  Donaldson conjectured \cite{Dona96} that the space of K\"ahler metrics is
geodesic convex by smooth geodesic and that it is a metric space. Following
Donaldson's program, we verify the second part of Donaldson's conjecture
completely and verify his first part partially. We also prove that the constant
scalar curvature metric is unique in {\bf each} K\"{a}hler class if the first
Chern class is either strictly negative or 0. Furthermore, if $C_1 \leq 0,$ the
constant scalar curvature metric realizes the global minimum of Mabuchi energy
functional; thus it provides a new obstruction for the existence of constant
curvature metric: if the infimum of Mabuchi energy (taken over all metrics in a
fixed K\"{a}hler class) isn't bounded from below, then there doesn't exist a
constant curvature metric. This extends the work of Mabuchi and
Bando\cite{Bando87}: they showed that Mabuchi energy bounded from below is a
necessary condition for the existence of K\"{a}hler-Einstein metrics in the
first Chern class.
