  We show that the standard picture regarding the notion of stability of
constant scalar curvature metrics in K\"ahler geometry described by S.K.
Donaldson, which involves the geometry of infinite-dimensional groups and
spaces, can be applied to the constant scalar curvature metrics in Sasaki
geometry with only few modification. We prove that the space of Sasaki metrics
is an infinite dimensional symmetric space and that the transverse scalar
curvature of a Sasaki metric is a moment map of the strict contactomophism
group.
