere of radius r, A_n(r) = A_n r^n. The volume of the unit (n+1)-ball is V_{n+1} = π^{(n+1)/2} / Γ((n+3)/2).

In applications, n-spheres serve as fundamental examples in topology and geometry, provide the domain for spherical harmonics on S^n, and arise in physics as simple compact spaces and as target spaces in field theories.