Definition and Construction
For any natural number n, an n-dimensional sphere, or n-sphere, can be defined as the set of points in an (n+1)-dimensional space which are a fixed distance from a central point. For concreteness, the central point can be taken to be the origin, and the distance of the points on the sphere from this origin can be assumed to be a unit length. With this convention, the n-sphere, Sn, consists of the points (x1, x2, …, xn+1) in Rn+1 with x12 + x22 + ⋯+ xn+12 = 1. For example, the 3-sphere consists of the points (x1, x2, x3, x4) in R4 with x12 + x22 + x32 + x42 = 1.
The Hopf fibration p: S3 → S2 of the 3-sphere over the 2-sphere can be defined in several ways.
Read more about this topic: Hopf Fibration
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