3-sphere two 3-balls. This construction is analogous to a construction of a 2-sphere, performed by gluing the boundaries of a pair of disks. A disk is a 2-ball, and the boundary of a disk is a circle (a 1-sphere). Let a pair of disks be of the same diameter. Superpose them and glue corresponding points on their boundaries. Again one may think of the third dimension as temperature. Likewise, we may inflate the 2-sphere, moving the pair of disks to become the northern and southern hemispheres. After removing a single point from the 2-sphere, what remains is homeomorphic to the Euclidean
Riemann sphere that group is called SO(3). In this sense, this is by far the most symmetric metric on the sphere. (The group of all isometries, known as O(3), is also 3-dimensional, but unlike SO(3) is not a connected space.) Conversely, let "S" denote the sphere (as an abstract smooth or topological manifold). By the uniformization theorem there exists a unique complex structure on "S", up to conformal equivalence. It follows that any metric on "S" is conformally equivalent to the round metric. All such metrics determine the same conformal geometry. The round metric is therefore not intrinsic to the Riemann sphere,