Groups of conformal transformations of Riemannian spaces

It is proved that if a Riemannian space $(M,g)$ of class $C^\infty$ has a connected group of conformal transformations which leaves no conformally given metric $e^\sigma_g$ invariant, then $(M,g)$ is globally conformal to a sphere $(S^n,g_0)$ or to Euclidean space $(E^n,g_0)$.
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