one can basically describe each of the classical geometries (Euclidean, affine, projective,spherical, hyperbolic, Minkowski, 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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