Riemann sphere
Stereographic projection identifies formula_3 with a sphere, which is then called the Riemann sphere; alternatively, formula_3 can be thought of as the complex projective line formula_5. The Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold; alternatively, they are the automorphisms of formula_5 as an algebraic variety. Therefore, the set of all Möbius transformations forms a group under composition. This group is called the Möbius group, and is sometimes denoted formula_7.