subspheres

In geometry and topology, a subsphere of the sphere S^n is a subset that is itself a sphere S^k for some k ≤ n. The standard construction realizes subspheres by intersecting S^n with a linear subspace V of R^{n+1} of dimension k+1; the intersection S^n ∩ V is a copy of S^k embedded in S^n. If the subspace passes through the origin, the subsphere is often called a great subsphere, while intersections with lower-dimensional subspaces yield smaller embedded spheres.

Examples include the equator of S^n, obtained by intersecting with a coordinate hyperplane through the origin, producing a subsphere S^{n-1}. On S^2, a circle obtained by fixing a linear relation among the coordinates is a subsphere homeomorphic to S^1. In general, choosing different subspaces gives subspheres of various dimensions embedded in S^n.

Properties of subspheres are straightforward: they are smooth submanifolds of S^n with the induced metric from the ambient space, and they inherit the topological type of a sphere of the corresponding dimension. They are closed subsets of S^n and the inclusion maps induce homology and homotopy information relevant to the ambient sphere.

Subsurfaces of this kind play a role in a range of topics, including the construction of cellular decompositions and CW-structures on spheres, arguments based on equators and sections, and the study of embedded submanifolds in spherical geometry. See also: sphere, great circle, S^k, submanifold.