×R×Rn×Sn

×R×Rn×Sn denotes the Cartesian product of three mathematical spaces: the real line ℝ, the n‑dimensional Euclidean space ℝⁿ, and the n‑dimensional sphere Sⁿ. In symbols the space can be written as ℝ × ℝⁿ × Sⁿ. Each factor is standard in mathematics and topology: ℝ is the one‑dimensional real line, ℝⁿ is the set of n‑tuples of real numbers equipped with the usual Euclidean metric, and Sⁿ is the set of points in ℝⁿ⁺¹ at unit distance from the origin. The product space inherits a natural product topology and metric, making it a smooth manifold of dimension \(1 + n + n\) = \(2n +1\). Since each component is a manifold, the product is also a manifold, and its tangent bundle is the direct sum of the tangent bundles of the factors. The space can be used to model configurations in physics where one has a temporal dimension (ℝ), a spatial n‑dimensional component (ℝⁿ), and an angular or constraint component represented by Sⁿ. For example, in classical mechanics a point particle moving on a sphere with an additional time parameter can be described within this product space. In differential geometry, the product is a standard example of a product manifold, and many constructions such as Riemannian metrics or differential forms can be defined component‑wise. The space is also employed in algebraic topology to study product cohomology and homotopy groups; because ℝ is contractible, the homotopy type of ℝ × ℝⁿ × Sⁿ is the same as that of ℝⁿ × Sⁿ, which is homotopy equivalent to Sⁿ. For specific values of n, such as n=1 giving ℝ × ℝ × S¹, or n=2 giving ℝ × ℝ² × S², the space simplifies to familiar spaces used in the study of surfaces, fiber bundles, and more. The notation ×R×Rn×Sn is succinct and often appears in advanced texts on manifolds, product topologies, and geometric analysis.