S2×R

S^2 × R is a three-dimensional Riemannian manifold formed by the Cartesian product of the 2-sphere S^2 with the real line R, equipped with the standard product metric. As a result, it is a non-compact, complete, simply connected space.

Geometrically, S^2 × R is homogeneous and locally isometric to the product of a unit sphere with a line. The isometry group is Isom(S^2) × Isom(R) ≈ O(3) × R, acting by diffeomorphisms that preserve the product structure. The sectional curvature is 1 on planes tangent to S^2, 0 on planes tangent to R, and 0 on mixed planes, so the curvature is not constant. The scalar curvature of the product metric is 2.

Topologically, S^2 × R is diffeomorphic to the trivial S^2 bundle over R and serves as the universal cover of S^2 × S^1; indeed, S^2 × S^1 = (S^2 × R)/Z where Z acts by translations along the R-factor. Consequently, many closed 3-manifolds modeled on the S^2 × R geometry are quotients of S^2 × R by discrete groups acting freely and properly discontinuously, with S^2 × S^1 being a primary example.

S^2 × R is one of the eight Thurston geometries, collectively called the S^2 × R geometry. It provides a simple model for studying geometric structures on 3-manifolds that split into a spherical factor and a one-dimensional Euclidean factor.

See also: Thurston geometries, S^2, R, S^2 × S^1.