  We show that if the group of holomorphic automorphisms of a connected complex
manifold $M$ of dimension $n$ is isomorphic as a topological group equipped
with the compact-open topology to the automorphism group of the unit ball
$B^n\subset\CC^n$, then $M$ is biholomorphically equivalent to either $B^n$ or
$\CC\PP^n\setminus\bar{B^n}$.
