one can basically describe each of the classical geometries (Euclideanaffineprojective,sphericalhyperbolicMinkowski, etc.) as a homogeneous space for its structure group.

The structure group (or gauge group) of the class of geometric objects arises from isomorphisms of one geometric object to the standard object of its class.

For example,

  • • the structure group for lengths is ℝ⁺;
  • the structure group for angles is ℤ/2ℤ;
  • the structure group for lines is the affine group Aff(ℝ);
  • the structure group for n-dimensional Euclidean geometry is the Euclidean group E(n);
  • • the structure group for oriented 2-spheres is the (special) orthogonal group SO(3).

Terence Tao

(I rearranged his text freely.)

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