  In this paper we address the problem of studying those complex manifolds $M$
equipped with extremal metrics $g$ induced by finite or infinite dimensional
complex space forms. We prove that when $g$ is assumed to be radial and the
ambient space is finite dimensional then $(M, g)$ is itself a complex space
form. We extend this result to the infinite dimensional setting by imposing the
strongest assumption that the metric $g$ has constant scalar curvature and is
well-behaved (see Definition 1 in the Introduction). Finally, we analyze the
radial Kaehler-Einstein metrics induced by infinite dimensional elliptic
complex space forms and we show that if such a metric is assumed to satisfy a
stability condition then it is forced to have constant non-positive holomorphic
sectional curvature.
