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Could somebody please clarify the following for me?

I am not too clear about the relationship between the Riemann sphere and Möbius maps. I know that we can through projection make some Möbius maps correspond to isometries of the sphere. But it is not a bijection right? Which maps have corresponding isometries and which don't, vice versa?

Thanks

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    Only some Möbius maps give isometries of the sphere. What's really going on is that the Möbius maps are the automorphisms of the Riemann sphere, considered as a one-dimensional complex manifold.2012-04-27

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