2Sphäre

2Sphäre, in mathematics often written as S^2, denotes the two-dimensional sphere. It is defined as the set of all points in three-dimensional Euclidean space at unit distance from the origin, i.e., S^2 = { (x, y, z) in R^3 : x^2 + y^2 + z^2 = 1 }. In deutschsprachiger Terminologie wird sie oft als 2-Sphäre bezeichnet.

As a geometric object, S^2 is a smooth, closed surface and thus a two-dimensional manifold embedded in R^3. It has constant positive Gaussian curvature equal to 1 for the unit sphere. Topologically, S^2 is compact, connected and simply connected. Its homology groups are H0 ≅ Z, H2 ≅ Z, with other groups zero, giving an Euler characteristic χ = 2. The standard metric is the one induced from R^3, and in spherical coordinates (θ, φ) the line element on the unit sphere is ds^2 = dθ^2 + sin^2 θ dφ^2.

Stereographic projection provides a conformal diffeomorphism between S^2 minus a point and the plane R^2, illustrating its role as a model surface in complex analysis on the sphere. The sphere has a rich symmetry: the rotation group SO(3) acts transitively on S^2, making S^2 ≅ SO(3)/SO(2).

Geodesics on S^2 are great circles, and the sphere serves as a standard domain for spherical harmonics Y_l^m in physics and engineering. It appears in computer graphics, geodesy, and various areas of geometry and topology as a fundamental example of a compact 2-manifold with constant curvature.