Geometric structure of higher-dimensional spheres

We explain in some detail the geometric structure of spheres in any dimension. Our approach may be helpful for other homogeneous spaces (with other signatures) such as the de Sitter and anti-de Sitter spaces. We apply the procedure to the recently proposed division-algebras/Poincar\'e-conjecture correspondence. Moreover, we explore the possibility of a connection between N-qubit system and the Hopf maps. We also discuss the possible links of our work with squashed-spheres in supergravity and pseudo-spheres in oriented matroid theory.