What I can understand is global conformal transformation on $M$ is the subgroup of Diff(M), so at least global conformal transformation is well-defined on every points of $M$.
I haven't found a book offering an exact definition of "local" conformal transformation although they all use it. And seems different people even use different deinition?
For example, my teacher told me yesterday that holomorphic function group on C is the global conformal group on C. Its algebra is the Witt algebra. And it is local conformal transformation group on the Riemann Sphere.
However, Witt algebra has generators like $z^{-2}$ which are even not defined on C.
So can someone tell me the exact definition of local conformation transformation and infinitesimal conformal transformation?
